Is 1 = 0.999... ? Really?

[gib]

You need to address this argument:

Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as before except that you're no longer part of it. Noone joined the line, noone left the line -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.

What will be accomplished if I address this? What are you looking for? Do you want me to show how it's not really a contradiction? And if I do, what are you expecting to change in your thinking or the discussion?

Anyway, I'll do my best.

First of all, it's a loaded question. You're already presupposing that the notion that the line remains the same means one or more of your premises has been negated. You're boxing me in. But you know very well what my view is: the line remains the same despite that you really did take me out of it and no one else joined. The question you should be asking is: how is that possible? <-- I'll address this question.

The truth is, I'm not sure what to say about the number of people in the line. I understand that it seems intuitive to say: if you add someone to an infinite line of people, it has one more person. And if you take a person away, it has one less person. And if the notion of adding or subtracting from infinity wasn't so problematic, I would have no problem with this. But it is problematic. It's problematic because if you accept the notion that adding or subtracting a person changes the number of people in the line, it may square well with your intuition (thereby satisfying that aspect of the problem intellectually), but then you have to address other problems that crop up. For me, it's terribly problematic to do arithmetic with infinity. As soon as I start thinking about doing arithmetic on infinity, I can't help but to think of it as a quantity. Then I ask: well, what is that quantity? Is it bigger than a thousand? Less? Is it bigger than a million? Less? I don't know how to understand quantities except as existing one the number line, as existing between greater quantities and lesser quantities. But that, to me, defies the definition of infinity. Infinity means endlessness. It means: as soon as you've got a quantity, think bigger. Think: it's greater than this quantity. So if a line has an infinite number of people, and you add one more person to it, and you say that gives you \(\infty\) + 1, these are the thoughts that are going to come to my mind. They are problematic. I can't accept the notion that adding/subtracting a person from the line changes the number of people in the line without being bothered by these other considerations.

In other words, even if I were to agree that thinking of infinity as something to which you can add or subtract solves the problem of how counterintuitive it seems to say nothing in the line changes, I see this more as a trade of one problem for another, and in the larger picture, you still have a problem. You don't seem to be bothered by this. You're either dismissing the other problems I bring up as if they aren't there, or you have a different understanding of these problems such that they aren't really problems. So far, I've gathered that you think of infinity as a quantity that finds a place on the number line but is an infinite distance away from the finite numbers. That certainly solves the problem of how infinity can have a place on the number line (so we don't have to ask: is it greater than a thousand? Less? etc.), but for me it raises yet another problem: it seems to suggest that infinity has an end, or an "after". You find \(\infty\) + 1 after \(\infty\). But that, for me, flies in the face of the very definition of infinity: endlessness... infinity means no end. So how can there be an "after"?

Maybe you have a solution to this problem as well, maybe not. But as you can see, the problem to me is like a balloon. You try to solve little bits of it by squeezing on those bits. But this only results in other bits inflating, making other sides of the problem more emphatic. This is what I've been seeing throughout this whole discussion. I've been watching you squeeze certain parts of the balloon and seeing other parts inflating. And I hear from you: what inflation? There is no inflation!

Sorry that's such a long winded answer to your question. The short answer is: I don't know what to say about what happens to the number of people in the line when you add or subtract from it. If I agree with you, I see other problems crop up. If I don't, I have to deal with this counterintuitive notion that adding or subtracting doesn't change the quantity. At least I can say that intuition isn't the same as logic, so if it seems counterintuitive that adding or subtracting from the line doesn't change the line, maybe it's still logical. Therefore, I lean towards saying it doesn't change the line.

I think a huge part of the problem has to do with something you said: most of infinity lies outside what we can see. When we visualize the infinite line of people, we can't help but to visualize something that looks finite. We keep in mind that it's supposed to be infinite, but this is more how we conceptualize it, not how we visualize it. Because the visualization of the line is inevitably finite, it gives rise to the intuition that adding and subtracting from the line must change the number of people in the line (because that's how it works with finite things). I think if we had the capacity to visualize the entirety of an infinity of things, we would see how it really works, and the notion that adding or subtracting from an infinite set doesn't change the quantity of the set wouldn't seem nearly as problematic.

[Ecmandu]

Just a nitpick Magnus...

Infinity has a beginning and never ends

Eternity has no beginning and never ends

Gib,

I think what Magnus is saying is that a finite fraction (rational number), like 1/9....

0.111...

If you take away 1/10,000th away

The number is now...

0.111 0! (111....)

Thus is is less.

There’s a problem with infinity that nobody on this board has discussed yet:

The probability of a finite instance occurring in infinity is zero percent!

Let’s say you have the powerball competition, except that there are an infinite amount of numbers to choose from (any of the real numbers) and it’s completely random...

Magnus argues that someone can win. I argue that the odds are zero percent.

1 number compared to infinity is so infitessimally small that the odds of picking it are literally 0 percent, even if you played the game forever!

[gib]

Gib,

I think what Magnus is saying is that a finite fraction (rational number), like 1/9....

0.111...

If you take away 1/10,000th away

The number is now...

0.111 0! (111....)

Thus is is less.

Yes, I agree. 1/9 > 1/9 - 1/10000.

There’s a problem with infinity that nobody on this board has discussed yet:

The probability of a finite instance occurring in infinity is zero percent!

Let’s say you have the powerball competition, except that there are an infinite amount of numbers to choose from (any of the real numbers) and it’s completely random...

Magnus argues that someone can win. I argue that the odds are zero percent.

1 number compared to infinity is so infitessimally small that the odds of picking it are literally 0 percent, even if you played the game forever!

And yet one ball is bound to be picked.

[Magnus Anderson]

What will be accomplished if I address this? What are you looking for?

You said that the reason you think \(0.\dot9 = 1\) is because of this proof. I think there's a flaw in that proof and that's based on the premise that adding an element to an infinite set of elements increases the size of that set.

Do you want me to show how it's not really a contradiction?

Yes.

And if I do, what are you expecting to change in your thinking or the discussion?

It won't change my belief that \(0.\dot9 \neq 1\) but I'd no longer have an argument against your proof.

First of all, it's a loaded question. You're already presupposing that the notion that the line remains the same means one or more of your premises has been negated. You're boxing me in.

Well, if you think that my argument is a non-sequitur, you can always explain why (:

The truth is, I'm not sure what to say about the number of people in the line. I understand that it seems intuitive to say: if you add someone to an infinite line of people, it has one more person. And if you take a person away, it has one less person. And if the notion of adding or subtracting from infinity wasn't so problematic, I would have no problem with this. But it is problematic. It's problematic because if you accept the notion that adding or subtracting a person changes the number of people in the line, it may square well with your intuition (thereby satisfying that aspect of the problem intellectually), but then you have to address other problems that crop up. For me, it's terribly problematic to do arithmetic with infinity. As soon as I start thinking about doing arithmetic on infinity, I can't help but to think of it as a quantity. Then I ask: well, what is that quantity? Is it bigger than a thousand? Less? Is it bigger than a million? Less? I don't know how to understand quantities except as existing one the number line, as existing between greater quantities and lesser quantities.

Infinity is a number greater than every integer, so it's greater than 1,000 as well as 1,000,000. And yes, infinity (if it refers to a specific quantity greater than every integer, and not merely to any such quantity) has a place on the number line.

But that, to me, defies the definition of infinity. Infinity means endlessness. It means: as soon as you've got a quantity, think bigger. Think: it's greater than this quantity.

It means "as soon as you've got an integer, think bigger". In other words, it's a number greater than every integer. But it's not a number greater than every quantity. Only the largest number is greater than every quantity.

So far, I've gathered that you think of infinity as a quantity that finds a place on the number line but is an infinite distance away from the finite numbers. That certainly solves the problem of how infinity can have a place on the number line (so we don't have to ask: is it greater than a thousand? Less? etc.), but for me it raises yet another problem: it seems to suggest that infinity has an end, or an "after". You find \(\infty\) + 1 after \(\infty\). But that, for me, flies in the face of the very definition of infinity: endlessness... infinity means no end. So how can there be an "after"?

What you're saying, but without realizing it, is that the word "infinity" means the same thing as "the largest number". The largest number is the only number for which there is no greater number, no "after". But that's not what infinity means. It means "a number greater than every integer". There isn't one such number and the largest number is only one of them.

Perhaps we should ask what does it mean for something to have no end. What does the word "end" mean? Note that sets have no beginning and no end. The set of numbers \(A = \{1, 2, 3, 4, 5\}\) has no first and no last element even though it looks like it does because its visual representation starts with \(1\) end ens with \(5\) (in reality, it could have started and ended with any other number.)

Sequences, on the other hand, can have a beginning and an end, and in their case, the word "beginning" refers to the position with the lowest index and the word "end" refers to the position with the highest index. These terms have nothing to do with how many positions there are in the sequence. To say that a sequence has an end is to say there is a position with the highest index, it is not to say that the number of positions is finite (i.e. less than some integer.)

Sorry that's such a long winded answer to your question. The short answer is: I don't know what to say about what happens to the number of people in the line when you add or subtract from it. If I agree with you, I see other problems crop up. If I don't, I have to deal with this counterintuitive notion that adding or subtracting doesn't change the quantity. At least I can say that intuition isn't the same as logic, so if it seems counterintuitive that adding or subtracting from the line doesn't change the line, maybe it's still logical. Therefore, I lean towards saying it doesn't change the line.

But why lean in that direction when you can also lean in the other? (: You don't think that these other problems cropping up is merely your intuition rebelling?

[gib]

You said that the reason you think \(0.\dot9 = 1\) is because of this proof. I think there's a flaw in that proof and that's based on the premise that adding an element to an infinite set of elements increases the size of that set.

Well, let's bring it up again:

1. \(x = 0.\dot9\)
2. \(10x = 9.\dot9\)
3. \(10x = 9 + 0.\dot9\)
4. \(10x = 9 + x\)
5. \(9x = 9\)
6. \(x = 1\)

At what step is the flaw?

Keep in mind that this proof is following all the laws of algebra. It's pretty tight. If you want to say there's a flaw, you'd have to show how the laws of algebra don't really hold.

Infinity is a number greater than every integer, so it's greater than 1,000 as well as 1,000,000. And yes, infinity (if it refers to a specific quantity greater than every integer, and not merely to any such quantity) has a place on the number line.

You are, once again, making up definitions. No one has ever said "infinity is a number greater than any integer (as opposed to any real number)." But that's fine. I don't mind new and exotic definitions, as long as the logic is consistent. But here's the rub: what does it mean to say: greater than any integer but not necessarily any quantity? Are you saying that at some point on the number line, the integers end (i.e. there is a largest integer) and after that there continues to be quantities but not integers?

(This would be a great place to bring in hyperreals but you seem curiously resistant to that for some reason.)

It means "as soon as you've got an integer, think bigger". In other words, it's a number greater than every integer. But it's not a number greater than every quantity. Only the largest number is greater than every quantity.

So what happens if you add 1 to the largest number?

What you're saying, but without realizing it, is that the word "infinity" means the same thing as "the largest number". The largest number is the only number for which there is no greater number, no "after". But that's not what infinity means. It means "a number greater than every integer". There isn't one such number and the largest number is only one of them.

That's not quite right. The reason there is no "after" infinity is because infinity means "no end". You have to have an end before you can have an "after". That's different from the concept of a largest number. A largest number represents an end. The reason it doesn't have an "after" is because "largest" means "nothing after".

Perhaps we should ask what does it mean for something to have no end. What does the word "end" mean? Note that sets have no beginning and no end. The set of numbers \(A = \{1, 2, 3, 4, 5\}\) has no first and no last element even though it looks like it does because its visual representation starts with \(1\) end ens with \(5\) (in reality, it could have started and ended with any other number.)

This is true, but there's still a subtle difference. To say a set has no end is just to say the element we pick to be the end is arbitrary. When we say the number line has no end, on the other hand, we mean there is no number that can be picked as the end. I mean, I suppose we could say, we pick 5 as the last number, and to ensure every other number is still in there, we'll skip 5 as we're counting up and come back to it when we're done. But you'd still have the problem of the remaining numbers (6 onwards) having no end and therefore not being able to ever come back to 5.

In other words, I don't think you can make this problem go away just by imagining the integers as a set. You'd have to show how the problem is solved in all cases--sets, sequences, and anything else.

Sequences, on the other hand, can have a beginning and an end, and in their case, the word "beginning" refers to the position with the lowest index and the word "end" refers to the position with the highest index. These terms have nothing to do with how many positions there are in the sequence. To say that a sequence has an end is to say there is a position with the highest index, it is not to say that the number of positions is finite (i.e. less than some integer.)

This presupposes that infinity has a place on the number line, but this is precisely what's in dispute. As far as I'm concerned, infinity means no end and doesn't have a place on the number line. Therefore, it can't be at the position with the highest index.

But why lean in that direction when you can also lean in the other? (: You don't think that these other problems cropping up is merely your intuition rebelling?

No, I think they are logical problems. I gave an example of how the problem of adding/subtracting from an infinite set might only be intuition: it might be a consequence of not being able to fully visualize an infinite set. It might be that being limited to visualizing an infinite set as finite forces us to feel the size of the set must increase upon adding to it or decrease upon subtracting from it.

Imagine it this way:

Suppose you had a set of 5 elements. You add 1 to it. You now have 6 elements. That's an increase of 20%.

Suppose you had a set of 10 elements. You add 1 to it. You now have 11 elements. That's an increase of 10%.

Suppose you had a set of 20 elements. You add 1 to it. You now have 21 elements. That's an increase of 5%.

Notice that as the size of the set approaches infinity, the amount by which it increases approaches 0. Therefore, it stands to reason that if the size of the set is infinite, any increase in size after adding an element to it will be 0. That obviously doesn't get rid of the intuition, but it shows a mathematical way of conceptualizing how the intuition might be wrong.

[Magnus Anderson]

Well, let's bring it up again

If you can't see the flaw in Hilbert's paradox (and other simpler sophisms) there is no way in hell you can see the flaw in that Wikipedia proof. That's why it's utterly pointless to bring it up.

So what happens if you add 1 to the largest number?

You can't add 1 to the largest number.

You are, once again, making up definitions. No one has ever said "infinity is a number greater than any integer (as opposed to any real number)." But that's fine. I don't mind new and exotic definitions

It's not a new and exotic definition.

But here's the rub: what does it mean to say: greater than any integer but not necessarily any quantity? Are you saying that at some point on the number line, the integers end (i.e. there is a largest integer) and after that there continues to be quantities but not integers?

"Greater than every integer" does not imply the existence of the largest integer. It merely describes something that is greater than every integer.

To say that something is greater than every number of some infinite set \(S\) is not to say there is a number in that set greater than every other number. It simply does not follow.

(This would be a great place to bring in hyperreals but you seem curiously resistant to that for some reason.)

There's no need to do so.

This presupposes that infinity has a place on the number line, but this is precisely what's in dispute. As far as I'm concerned, infinity means no end and doesn't have a place on the number line. Therefore, it can't be at the position with the highest index.

Let's keep things simple. I am not talking about the number line (which is a specific sequence.) I am talking about sequences in general.

Sequences, on the other hand, can have a beginning and an end, and in their case, the word "beginning" refers to the position with the lowest index and the word "end" refers to the position with the highest index. These terms have nothing to do with how many positions there are in the sequence. To say that a sequence has an end is to say there is a position with the highest index, it is not to say that the number of positions is finite (i.e. less than some integer.)

Basically, what I'm doing here is explaining what it means for a sequence to have a beginning and an end. To say that a sequence has an end is NOT to say that the number of its positions is finite. Rather, it is to say that there is a position with the highest index. What makes you think that infinity can't be used to represent the index of such a position?

Note that a sequence is nothing more than a relation between the set of positions and the set of elements that occupy these positions. \(S = (1, 2, 3, \dotso, 0)\) is the same as \(S = \{(\infty, 0), (1, 1), (2, 2), (3, 3), \dotso\}\). Every sequence can be represented as a set of ordered pairs.

[iambiguous]

I think there's a flaw in that proof and that's based on the premise that adding an element to an infinite set of elements increases the size of that set.

For those of us without a background in math, the idea of establishing a flawless proof here becomes almost incomprehensible.

We think more in the way in which mathematical equations were used to, say, send astronauts to the Moon. The proof that they were correct is clearly embodied in Neil Armstrong's, "one small step for a man, one giant leap for mankind.”

[Assuming of course it wasn't all filmed in Hollywood]

What is it about the equation, "1 = 0.999..." that, after 72 pages, it is still not able to be pinned down and accepted as either one or the other by all.

Let alone a cogent and definitive explanation for what James meant by, "[t]his is one of those issues that display the clear distinction between a good philosopher and a expert mathematician."

[Magnus Anderson]

To say a set has no end is just to say the element we pick to be the end is arbitrary. When we say the number line has no end, on the other hand, we mean there is no number that can be picked as the end. I mean, I suppose we could say, we pick 5 as the last number, and to ensure every other number is still in there, we'll skip 5 as we're counting up and come back to it when we're done. But you'd still have the problem of the remaining numbers (6 onwards) having no end and therefore not being able to ever come back to 5.

That would be \(S = (1, 2, 3, 4, 6, 7, 8, \dotso, 5)\). If you want to go from \(1\) to \(5\), it's not enough to make a finite number of steps. You need to make an infinite number of them.

Your argument is that this can't happen because it is a logical contradiction to say that someone has completed an infinite number of steps.

Basically:

1) If something is infinite, it has no end.
2) If something has been completed, it has an end.
3) Therefore, if something that is infinite has been completed, that something has an end and does not have an end (\(P\) and \(\neg P\) a.k.a. logical contradiction.)

But this is an example of equivocation (the equivocated term being "end".)

What does it mean to say that someone has completed an infinite number of steps?

It means that they performed a number of steps before certain point in time (that's what the word "completed" indicates: the presence of a temporal end) and that the number of steps they performed has no integer greater than it (that's what the word "infinite" indicates: the lack of quantitative end.)

They are two DIFFERENT ends. They AREN'T one and the same end. Hence why there's no contradiction.

Perhaps the best way to realize that the above sequence is not a logical contradiction is to represent it as a set of ordered pairs \(S = \{(\infty, 5), (1, 1), (2, 2), (3, 3), (4, 4), (5, 6), (6, 7), \dotso\}\).

The sequence has no end in the sense that the number of its element is infinite but it has an end in the sense that it has a position with the highest index.

[Ecmandu]

Magnus,

.... so, you did go to hyper reals.

If you have (infinity, 5) as any member of the set, the rest of the members don’t get expressed.

You think that because you can type contradictions that they must be true:

“Married bachelor” MUST be true!!! Because you typed it.

You’ve been doing that song and dance this entire thread.

[gib]

Well, let's bring it up again

If you can't see the flaw in Hilbert's paradox (and other simpler sophisms) there is no way in hell you can see the flaw in that Wikipedia proof. That's why it's utterly pointless to bring it up.

You are such an avoider.

So what happens if you add 1 to the largest number?

You can't add 1 to the largest number.

Just like that? No reasoning? No argument? If it's a number, you can add to it. Every number represents a certain quantity. If it's a quantity, you can have more than it or less than it. Thus, you can add 1 to any number. It makes no sense to say there is some magic number that defies addition. What supernatural power prevents 1 from being added to it? Don't tell me: it's the largest number, so by definition you can't add 1 to it--because I'll tell you that the inability to add 1 to it is why we should doubt the existence of a largest number in the first place.

You are, once again, making up definitions. No one has ever said "infinity is a number greater than any integer (as opposed to any real number)." But that's fine. I don't mind new and exotic definitions

It's not a new and exotic definition.

Right because the word "integer" is featured in every dictionary definition of infinity and no mathematician would ever fail to mention "integers" when defining infinity. Why do you even feel compelled to dwell on this point?

But here's the rub: what does it mean to say: greater than any integer but not necessarily any quantity? Are you saying that at some point on the number line, the integers end (i.e. there is a largest integer) and after that there continues to be quantities but not integers?

"Greater than every integer" does not imply the existence of the largest integer. It merely describes something that is greater than every integer. WTF?!?!

To say that something is greater than every number of some infinite set \(S\) is not to say there is a number in that set greater than every other number. It simply does not follow.

So you're saying infinity is greater than any integer but it is not among the set of all integers. And at the same time, there is no greatest integer. <-- This is actually quite sensible. It's the common definition of infinity. Where I get thrown for a loop is when you say infinity is not necessarily greater than any quantity. If we leave out the part about there being a greatest integer, we still have the idea that there continues to be quantities after the integers. Now, I can't help but to go into hyperreals at this point. It's the only way I can make sense out of this. So...

(This would be a great place to bring in hyperreals but you seem curiously resistant to that for some reason.)

There's no need to do so.

There's every need to do so. It's only way your argument makes any sense (granting some pretty generous claims). Everything you've been arguing revolves around hyperreals. You've just been avoiding calling them that. I don't know how we can continue down this line of the discussion without talking about hyperreals, so if you won't, I will. I know hyperreals are big, bad, scary numbers--you must have had nightmares of hyperreals dwelling under your bed as a kid--but don't be afraid of them. I'll hold your hand.

In this post, I'll use hyperreals to address your point about quantities greater than any integer even though there is no greatest integer, but in a later post, I'll express my more general thoughts on them.

I've seen you use graphs like the following:

\(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots\)

...which indicates to me that you believe in the continuation of quantities after infinity. I'm sorry, Magnus, but that's hyperreals. It's not much different than:

1, 2, 3, ... \(\infty\) ... R-2, R-1, R, R+1, R+2 ... <-- where R is a hyperreal number.

One point that can be brought up is that it is completely arbitrary what hyperreal number we say we land on after we've skipped over all the reals on the number line. There's no "first" hyperreal number after infinity. There's no more reason to say we land on R than R+2 or R-2. The hyperreals extend infinitely in both directions just like the reals. Therefore, it stands to reason that there isn't even a determined point on the hyperreal section of the number line where you have integers. IOW, if it's arbitrary to say we land on R after we skip infinity, it's also arbitrary to say we land on R + 0.5. We could say we land on R + 0.5, but the point is R + 0.5 could have been labeled R all the same. Therefore, where does R the integer end and R the fraction begin? It's arbitrary. All we know is that R + 0.5 and R+1 are greater than R. Therefore, it is meaningless to say there are integers once we enter the realm of the hyperreals. <-- In that sense, I can understand why you would say there are quantities greater than any integer without there being a greatest integer.

The one qualm I have with this is what to say about the fuzzy region between the reals and the hyperreals--the ...\(\infty\)... that separates them. Most considerations of hyperreals kind of take this region for granted and therefore treats it as ignorable. But I can't. It implies a continuum along the number line between the reals and the hyperreals, yet at the same time blurs the point at which they meet. If there is no greatest integer, then I question what's happening in this blurred region such that you have integers going in and non-integer hyperreals coming out. What happened to the integers? What kind of transformation did they go through? If there is no greatest integer, then I think definitely we are saying the integers don't suddenly stop and the hyperreals begin. But then what? Do they slowly "morph" into hyperreals? Do they start appearing interspersed between hyperreals, getting farther and farther apart, like prime numbers as you go up? If that were the case, we'd have to say there is still the occasional integer found among the hyperreals, but obviously, we don't want to say that. From what I can tell, we just don't ask the question. We just accept the obscure fuzziness of what's happening in that region and try not to think about it.

This presupposes that infinity has a place on the number line, but this is precisely what's in dispute. As far as I'm concerned, infinity means no end and doesn't have a place on the number line. Therefore, it can't be at the position with the highest index.

Let's keep things simple. I am not talking about the number line (which is a specific sequence.) I am talking about sequences in general.

Meaning there is a definite order to the elements, each one coming "before" or "after" others.

Sequences, on the other hand, can have a beginning and an end, and in their case, the word "beginning" refers to the position with the lowest index and the word "end" refers to the position with the highest index. These terms have nothing to do with how many positions there are in the sequence. To say that a sequence has an end is to say there is a position with the highest index, it is not to say that the number of positions is finite (i.e. less than some integer.)

Basically, what I'm doing here is explaining what it means for a sequence to have a beginning and an end. To say that a sequence has an end is NOT to say that the number of its positions is finite. Rather, it is to say that there is a position with the highest index. What makes you think that infinity can't be used to represent the index of such a position?

Note that a sequence is nothing more than a relation between the set of positions and the set of elements that occupy these positions. \(S = (1, 2, 3, \dotso, 0)\) is the same as \(S = \{(\infty, 0), (1, 1), (2, 2), (3, 3), \dotso\}\). Every sequence can be represented as a set of ordered pairs.

You're kinda cheating there. If (\(\infty\), 0) is supposed to be the "last" item in the list (or the item with the "highest index"), then you're representing it out of sequence. The proper representation of the sequence should lay out the sequence, well, in sequence. It's easy to take the infinitieth item and place it in the front (which, btw, I noticed is 0, which means you're not using the index properly), but when laid out according to the actual sequence, you see you can never get to (\(\infty\), 0). Which reinforces what I said: \(\infty\) is not a point on the number line, and therefore can't be used as an index.

[gib]

To say a set has no end is just to say the element we pick to be the end is arbitrary. When we say the number line has no end, on the other hand, we mean there is no number that can be picked as the end. I mean, I suppose we could say, we pick 5 as the last number, and to ensure every other number is still in there, we'll skip 5 as we're counting up and come back to it when we're done. But you'd still have the problem of the remaining numbers (6 onwards) having no end and therefore not being able to ever come back to 5.

That would be \(S = (1, 2, 3, 4, 6, 7, 8, \dotso, 5)\). If you want to go from \(1\) to \(5\), it's not enough to make a finite number of steps. You need to make an infinite number of them.

Your argument is that this can't happen because it is a logical contradiction to say that someone has completed an infinite number of steps.

Sure, we'll go with that.

Basically:

1) If something is infinite, it has no end.
2) If something has been completed, it has an end.
3) Therefore, if something that is infinite has been completed, that something has an end and does not have an end (\(P\) and \(\neg P\) a.k.a. logical contradiction.)

That would indeed be a contradiction but not quite what I had in mind. I'm not thinking of temporal processes. I'm thinking that for the same reason I said the number sequence doesn't have an end, the same must be said of the remaining number sequence even if we assigned 5 as the last number. In other words, there can't be a second last number.

But this is an example of equivocation (the equivocated term being "end".)

What does it mean to say that someone has completed an infinite number of steps?

It means that they performed a number of steps before certain point in time (that's what the word "completed" indicates: the presence of a temporal end) Yes, in particular, the past. and that the number of steps they performed has no integer greater than it (that's what the word "infinite" indicates: the lack of quantitative end.)

They are two DIFFERENT ends. They AREN'T one and the same end. Hence why there's no contradiction.

Yes, there is a difference between the end of a temporal process or task, and the end of a sequence. However, I did not mean the end of a process or task, I meant the end of a sequence. As per my clarification above, my argument was that even if you assigned 5 to be the last number, the problem is simply shifted to the second last number, of which there isn't one, and therefore neither can there be a last one.

Perhaps the best way to realize that the above sequence is not a logical contradiction is to represent it as a set of ordered pairs \(S = \{(\infty, 5), (1, 1), (2, 2), (3, 3), (4, 4), (5, 6), (6, 7), \dotso\}\).

Right, now what index would you assign the second last number in the sequence? Don't say \(\infty\) - 1 because that would count as a hyperreal number and the sequence consists of only real integers.

The sequence has no end in the sense that the number of its element is infinite but it has an end in the sense that it has a position with the highest index.

Those are indeed two different meanings to the word "end", but they are not compatible. The first sense of the word "end" in your statement precludes the second. Not having an end in the sense that the number of elements is infinite means you can't have an index which is the "ending" index ("last", "highest" <-- these all mean the same thing). You'd have to skip the entire infinity of numbers in the sequence to get to that index, but that's just another way of saying "skip to the end" which contradicts the first meaning of "end".

[gib]

Now...

Let me write down my thoughts on hyperreals.

I don't think the concept requires any introduction. Anybody who's been reading along so far probably gets the gist that they are numbers bigger than infinity or smaller than infinitesimals (and by "bigger than infinity" we mean greater in absolute magnitude so that negative infinity is included).

I only grant the validity of hyperreal numbers given an incoherent assumption: that you can have quantities greater than infinity. I won't go into the reasons why I think this is incoherent (anyone can read my numerous arguments throughout this thread), and given that I am granting that assumption in this post (and any discussion on hyperreals), it would be unnecessary anyway.

But if we grant that assumption, a whole world of possibilities opens up with hyperreals, a whole lota math and, yes, even arithmetic. Mathematicians wouldn't have been able to establish a whole branch of mathematics if this were not so. It should be noted that hyperreals aren't universally accepted even among mathematicians, but there are a sizable number of them who do accept them for them to warrant serious consideration.

Here are a few assumptions I hold which will likely come up for debate:

1) Hyperreals don't represent specific quantities.

The vsauce video which I've been trying to urge Magnus to watch talks about a similar concept. Michael Stevens of vsauce talks about counting past infinity with ordinal numbers. Ordinals are numbers used to describe the order of things. Contrast them with cardinals which are used to describe the quantity of things. Examples of cardinals are 1 apple, 2 oranges, 3 shoes, etc. Examples of ordinals are 1st place, 2nd best, 3rd contestant, etc. Stevens makes the point that you can't count past infinity with cardinals, that there is no quantity greater than infinity that can be represented by a cardinal. But you can represent numbers greater than infinity with ordinals. If your purpose is to count items in the order they are arranged or added or appear, then it is sometimes impossible to do so when you start with an already infinite set unless you have omega, the first ordinal number. The idea is that if you have an infinite number of things, and then one additional thing shows up, you can't use cardinal numbers to count that additional thing since all the cardinals would have been used up in counting the initial infinity of things. So you need to bring in omega. If order didn't matter, you could count that additional thing as item #1, and then begin counting the infinite set starting at 2, 3, 4,... which would never give you a result greater than \(\infty\). But if order does matter, you have to count all the items in the original set first, then move onto the additional item.

Stevens stresses that once you get into omega, you are no longer talking about quantities. You are simply talking about order. The omegath item does not make the sum of items \(\infty\) + 1, it is simply the item that comes last. Likewise if you have a (omega+1)th item, that is simply the item that comes after the omegath item, and so on for omega+2, omega+3, etc.

I don't know if Stevens would agree that ordinals greater than infinity are synonymous with hyperreals (even if we focus strictly on the hyperreals greater than \(\infty\)), but I think there's more to hyperreals than Stevens' treatment of ordinals greater than infinity. For one thing, hyperreals are on the number line and form a continuum with the reals. This makes it awefully difficult to dismiss their function in representing quantities. It seems we'd have to at least say they represent quantities greater than infinity, and that if R is a hyperreal number, R+1 is greater than R by 1. But I would agree with Stevens if he said that hyperreals don't represent a specific quantity. They can't. Exactly how much greater than infinity is some arbitrary hyperreal number R? If the hyperreal numbers extend infinitely in both directions, then the only answer to this questions seems to be "infinity". R is infinitely larger than infinity. So whereas Stevens might say hyperreals don't represent quantities at all (assuming his treatment of ordinals greater than infinity can be carried over to hyperreals), I'm willing to be a bit more leneant and say they do represent quantities, just not specific ones (other than being greater than infinity).

2) Infinitesimals don't represent the smallest numbers possible.

Just as the infinitely large hyperreals extend infinitely in both directions, infinitesimals also extend infinitely in both directions, but in terms of scale rather than position on the number line (though they extend infinitely in that sense too). By scale, what I mean is the amount of division one must do to get to infinitesimals. Take any real number and divide it infinitely many times, and you get to an infinitesimal. But that doesn't get you to "the end of the line" so to speak. Infinitesimals can still be divided further to get infinitesimals that are even infinitesimal in relation to the infinitesimal you started with. And it goes without saying that you can multiply any infinitesimal upward an infinite number of times to get back to the reals.

It also goes without saying that infinitesimals don't stand for specific quantities for similar reasons that infinitely large hyperreals don't stand for specific quantities. It is completely arbitrary where you land after dividing a real number an infinite number of times. If we call the infinitesimal where you land \(\epsilon\), is this where you land after dividing \(\infty\) times? \(\infty\) + 1 times? \(\infty\) + \(\infty\) times? Since we can't say specifically where you land, we can't say how much smaller \(\epsilon\) is from 1 (or any real number), not specifically. We can, of course, say how much \(\epsilon\) is relative to other multiples of \(\epsilon\). For example \(\epsilon\)2 is twice as large as \(\epsilon\). We can even say how much larger \(\epsilon\) is from 0--it's excatly \(\epsilon\) larger than 0. But beyond that, we can't say specifically how much a given infinitesimal is.

3) You can do arithmetic with hyperreals but you might get unexpected results.

For example, multiplying an infinitely large hyperreal R by 2 gives you a hyper-hyperreal number. That is, a hyperreal number that is hyperreal even relative to R--i.e. R2 is infinitely larger than R. You can see why just by giving it a moment's thought. If R is already infinite in size, then multiplying it by 2 should give you a result that is twice as infinite.

Unlike the infinitely large hyperreals, infinitesimals play by the opposite principle. Multiplying them by even astronomically large real numbers still gives you an infinitesimal on the same order. But doing something as simple as adding 1 to an infinitesimal causes it to take an infinite leap to another spot on the real number line. And it can be seen why: if you have \(\epsilon\) (the first infinitesimal after 0 at a given scale), and you add 1 to it, you get 1 + \(\epsilon\), which, relative to \(\epsilon\), is an infinite distance away.

If X = 1, then you can count up by multiplying X by the natural numbers or by adding the natural numbers:

X1, X2, X3, X4... = 1, 2, 3, 4...

X+1, X+2, X+3, X+4... = 1, 2, 3, 4...

If R is an infinitely large hyperreal, you cannot count up by multiplying it by the natural numbers (as each multiplication after 1 would give you a hyper-hypereal number), but you can count up by adding the natural numbers:

R, R+1, R+2, R+3...

If \(\epsilon\) is an infinitesimal, you cannot count up by adding the natural numbers (as each addition after 0 would give you a number an infinite distance away from \(\epsilon\) relative to \(\epsilon\)), but you can count up by multiplying by the natural numbers:

\(\epsilon\)1, \(\epsilon\)2, \(\epsilon\)3, \(\epsilon\)4...

4) The standard treatment of infinity we usually give before bringing hyperreals onto the table carries over to hyperreals after bringing them to the table.

For example, we say there is no largest real number, that the reals just go on infinitely. And while we don't stop saying this once we bring hyperreals to the table, we must say the same thing of hyperreals. There is no more reason to say there is a largest hyperreal number than there is to say there is a largest real number. This reason applies repeated to any higher order of hyperreals (so hyper-hyperreals, hyper-hyper-hyperreals, etc.). Similarly for infinitesimals. Just as we say there is no smallest number (down in scale, though also down through the negatives), we must also say there is no smallest infinitesimal.

What this means is that a lot of the issues that makes the concept of infinity so problematic won't necessarily go away just because we bring hyperreals to the table.

For example, in an earlier discussion I had with Magnus, I argued that removing every odd point from an infinite line and shifting the remaining points to fill the gaps would give you a perfectly identical line. Magnus's response was to argument that beyond the infinitieth point, you would see a difference. The line would go from this:

\(\bullet \bullet \bullet ... + \bullet \bullet \bullet ...\)

... to this:

\(\bullet \bullet \bullet ... + \circ \circ \circ ...\)

He suggested that all the points beyond infinity would have to be used up to fill the gaps on this side of infinity.

But if the hyperreals have their own infinity (beyond which there are the hyper-hyperreals), and indeed there is no end to the orders of hyperreals, then a more extended picture of the line might look like this:

\(\bullet \bullet \bullet ... + \bullet \bullet \bullet ... + \bullet \bullet \bullet ...\)

And if you remove every odd point from the line (at least on this side of infinity) and shift the remaining points into the gaps, then maybe the result would look like this:

\(\bullet \bullet \bullet ... + \bullet \bullet \bullet ... + \circ \circ \circ ...\)

...but of course, this line of reasoning carries on indefinitely, meaning that you could have an infinite string of infinities, none of which are ever depleted of points because the points from the next infinity over get shifted down.

One eventually wonders: why not just say the entire series of infinities is one great infinity? And you're back at square one. This is just one example of how introducing hyperreals won't necessarily solve all the problems plagued by infinity.

I'll stop there. I could go one with several more points, but I believe this is enough to get a fruitful discussion going, if not between Magnus and I (because he's scared of hyperreals even though that's what he's talking about without knowing it) then between other members of ILP. Should be fun.

[Magnus Anderson]

You are such an avoider.

Actually, it is you who are the avoider. I am merely avoiding to go down the path of the avoider.

Avoiding avoidance =/= avoidance (:

So what happens if you add 1 to the largest number?

You can't add 1 to the largest number.

Just like that? No reasoning? No argument?

By definition, the largest number is a number greater than every other number. This means that there is no number greater than it. Thus, it is a contradiction in terms to speak of \(L + 1\) where \(L\) is the largest number.

If it's a number, you can add to it. Every number represents a certain quantity. If it's a quantity, you can have more than it or less than it. Thus, you can add 1 to any number.

Not really. Once again, you are limiting yourself to numbers that are familiar to you (integers, rationals, reals, etc.)

If something is a number, that means it's greater than AND/OR less than something else. It does not mean there's something greater than it AND something less than it.

What supernatural power prevents 1 from being added to it? Don't tell me: it's the largest number, so by definition you can't add 1 to it--because I'll tell you that the inability to add 1 to it is why we should doubt the existence of a largest number in the first place.

I'm afraid that, in order to resolve this disagreement, we'll have to discuss subjects more fundamental than this one such as logic.

Let's adopt your definition of the word "number". Note that I do not accept this definition, I'm merely adopting it for the purpose of this post; I want to explore where it leads. Let the word "number" denote something for which there is always something greater than it and something less than it.

With that in mind, to speak of a number greater than every other number is a logical contradiction. But that does not mean we can't speak of something that is greater than every number. Of course we can. And the fact it's not a number (as per Gib's definition) does not mean we can't do arithmetic with it. Of course we can. And the fact it's not a number (as per Gib's definition) does not mean it doesn't have a place on the number line. Of course it does.

Right because the word "integer" is featured in every dictionary definition of infinity and no mathematician would ever fail to mention "integers" when defining infinity. Why do you even feel compelled to dwell on this point?

Why are you bringing it up?

There's nothing new and exotic about the definition (I'm neither the first nor the last person to define infinity that way), but most importantly, there's nothing wrong with it. In fact, it's less deceptive than the usual "Infinity is something without an end."

So you're saying infinity is greater than any integer but it is not among the set of all integers. And at the same time, there is no greatest integer. <-- This is actually quite sensible. It's the common definition of infinity.

That's correct. Infinity is greater than every integer but it is not an integer itself. And there is no largest integer (that's a contradiction in terms.)

Where I get thrown for a loop is when you say infinity is not necessarily greater than any quantity. If we leave out the part about there being a greatest integer, we still have the idea that there continues to be quantities after the integers. Now, I can't help but to go into hyperreals at this point. It's the only way I can make sense out of this. So...

It seems to me the only way for you to make sense out of anything is to defer to authority.

You're kinda cheating there. If (\(\infty\), 0) is supposed to be the "last" item in the list (or the item with the "highest index"), then you're representing it out of sequence. The proper representation of the sequence should lay out the sequence, well, in sequence. It's easy to take the infinitieth item and place it in the front (which, btw, I noticed is 0, which means you're not using the index properly), but when laid out according to the actual sequence, you see you can never get to (\(\infty\), 0). Which reinforces what I said: \(\infty\) is not a point on the number line, and therefore can't be used as an index.

You just forgot that sets have no order and that it does not matter how you're visually representing them. You can place \((\infty, 0)\) anywhere you want. You can place it at the beginning, you can place it at the end, you can place it at somewhere in the middle; it does not matter. In each case, it represents \(0\) occupying the position the index of which is \(\infty\). That's what the pair is telling you. The first element in the pair represents the position in the sequence, the second element represents the element occupying that position.

So no, I'm not cheating, it's merely you not being able to think properly.

[Magnus Anderson]

In other words, there can't be a second last number.

There can be such a number but in the case of \(S = (1, 2, 3, 4, 6, 7, 8, \dotso, 5)\) there is no such a number. There is no "second last number". There is only "last number".

the end of a sequence

I wasn't talking about the end of a sequence.

As per my clarification above, my argument was that even if you assigned 5 to be the last number, the problem is simply shifted to the second last number, of which there isn't one, and therefore neither can there be a last one.

If there is no second last number, it does not follow there is no last number.

"Last number", in the case of sequences of numbers, simply means "number occupying the position with the highest index".

"Second last number", in the case of sequences of numbers, simply means "number occupying the position with the second highest index".

In the case of \(S = (1, 2, 3, \dotso, 0)\), there is a position with "the highest index" but no position with "the second highest index".

On the other hand, \(S_2 = (1, 3, 5, \dotso, 6, 4, 2)\) does have the second last number, as well as the third last number and so on. But it does not have a middle point. On the other hand, \(S_3 = (1, 3, 5, \dotso, 0, \dotso, 6, 4, 2)\) does. And so on.

Right, now what index would you assign the second last number in the sequence? Don't say \(\infty\) - 1 because that would count as a hyperreal number and the sequence consists of only real integers.

There is NO second last number in the sequence.

The set \(S = \{(\infty, 5), (1, 1), (2, 2), (3, 3), (4, 4), (5, 6), (6, 7), \dotso\}\) contains no pair of the form \((\infty - a, b)\) where \(a\) is a number greater than \(0\) but less than \(\infty\) and \(b\) is an integer or a hyperreal (or any other number for that matter.)

[Ecmandu]

Magnus,

Honestly!

There is no last number but there is a last number?

You’re making no sense here.

“Completed infinity” is “married bachelor”

Accept it and move on.

[Silhouette]

Magnus,

Honestly!

There is no last number but there is a last number?

You’re making no sense here.

“Completed infinity” is “married bachelor”

Accept it and move on.

I think the following sums up the last month of 30 pages worth of getting nowhere, since this non-issue thread was resurrected by our primary offender:

It won't change my belief that \(0.\dot9 \neq 1\) but I'd no longer have an argument against your proof.

You can't reason with a zealot.

[gib]

Actually, it is you who are the avoider. I am merely avoiding to go down the path of the avoider.

Avoiding avoidance =/= avoidance (:

Ooooh, that's what this is. Avoidig avoidance--which is not avoidance at all. So what exactly am I avoiding? Not talking about hyperreals? Do I lack the courage to not talk about hyperreals? Is that what you're saying?

So tell me, how do we know what counts as "courageous" and what counts as "cowardice"? Seems to me like you can arbitrarily call any form of avoidance "courage" by calling it "avoiding avoidance".

Fact of the matter is, Anderson, you are talking about hyperreals. The only thing you're avoiding is calling them that.

By definition, <-- OMG, you said it. the largest number is a number greater than every other number. This means that there is no number greater than it. Thus, it is a contradiction in terms to speak of \(L + 1\) where \(L\) is the largest number.

Hey, Magnus, guess what! I define the "magic number" as a number that is both the greatest number and smallest number at the same time. It's also indivisible so you can't divide it. It also has a nasty drug habit, likes setting things on fire, and is gay. By definition, you cannot add 1 to it or subtract 1 from it. By definition, you can't divide it... not even a little bit. Also, by definition, it has a secret crush on the number 2 who happens to be male.

These things have to be true of the magic number 'cause that's how it's defined. But I'm going to venture a guess that you don't believe in the magic number. You'd press me to prove that it exists, right? Well, here's one last part of the definition: it exists! The magic number, by definition, exists. There! I defined it into existence, just like the being greater than which can be conceived.

What can we say about the largest number L? Can you prove such a number exists, or are you just going to define it into existence?

If it's a number, you can add to it. Every number represents a certain quantity. If it's a quantity, you can have more than it or less than it. Thus, you can add 1 to any number.

Not really. Once again, you are limiting yourself to numbers that are familiar to you (integers, rationals, reals, etc.)

If something is a number, that means it's greater than AND/OR less than something else. It does not mean there's something greater than it AND something less than it.

Definitions. You're defining "numbers" in such a way that they can include things which I wouldn't call numbers. Like I said earlier, I don't mind customized definitions (and you don't have to deny it's customized--let's not dwell on that distraction), just as long as they are coherent and you use it consistently. It's the coherency that I stuggle with here. What kind of thing is the "greatest possible number"? <-- This defies any definition of number I'm familiar with, but if you have your own definition, please help me understand what "greatest" means in this case? I can't make sense out of "greatest" with my concept of numbers, so help me understand your concept such that I understand how there can be a "greatest" and what "greatest" would mean in that case. Even you say that \(\infty\) + 1 > \(\infty\). I would have thought, had I agreed that \(\infty\) was a number, that if anything is the greatest number to which you cannot add 1, it would be \(\infty\). But apparently, you can go beyond endlessness according to you. What would be so great that you cannot go beyond it despite that you can go beyond endlessness?

Let's adopt your definition of the word "number". Note that I do not accept this definition, I'm merely adopting it for the purpose of this post; Fair enough. I want to explore where it leads. Let the word "number" denote something for which there is always something greater than it and something less than it.

Works for me.

With that in mind, to speak of a number greater than every other number is a logical contradiction. But that does not mean we can't speak of something that is greater than every number. Of course we can. Sure, but then we have to redefine "greater". And the fact it's not a number (as per Gib's definition) does not mean we can't do arithmetic with it. Of course we can. And the fact it's not a number (as per Gib's definition) does not mean it doesn't have a place on the number line. Of course it does.

You're gonna have to work a bit harder to convince me of those last two points. You know my argument about doing arithmetic with non-numbers. It's the equivalent of doing arithmetic with "cow". Or, alternatively, you can do arithmetic but then you have to decide upon a whole different set of rules. To say we can do arithmetic with something that's not a number (according to my definition) means that we have to decide what each arithmetic operation means. What does it mean to "add" something that isn't a number? To subtract? To divide? etc. As for the number line, it's made from numbers. The number line is essentially the sequence of real numbers (plus hyperreals if you believe in that), it's the order in which they are arranged. There's nothing that's not a number that has a place in the sequence. It makes no sense to say: the numbers go 1, 2, 3, cow, 4, ... If you want to say something that's not a number (according to my definition) belongs on the number line, you need to prove to me why it must fall on the number line.

Right because the word "integer" is featured in every dictionary definition of infinity and no mathematician would ever fail to mention "integers" when defining infinity. Why do you even feel compelled to dwell on this point?

Why are you bringing it up?

I respond to what I feel like.

There's nothing new and exotic about the definition (I'm neither the first nor the last person to define infinity that way), Then show me the plethora of cases that are out there. but most importantly, there's nothing wrong with it. <-- Yeah, that was my point... as long as it's coherent and used consistently. In fact, it's less deceptive than the usual "Infinity is something without an end."

Deceptive?!?! That's just the definition:

"The word is from a Latin word, which means "without end"." <-- https://simple.wikipedia.org/wiki/Infinity

"Definition of infinity
1a: the quality of being infinite
b: unlimited extent of time, space, or quantity : BOUNDLESSNESS" <-- https://www.merriam-webster.com/dictionary/infinity

"Infinity ...

... it's not big ...

... it's not huge ...

... it's not tremendously large ...

... it's not extremely humongously enormous ...

... it's ...

Endless!" <-- https://www.mathsisfun.com/numbers/infinity.html <-- I like this one.

The fact that you think it's deceptive speaks volumes. It means you have a specific concept in mind, and from that concept, it doesn't follow that infinity is endlessness, or at least that endlessness is not at the core of the concept. But we don't all necessarily have the same concept. Sure, it's possible that we make mistakes in trying to define our concepts--unraveling the content of our understandings and putting them into words can be challenging--but just because someone comes up with a different definition than you doesn't mean they're making a mistake. It can mean they simply have a different concept. More importantly, our concepts are derived just as much from definitions as definitions are derived from our concepts. So while we sometimes come up with definitions to describe our concepts, we learn concepts from being taught definitions. I would think the concept of infinity is learnt this way. We're taught, when young, that "infinity means no end," and we carry the concept derived from this definition forward with us.

Where I get thrown for a loop is when you say infinity is not necessarily greater than any quantity. If we leave out the part about there being a greatest integer, we still have the idea that there continues to be quantities after the integers. Now, I can't help but to go into hyperreals at this point. It's the only way I can make sense out of this. So...

It seems to me the only way for you to make sense out of anything is to defer to authority.

Did you read my post on hyperreals? Those are my thoughts. I did't borrow them from any authority. I didn't even verify them with any authority (they might tell me I'm out to lunch). The only thing I borrowed was the term "hyperreal", and given that hyperreals are really all you're talking about, you might as well too.

Furthermore, don't lose sight of the fact that I think hyperreals are a silly concept. I actually don't agree with the authorities on their reality. Not to mention that the "authorities" aren't even unanimous in agreeing with the validity of hyperreals.

Also, while there is a fallacy in logic called "appeal to authority", we shouldn't commit to opposite fallacy either: assuming that an appeal to authority automatically invalidates one's point. A lot of the time, the authorities are a good source because they do their job well--whether that's with scientific scrutiny, logical rigor, or scholarly discipline--and all one needs to do is look into the subject matter and judge for themselves whether it makes sense or not.

I might even add that almost everything we talk about can be traced back to an authority of one kind or another; you seem to believe in the basic rules of arithmetic. Did you not learn them from an authority in grade school?

Lastly, if I recall, it was you who brought up hyperreals to bolster your point. Hello kettle, I'm teapot.

You just forgot that sets have no order and that it does not matter how you're visually representing them. You can place \((\infty, 0)\) anywhere you want. You can place it at the beginning, you can place it at the end, you can place it at somewhere in the middle; it does not matter. In each case, it represents \(0\) occupying the position the index of which is \(\infty\). That's what the pair is telling you. The first element in the pair represents the position in the sequence, the second element represents the element occupying that position.

Let's not lose sight of the original objection. I was objecting to saying you can have an infinitieth place on the grounds that such a place would have to exist at the end of the sequence, and with an endless sequence, there is, by definition, no end.

What you're saying is that you can think of it as a set, and therefore you can tag the last element as the "infinitieth" element out of order (i.e. first, second, 50th, millionth, whatever). It's true that with sets, strictly speaking, there is no order. But it's cheating because treating it as a set doesn't allow you to say it's not a sequence. It is both a set and a sequence at the same time. By labeling a certain item in the set "infinity" what you are in fact saying is that it's the "infinitieth" (or last) item, which is to say, it is a sequence after all. And to say it is the "infinitieth" item in the sequence is to say it is the end item in an endless sequence--a contradiction in terms.

If you don't embrace both--that it is a set AND a sequence at the same time--then you're just tagging items with meaningless symbols. You might as well label them "cow", "pig", "stapler", "xyz", whatever.

There can be such a number but in the case of \(S = (1, 2, 3, 4, 6, 7, 8,...,5)\) there is no such a number. There is no "second last number". There is only "last number".

That is the most absurd thing I've ever heard. How can you have a last number if there is no second last number? What do you call the number that comes right before the last number? What do you go through in order to get to the last number?

Don't you think this goes completely against the definition of a sequence? Doesn't a sequence have to have a first item, a second item, a last item, and every item inbetween? Isn't that what a sequence is?

Besides that, if you think S doesn't have a second last item, why would you despute the claim that that \(S' = (1, 2, 3, 4, 5, 6 ... \infty)\) doesn't have a last item? Seems like the exact same logic to me.

the end of a sequence

I wasn't talking about the end of a sequence.

I was.

If there is no second last number, it does not follow there is no last number.

Uh... yeah, it actually does.

"Last number", in the case of sequences of numbers, simply means "number occupying the position with the highest index".

And there needn't be a second highest index? Remember, Magnus, these indexes are supposed to represent the order in which the numbers are arranged. It's impossible to have an order with positions in the order missing.

"Second last number", in the case of sequences of numbers, simply means "number occupying the position with the second highest index".

In the case of \(S = (1, 2, 3, \dotso, 0)\), there is a position with "the highest index" but no position with "the second highest index".

Why? Because there is no last item in the ...? Because it's endless? Isn't that what I was arguing to begin with? That an infinite sequence has no last item? Why you're suddenly singing that tune just because we're talking about a second last item is beyond me.

And how do you end up putting 0 as the last item when the ... implies no end, and therefore no possibility of anything coming after?

On the other hand, \(S_2 = (1, 3, 5, \dotso, 6, 4, 2)\) does have the second last number, as well as the third last number and so on. But it does not have a middle point. On the other hand, \(S_3 = (1, 3, 5, \dotso, 0, \dotso, 6, 4, 2)\) does. And so on.

You do see where this leads, right? You're always left with a ... in the sequence. That represents an endless sub-sequence. In that sub-sequence, there will be items that are an infinite distance away. Even the middle item (what is the middle of infinity if not an infinite distance away?). There will always be some items that cannot be indexed because they are at the end of an endless sequence (even half the ..., which is where the middle item would be, is an endless sequence), and the end of an endless sequence doesn't exist.

There is NO second last number in the sequence.

The set \(S = \{(\infty, 5), (1, 1), (2, 2), (3, 3), (4, 4), (5, 6), (6, 7), \dotso\}\) contains no pair of the form \((\infty - a, b)\) where \(a\) is a number greater than \(0\) but less than \(\infty\) and \(b\) is an integer or a hyperreal (or any other number for that matter.)

Right, but it's all fine and dandy to say there is a last number at infinity, because we can just label it as such in a set. I get that you're saying there is no second last item because we haven't labeled one as such, but like I said above, the natural numbers are not just a set, they are also a sequence. Whether or not we label them, each one does have a position in the sequence. If we say that infinity is at position \(\infty\), then there has to be an item at position \(\infty - 1\). So you can't just say "there is no second last number," because that defeats the purpose of labeling the numbers in the set according to their position in the sequence.

[Magnus Anderson]

Hey, Magnus, guess what! I define the "magic number" as a number that is both the greatest number and smallest number at the same time. It's also indivisible so you can't divide it. It also has a nasty drug habit, likes setting things on fire, and is gay. By definition, you cannot add 1 to it or subtract 1 from it. By definition, you can't divide it... not even a little bit. Also, by definition, it has a secret crush on the number 2 who happens to be male.

You're being overly dramatic. Typical for people drowning in frustration. Not sure how you can expect anyone to have any sort of discussion with you if you're lacking basic manners.

These things have to be true of the magic number 'cause that's how it's defined.

The magic number you speak of is a logical contradiction.

What can we say about the largest number L? Can you prove such a number exists, or are you just going to define it into existence?

Obviously, you know absolutely nothing about how logic works.

please help me understand what "greatest" means in this case

That's what this discussion boils down to. You not understanding things and begging for others (in a rather rude manner) to make you understand them within a period of time equal to zero. If they fail to do so, you get frustrated and start telling them they have no idea what they are talking about.

Let's just say that noone owes you anything. Don't expect much if you have no respect for others.

Then show me the plethora of cases that are out there.

How about the Wikipedia article on hyperreal numbers?
https://en.wikipedia.org/wiki/Hyperreal_number

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

\({\displaystyle 1+1+\cdots +1}\) (for any finite number of terms).

What are numbers of the form \({\displaystyle 1+1+\cdots +1}\) if not positive integers greater than \(0\) otherwise known as natural numbers?

You have no point whatsoever.

The fact that you think it's deceptive speaks volumes. It means you have a specific concept in mind, and from that concept, it doesn't follow that infinity is endlessness, or at least that endlessness is not at the core of the concept.

Let's say that you have absolutely no clue what I meant by that statement.

What you're saying is that you can think of it as a set, and therefore you can tag the last element as the "infinitieth" element out of order (i.e. first, second, 50th, millionth, whatever). It's true that with sets, strictly speaking, there is no order. But it's cheating because treating it as a set doesn't allow you to say it's not a sequence. It is both a set and a sequence at the same time. By labeling a certain item in the set "infinity" what you are in fact saying is that it's the "infinitieth" (or last) item, which is to say, it is a sequence after all. And to say it is the "infinitieth" item in the sequence is to say it is the end item in an endless sequence--a contradiction in terms.

You are missing the point.

The set of ordered pairs \(S = \{(\infty, 0), (1, 1), (2, 2), (3, 3), \dotso\}\) is the sequence \(S = (1, 2, 3, \dotso, 0)\). There is no difference between the two. They are merely two different symbols representing one and the same thing. The only difference is that the first representation is less deceptive than the second one. That's why I'm placing so much emphasis on it.

The point that your missing is the meaning of the term "the end of a sequence". What that means is simply "the position with the highest index". Both finite and infinite sequences can have such a position. Thus, there is ABSOLUTELY NOTHING contradictory about the idea of an infinite sequence that has an end (as well as the idea of an infinite sequence that has both a beginning and an end.)

That is the most absurd thing I've ever heard. How can you have a last number if there is no second last number?

You can have a last number without there being second last number.

What do you call the number that comes right before the last number?

There is NO such a number.

What do you go through in order to get to the last number?

You go through all of the infinitely many numbers that precede it.

Don't you think this goes completely against the definition of a sequence? Doesn't a sequence have to have a first item, a second item, a last item, and every item inbetween? Isn't that what a sequence is?

If it did, what would that mean? Absolutely nothing relevant.

In mathematics, you have infinite sequences that begin but that do not end (no first element) and infinite sequences that do not begin and do not end (no first and no last element.) You can expand this concept further to include sequences that begin and end but that are nonetheless infinite. Such a concept is not logically contradictory and that's precisely what I'm trying to show here (with the aim to show that there is absolutely nothing contradictory about the concept of "completed infinity".)

Uh... yeah, it actually does.

It does not.

And there needn't be a second highest index? Remember, Magnus, these indexes are supposed to represent the order in which the numbers are arranged. It's impossible to have an order with positions in the order missing.

You can represent order using "greater than" or "less than" operator.

\(a_1 > a_2 > a_3 > \dotso > a_L\)

We know that \(a_L\) has the greatest value EVEN THOUGH there is no \(a_x\) with the second-greatest value.

I get that you're saying there is no second last item because we haven't labeled one as such

There's no second last item because there's no position that comes before the last position but after every other position.

but like I said above, the natural numbers are not just a set, they are also a sequence

And every sequence is a set of order pairs \((a, b)\) where \(a\) represents the position in the sequence and \(b\) represents the element occupying that position.

Of course, you're working with an extremely limited concept of sequence. What do you think you can prove with it? Absolutely nothing.

The first thing you need to do is to STOP insisting that what I'm saying is a contradiction. It is not. You can say it's IRRELEVANT (though I don't agree it is) but there is no way in hell you can say it's a CONTRADICTION.

Understand that when I use the word "sequence" I am not using it in the narrow sense that you do.

[gib]

Hey, Magnus, guess what! I define the "magic number" as a number that is both the greatest number and smallest number at the same time. It's also indivisible so you can't divide it. It also has a nasty drug habit, likes setting things on fire, and is gay. By definition, you cannot add 1 to it or subtract 1 from it. By definition, you can't divide it... not even a little bit. Also, by definition, it has a secret crush on the number 2 who happens to be male.

You're being overly dramatic. Typical for people drowning in frustration. Not sure how you can expect anyone to have any sort of discussion with you if you're lacking basic manners.

And yet you and I keep going on and on and on, don't we.

The magic number you speak of is a logical contradiction.

So is the largest number.

(And there's nothing contradictory about being gay.)

What can we say about the largest number L? Can you prove such a number exists, or are you just going to define it into existence?

Obviously, you know absolutely nothing about how logic works.

Well, I'm convinced.

please help me understand what "greatest" means in this case

That's what this discussion boils down to. You not understanding things and begging for others (in a rather rude manner) to make you understand them within a period of time equal to zero. Come on, I gave you at least 2 seconds. If they fail to do so, you get frustrated and start telling them they have no idea what they are talking about.

Nothing could be further from the truth. I cordially asked you for clarification in the nicest possible way, giving you the benefit of the doubt that you might have a meaningful concept after all, one that I'm obviously too ill equipped to understand. It's the most reasonable thing one can do when one fails to understand the sense or meaning of another's point. It was an opportunity for you to show that you're making sense after all. But if all you can do is whine and bitch about how mean I'm being to you (can you imagine, asking you for clarification), it doesn't look good on you. It looks like you just can't rise to the occasion, and you'd rather lash out as a defense mechanism instead.

Let's just say that noone owes you anything. Don't expect much if you have no respect for others.

Hey, if you don't want to help me understand your point, maybe you shouldn't bother making it. Sounds to me like you just dig yourself deeper into a pit of frustration yourself. If I didn't know any better, I'd say you pretty desperately wish I could understand your point (you wouldn't have stuck it through this long otherwise). If I'm right, your unwillingness to help me understand your point indicates that you really don't know what you're talking about, and on some level you're painfully aware of it. I wasn't gonna say it, but there it is.

Then show me the plethora of cases that are out there.

How about the Wikipedia article on hyperreal numbers?
https://en.wikipedia.org/wiki/Hyperreal_number

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

\({\displaystyle 1+1+\cdots +1}\) (for any finite number of terms).

What are numbers of the form \({\displaystyle 1+1+\cdots +1}\) if not positive integers greater than \(0\) otherwise known as natural numbers?

You have no point whatsoever.

That's the best you can do?! The one source you dig up doesn't even mention integers.

If hyperreals are greater than:

1 + 1 + 1

...and hyperreals are greater than:

1 + 1 + 1 + 1

...don't you think they're also greater than:

1 + 1 + 1 + 0.5?

You wanted to say infinity is greater than all integers but not all quantities. I simply objected to the notion that infinity had to be defined in terms of integers. It's more comprehensive to say it's greater than all reals (watch, you'll argue even this). You can still have your quantities greater than infinity (certain infinities) since you obviously mean infinitely large hyperreals, which are by definition greater than all reals.

(Think this through Magnus, you don't actually have to fight this.)

The fact that you think it's deceptive speaks volumes. It means you have a specific concept in mind, and from that concept, it doesn't follow that infinity is endlessness, or at least that endlessness is not at the core of the concept.

Let's say that you have absolutely no clue what I meant by that statement.

I don't think anyone has any clue what you mean by anything.

What you're saying is that you can think of it as a set, and therefore you can tag the last element as the "infinitieth" element out of order (i.e. first, second, 50th, millionth, whatever). It's true that with sets, strictly speaking, there is no order. But it's cheating because treating it as a set doesn't allow you to say it's not a sequence. It is both a set and a sequence at the same time. By labeling a certain item in the set "infinity" what you are in fact saying is that it's the "infinitieth" (or last) item, which is to say, it is a sequence after all. And to say it is the "infinitieth" item in the sequence is to say it is the end item in an endless sequence--a contradiction in terms.

You are missing the point.

The set of ordered pairs \(S = \{(\infty, 0), (1, 1), (2, 2), (3, 3), \dotso\}\) is the sequence \(S = (1, 2, 3, \dotso, 0)\). <-- Impossible, but sure. There is no difference between the two. They are merely two different symbols representing one and the same thing. The only difference is that the first representation is less deceptive than the second one. That's why I'm placing so much emphasis on it.

Ok, but I thought we were talking about the number sequence. You know, the one where 0 comes before 1 and after -1?

The point that your missing is the meaning of the term "the end of a sequence". What that means is simply "the position with the highest index". Sure. Both finite and infinite sequences can have such a position. No. Thus, there is ABSOLUTELY NOTHING <-- Ooo, all caps. Must be true. contradictory about the idea of an infinite sequence that has an end (as well as the idea of an infinite sequence that has both a beginning and an end.)

Riiight, an endless sequence that has an end. <-- Nothing contradictory there.

That is the most absurd thing I've ever heard. How can you have a last number if there is no second last number?

You can have a last number without there being second last number.

I'm starting to notice a pattern: reiterating your statements as an answer to questions about those statements.

What do you call the number that comes right before the last number?

There is NO such a number.

What do you go through in order to get to the last number?

You go through all of the infinitely many numbers that precede it.

Is infinity like a box to you? You just go in and come out the other side, nothing to see in between?

Don't you think this goes completely against the definition of a sequence? Doesn't a sequence have to have a first item, a second item, a last item, and every item inbetween? Isn't that what a sequence is?

If it did, what would that mean? Absolutely nothing relevant.

You're right, nothing relevant at all. We're only talking about the number sequence, a series of numbers where every position is occupied by a specific number. How is that an example of what I'm talking about.

In mathematics, you have infinite sequences that begin but that do not end (no first element) Uh, I'm going to assume that was a typo, though you've said more absurd things. and infinite sequences that do not begin and do not end (no first and no last element.) You can expand this concept further to include sequences that begin and end but that are nonetheless infinite. Such a concept is not logically contradictory and that's precisely what I'm trying to show here (with the aim to show that there is absolutely nothing contradictory about the concept of "completed infinity".)

Yeah, Zeno's Paradox.

Uh... yeah, it actually does.

It does not.

Does too!

And there needn't be a second highest index? Remember, Magnus, these indexes are supposed to represent the order in which the numbers are arranged. It's impossible to have an order with positions in the order missing.

You can represent order using "greater than" or "less than" operator.

\(a_1 > a_2 > a_3 > \dotso > a_L\)

We know that \(a_L\) has the greatest value EVEN THOUGH there is no \(a_x\) with the second-greatest value.

I'm getting the impression that you're extremely easily deceived but visual notation. You're talking as if the ... is literally a collection of 3 dots, like it's not just a representation of an infinite sequence. I don't know about you, but whenever I see ..., I interpret that to mean: the series continues here--meaning that after \(a_3\) there's an implicite \(a_4\) and before \(a_L\) there's an implicite \(a_{L-1}\). <-- That would be your second greatest value.

(Actually, it would be the second smallest as you got the > sign wrong.)

On the other hand, if we go with your interpretation (that the ... is literally a collection of 3 dots squeezed in between \(a_3\) and \(a_L\)), we'd still necessarily have a second greatest value, and it would be the .... The expression above is literally saying the ... is greater than \(a_L\) and less than \(a_3\) (again, you got the sign wrong).

You can't get around it. The basic principle of counting once again rears its ugly head. If it's a sequence, you can count each item from beginning to end. Start from 1 and finish with n. Whatever number you count at any point, that's the item's position. So unless, for some reason, you feel compelled to skip a number while counting, every position should be accounted for, first, last, and yes, second last.

I get that you're saying there is no second last item because we haven't labeled one as such

There's no second last item because there's no position that comes before the last position but after every other position.

This is a classic Magnus Anderson quote. Can I borrow it for my sig?

but like I said above, the natural numbers are not just a set, they are also a sequence

And every sequence is a set of order pairs \((a, b)\) where \(a\) represents the position in the sequence and \(b\) represents the element occupying that position.

Of course, you're working with an extremely limited concept of sequence. What do you think you can prove with it? Absolutely nothing.

Well, there is the crazy idea that the number sequence doesn't have any positions missing, or maybe that there is no infinitieth position, but that's nothing.

The first thing you need to do is to STOP insisting that what I'm saying is a contradiction. Sorry, don't wanna speak untruths. It is not. You can say it's IRRELEVANT (though I don't agree it is) but there is no way in hell you can say it's a CONTRADICTION.

But Magnus, that's such a contradiction.

Understand that when I use the word "sequence" I am not using it in the narrow sense that you do.

You mean a sequence that has an infinitieth position and no second last position? You're right, that's not what I'm talking about.

[Magnus Anderson]

So is the largest number.

Not really.

"Magic number" is an obvious contradiction. You are saying it's both larger and smaller than every other number. A conjuction of \(a > b\) and \(a < b\) is a contradiction. The conjuction of \(P\) and \(\neg P\) is pretty evident here. Nothing of that sort applies to the concept of largest number.

The only reason you think the concept of largest number is a contradiction is because you think that I define the word "number" in such a way that the set of all numbers includes no more than the set of naturals, the set of integers, the set of rationals and the set of standard reals. Of course, if you define the word "number" in such a narrow way, then there is no largest number, since there is no largest natural, no largest integer, no largest rational and no largest standard real. But have you considered the possibility that I'm defining the word "number" in a different way?

Your number one problem appears to be "How can you do arithmetic with something that is not a natural, not an integer, not a rational and not a standard real?"

Not sure how to help you.

The one source you dig up doesn't even mention integers.

Does it have to?

"Larger than every number of the form \(1 + 1 + 1 + \cdots + 1\)" is closer to "Larger than every integer" than it is to "Larger than every real".

don't you think they're also greater than:

1 + 1 + 1 + 0.5?

Yes, they are. And your point is?

You wanted to say infinity is greater than all integers but not all quantities. I simply objected to the notion that infinity had to be defined in terms of integers. It's more comprehensive to say it's greater than all reals (watch, you'll argue even this). You can still have your quantities greater than infinity (certain infinities) since you obviously mean infinitely large hyperreals, which are by definition greater than all reals.

It's fine either way. You can say "Larger than every integer" or you can say "Larger than every real". It makes no difference. If something is greater than every integer it's also greater than every standard real and vice versa. And I never said that infinity HAD to be defined in terms of integers. I simply said that infinity is a number greater than every integer and not a number greater than every other number.

I don't know about you, but whenever I see ..., I interpret that to mean: the series continues here--meaning that after \(a_3\) there's an implicite \(a_4\) and before \(a_L\) there's an implicite \(a_{L-1}\). <-- That would be your second greatest value.

The ellipsis means there is \(a_4\) after \(a_3\) and \(a_5\) after \(a_4\) and so on but it does not mean there is \(a_{L-1}\). There is, in fact, no such element. That's you being deceived by the visual notation (which means you're guilty of your own accusation.)

On the other hand, if we go with your interpretation (that the ... is literally a collection of 3 dots squeezed in between \(a_3\) and \(a_L\)), we'd still necessarily have a second greatest value, and it would be the .... The expression above is literally saying the ... is greater than \(a_L\) and less than \(a_3\) (again, you got the sign wrong).

That's not my interpretation.

[Magnus Anderson]

\(a_1 < a_2 < a_3 < \cdots < a_L\)

This expression does not mention \(a_{L-1}\) at all. To think that it does is to misunderstand it.

But let's put that aside and consider the set of numbers that includes \(0\) and \(1\) as well as all decimal numbers between \(0\) and \(1\).

The smallest number in the set is \(0\). The largest is \(1\). There is no second smallest, third smallest and so on just as there is no second largest, third largest and so on.

This represents a small part of the real number line and as anyone can see it's an infinite sequence with a beginning and an end.

Now suppose that someone wants to move from \(0\) to \(1\). According to some, you can't do so, because there is no second smallest number. In other words, there is no number that immediately follows \(0\). Is it \(0.1\) or \(0.01\) or \(0.001\) or \(0.0001\) or \(\dotso\)?

This is reminiscent of Zeno's paradoxes. Motion is impossible because 1) space is infinitely divisible, and 2) if space is infinitely divisible, there are no immediately adjacent points.

The hidden assumption being that you can't move from point A to point B unless point B is immediately adjacent to point A. (Two points A and B are said to be "immediately adjacent" if there are no points between them.)

In reality, it might be impossible to move from point \(A\) to point \(B\) if they are not immediately adjacent but in concept this is not the case.

[Magnus Anderson]

So is the largest number.

"The largest number" does not mean "A natural, integer, rational or a standard real greater than every other number". It means "A number greater than every other number".

It's impossible to have an order with positions in the order missing.

Order is merely about what comes before and what comes after.

The word "first" refers to something that comes before all other things. That means there are no things that come before it. There can be order without there being something that is "first". For example, \(\cdots < a_3 < a_2 < a_1\) represents an order without something that is "first" (i.e. there is no \(a\) for which there is no some other \(a\) that comes before it.)

The word "last" refers to something that comes after all other things. That means there are no things that come after it. There can be order without there being something that is "last". For example, \(a_1 < a_2 < a_3 < \cdots\) represents an order without something that is "last" (i.e. there is no \(a\) for which there is no some other \(a\) that comes after it.)

The term "second last" refers to something that comes after all other things except for one thing. This means that nothing comes after it except for that one thing. For example, \(a_1 < a_2 < a_3 < \cdots < b\) represents an order without something that is "second last" (i.e. there is no \(a\) that comes before \(b\) for which there is nothing other than \(b\) that comes after it.)

So YES, there CAN be order without there being "first", "last" and "second last".

[gib]

So is the largest number.

Not really.

"Magic number" is an obvious contradiction. You are saying it's both larger and smaller than every other number. A conjuction of \(a > b\) and \(a < b\) is a contradiction. The conjuction of \(P\) and \(\neg P\) is pretty evident here. Nothing of that sort applies to the concept of largest number.

Glad you see that. Now let's do the same for the largest number:

If L is the largest number, then it is a number. If it's a number, it represents a certain quantity of things. So if you had a set consisting of that many things, there's no reason to say you cannot add one more thing to the set. If you add one more thing to it, you will have L + 1 things (this was the argument you so adamantly insisted was true about adding to infinity). But if L is the largest number, you cannot have a larger number L + 1. Therefore, adding one more item to a set of items whose quantity is L will not give you L + 1 items. <-- A Contradiction.

The only reason you think the concept of largest number is a contradiction is because you think that I define the word "number" in such a way that the set of all numbers includes no more than the set of naturals, the set of integers, the set of rationals and the set of standard reals. Of course, if you define the word "number" in such a narrow way, then there is no largest number, since there is no largest natural, no largest integer, no largest rational and no largest standard real. But have you considered the possibility that I'm defining the word "number" in a different way?

Of course I have. There's no other possibility. Unfortunately, you refuse to explain to me what your concept of number is. The best I can do, therefore, is to respond within the context of the standard definition of number.

Your number one problem appears to be "How can you do arithmetic with something that is not a natural, not an integer, not a rational and not a standard real?"

Not sure how to help you.

Well, you could try, oh I don't know, explaining what you mean by "number"?

The one source you dig up doesn't even mention integers.

Does it have to?

"Larger than every number of the form \(1 + 1 + 1 + \cdots + 1\)" is closer to "Larger than every integer" than it is to "Larger than every real".

Yeah, well, when the only example you can dig up for the definition of infinity has to be described as "closer" to the definition you have in mind, you know you're reaching. On the other hand, if you type "definition of infinity" into google, you get a whole ream of exact definitions, and none of them mention integers.

don't you think they're also greater than:

1 + 1 + 1 + 0.5?

Yes, they are. And your point is?

That you're better off defining infinity in terms of reals rather than integers. You stated earlier: "Infinity is a number greater than every integer, so it's greater than 1,000 as well as 1,000,000. And yes, infinity (if it refers to a specific quantity greater than every integer, and not merely to any such quantity) has a place on the number line." <-- If you mean that infinity falls on the hyperreal section of the number line, you could state this more clearly by saying infinity is greater than all reals. Otherwise, it leads one to wonder whether you think there are real numbers not only greater than infinity but all integers (and given some of the ideas you've defended in this thread, I wouldn't put it passed you).

You wanted to say infinity is greater than all integers but not all quantities. I simply objected to the notion that infinity had to be defined in terms of integers. It's more comprehensive to say it's greater than all reals (watch, you'll argue even this). You can still have your quantities greater than infinity (certain infinities) since you obviously mean infinitely large hyperreals, which are by definition greater than all reals.

It's fine either way. You can say "Larger than every integer" or you can say "Larger than every real". It makes no difference. If something is greater than every integer it's also greater than every standard real and vice versa. And I never said that infinity HAD to be defined in terms of integers. I simply said that infinity is a number greater than every integer and not a number greater than every other number.

The problem is that you refuse to acknowledge you're talking about hyperreals. The only way your statement can be true is if we allow for hyperreals. Since you refuse, it leads one to wonder whether you think "every other number" means real numbers other than the integers. And if you think infinity is greater than every integer but not every real number, it follows that you think there are infinitely large real numbers greater than every integer.

I don't know about you, but whenever I see ..., I interpret that to mean: the series continues here--meaning that after \(a_3\) there's an implicite \(a_4\) and before \(a_L\) there's an implicite \(a_{L-1}\). <-- That would be your second greatest value.

The ellipsis means there is \(a_4\) after \(a_3\) and \(a_5\) after \(a_4\) and so on but it does not mean there is \(a_{L-1}\). There is, in fact, no such element. That's you being deceived by the visual notation (which means you're guilty of your own accusation.)

Woaw, now that's an odd concept. This scares me because I'm going to have to ask, once again, how you're thinking about this, and we've seen how that goes down. But I'm going to venture a guess: you're thinking of the elipses as saying something like "goes off to infinity", as in the way we might write: 1, 2, 3,... Which is fine, but then you still want to add another number "after" infinity. At least with hyperreals, one could grant the existence of infinitely large hyperreal numbers after infinity by imagining a continuum between the reals and the hyperreals, but a continuum implies a merging from things on one side with things on the other (such that you could start on either side and count up or down to the other side). So you coud start with some arbitrary hyperreal R and count down to R-1, R-2, R-3, etc., indefinitely.

But what you seem to be saying is that ... means "goes on without end" where "without end" this times means you can't have a last item. And when you place L after the ..., it becomes the last item of the entire sequence and the "last item" of the ... (which doesn't exist) becomes the "second last item" of the entire sequence (which doesn't exist). This is unlike the case of hyperreals in that you are breaking the continuity between the one end and the other. You are essentially saying there is no continuity between \(a_1\), \(a_2\), \(a_3\), ... and \(a_L\). \(a_L\) is essentially the first item of a new continuum (and maybe the only item). But then in what sense is \(a_L\) part of the same sequence. A sequence implies continuity between all its members. All you seem to be doing is denoting an endless sequence as \(a_1\), \(a_2\), \(a_3\), ... and then just plopping \(a_L\) to the right of it (strictly at the level of notation only). But this cannot represent a single sequence. You're essentially saying the ... represents a section of the sequence without any end, yet \(a_L\) comes after the end.

On the other hand, if we go with your interpretation (that the ... is literally a collection of 3 dots squeezed in between \(a_3\) and \(a_L\)), we'd still necessarily have a second greatest value, and it would be the .... The expression above is literally saying the ... is greater than \(a_L\) and less than \(a_3\) (again, you got the sign wrong).

That's not my interpretation.

You know, you're really, really good at telling me what your interpretation is not.

\(a_1 < a_2 < a_3 < \cdots < a_L\)

This expression does not mention \(a_{L-1}\) at all. To think that it does is to misunderstand it.

It doesn't mention \(a_4\) either yet you agreed \(a_4\) was implicite.

But let's put that aside and consider the set of numbers that includes \(0\) and \(1\) as well as all decimal numbers between \(0\) and \(1\).

The smallest number in the set is \(0\). The largest is \(1\). There is no second smallest, third smallest and so on just as there is no second largest, third largest and so on.

This represents a small part of the real number line and as anyone can see it's an infinite sequence with a beginning and an end.

Now suppose that someone wants to move from \(0\) to \(1\). According to some, you can't do so, because there is no second smallest number. In other words, there is no number that immediately follows \(0\). Is it \(0.1\) or \(0.01\) or \(0.001\) or \(0.0001\) or \(\dotso\)?

This is reminiscent of Zeno's paradoxes. Motion is impossible because 1) space is infinitely divisible, and 2) if space is infinitely divisible, there are no immediately adjacent points.

The hidden assumption being that you can't move from point A to point B unless point B is immediately adjacent to point A. (Two points A and B are said to be "immediately adjacent" if there are no points between them.)

In reality, it might be impossible to move from point \(A\) to point \(B\) if they are not immediately adjacent but in concept this is not the case.

There is that. However, there is a common but subtle misconception about the "existence" of reals compared to integers, which effects how we understand the sense in which reals form a sequence compared to the sense in which integers form a sequence. This in turn is effected by how the brain develops the concepts of reals compared to integers (take a seat).

There is a difference between what I call counting numbers and measuring numbers. Counting numbers are pretty much integers. They are the numbers we invoke in order to count. We say 1, 2, 3,... as we count the number of things in a group. Counting numbers are discrete. There is a definite number of them between any two points on the number line. Children learn to grasp counting numbers first, meaning that the idea of counting numbers evolves at an early stage. Then you have measuring numbers. These are the reals. I call them measuring numbers because they describe amounts that can't always be derived from counting, such as how many liters of water are in a jug, or how many pounds of dirt. They represent numbers that are not discrete but rather form a smooth continuum with each other. Children can't easily grasp this concept and only learn to wrap their heads around it in mid- to late-childhood, indicating that this idea evolves at a later stage.

The key difference between these two types of numbers is discreteness. Integers are discrete units whereas reals are not. Real numbers form a smooth continuum on the number line. They aren't discrete entities that are all lined up in a row along the number line. Even if you want to talk about infinitesimals, infinitesimals are always further divisible so you can never quite zoom in far enough so that you finally find the elementary numbers that make up the number line. Some might compare reals to the geometric concept of a point. They say that lines are made of an infinity of points. But points are typically imagined as infinitesimal objects that are arranged adjacently in a linear row that, on a larger scale, make up the line. This is a bad way of imagining reals, not only because they are infinitely divisible even beyond the point of infinitesimals (although that would bring you into the hyperreals), but because they don't exist as discrete units that are arranged side-by-side. Rather, we should think of them as measures of the distance from zero to some point on the number line. That's what measuring numbers do after all, measure amounts, not count things. So if you mark a point on the number line and label it 0.1234... (whether the decimal expansion is finite or infinite, whether the number is rational or irrational), what you're saying is not that, right here is the number 0.1234.... You are saying, this point is a distance of 0.1234... from 0. The mark is more of a divide between the side of the number line containing 0 and the other side. Thinking of reals in this way (as measures of distance from 0 as opposed to little discrete objects aligned in a row), what does it mean to say there is an infinite number of reals between 0 and 1? It means we are not limited in the number of reals we have at our disposal for marking out points on the number line between 0 and 1 to represent the distance from 0. The infinity here denotes our options for dividing the number line up, not the number of things on the number line. As far as "things" go, the number line is not made of discrete things. It is a smooth continuum. We divide it up into discrete units. The brain naturally develops a default division scheme when it learns to count. All the integers are is a natural scheme for dividing the number line into equal discrete consecutive units, and this is so natural, we come to recognize this division scheme before we come to recognize the smooth continuity of quantities for what it really is. The smooth continuity of quantities, in other words, is actually more primordial than integers. Quantities are not discrete.

The question of how many there are, is therefore misguided. There aren't a specific number of reals. There are only specific number of reals we "pick out" (i.e. that we mark on the number line). But it become especially deceptive when we think of the reals as belonging to a set. When we talk about sets, we are talking about a collection of members, and members are conceptualized as discrete units. It's all fine and dandy to talk about the set of integers because these are already conceptualized as discrete units. But when it comes to reals, the idea of the set of real numbers presupposes that you can add them to the set as discrete units. This forces us to imagine them as little entities that form a sequence on the number line, each one coming after its predecessor, like geometric points forming a line, and this can lead to a lot of confusion.

You would be right to say there is no second last real number between 0 and 1, but I question whether it makes sense to talk about them as members of a set. If we can't talk about them as members of a set, then we certainly can't talk about them as members of a sequence (since sequences are just ordered sets). I was ok talking about the sequence of all integers, or even a sequence of real numbers so long as we picked out specific real numbers to belong to the sequence, but when it comes to all the real numbers between two points on the number line (or the whole number line), I'm not even sure we can talk about them as belonging to a set. They aren't discrete enough; they form a smooth continuum. The number line is composed of arbitrary segments, each one merging into its neighbors, no definite point where it begins and the next ends--all except if we divide it up a specific way--but we can't do that infinitely.

"The largest number" does not mean "A natural, integer, rational or a standard real greater than every other number". It means "A number greater than every other number".

Ok, well, I don't believe in it.

Order is merely about what comes before and what comes after.

Sure.

The word "first" refers to something that comes before all other things. That means there are no things that come before it. There can be order without there being something that is "first". For example, \(\cdots < a_3 < a_2 < a_1\) represents an order without something that is "first" (i.e. there is no \(a\) for which there is no some other \(a\) that comes before it.)

Works for me.

The word "last" refers to something that comes after all other things. That means there are no things that come after it. There can be order without there being something that is "last". For example, \(a_1 < a_2 < a_3 < \cdots\) represents an order without something that is "last" (i.e. there is no \(a\) for which there is no some other \(a\) that comes after it.)

This also works for me.

The term "second last" refers to something that comes after all other things except for one thing. This means that nothing comes after it except for that one thing. For example, \(a_1 < a_2 < a_3 < \cdots < b\) represents an order without something that is "second last" (i.e. there is no \(a\) that comes before \(b\) for which there is nothing other than \(b\) that comes after it.)

I can't address this without understand what the ... means to you. We've already ruled out a continuum, at least on the high end, but then I'm at a loss to understand what this could possibly mean.

So YES, there CAN be order without there being "first", "last" and "second last".

[/quote]

I didn't say there couldn't be order without "first", "last", and "second last". I said "It's impossible to have an order with positions in the order missing."

It's ludicrous to say there is a queue of 10 people but the last one in the queue is the 11th one because there is no 7th person. If there were 11 people initially, and person #7 left, then the 8th person simply becomes the 7th, the 9th becomes the 8th, and so on.

[Magnus Anderson]

I can't address this without understand what the ... means to you. We've already ruled out a continuum, at least on the high end, but then I'm at a loss to understand what this could possibly mean.

The ellipsis indicates that the pattern continues indefinitely. In the case of \(a_1 < a_2 < a_3 < \cdots < b\), the pattern is \(a_n < a_{n+1}\).

The first part of the expression, namely \(a_1 < a_2 < a_3 < \cdots\), is equivalent to the following list of statements:

\(a_1 < a_2\)
\(a_2 < a_3\)
\(a_3 < a_4\)
\(\dotso\)

The list is made out of an infinite number of statements. It contains every \(a_n, n \in N\) and it basically says that every \(a_n, n \in N\) is less than \(a_{n+1}\).

The second part of the expression, namely \(\cdots < b\), is equivalent to the following list of statements:

\(a_1 < b\)
\(a_2 < b\)
\(a_3 < b\)
\(\dotso\)

Like the previous one, the list is made out of an infinite number of statements. It basically says that every \(a_n, n \in N\) is less than \(b\).

You can take every variable in that expression and replace it with a standard real while preserving the truth value.

For example:

\(0.9 < 0.99 < 0.999 < \cdots < 1\)

In this case, \(a_n = \sum_{i=1}^{n} \frac{9}{10^i}, n \in \{1, 2, 3, \dotso\}\) and \(b = 1\).

[Magnus Anderson]

It doesn't mention \(a_4\) either yet you agreed \(a_4\) was implicit.

It does mention \(a_4\) but not explicitly i.e. it mentions it implicitly. \(a_{L-1}\), on the other hand, is not mentioned at all -- explicitly or implicitly.

I didn't say there couldn't be order without "first", "last", and "second last". I said "It's impossible to have an order with positions in the order missing."

Maybe you should try explaining what it means "to have an order with positions in the order missing".

It's ludicrous to say there is a queue of 10 people but the last one in the queue is the 11th one because there is no 7th person. If there were 11 people initially, and person #7 left, then the 8th person simply becomes the 7th, the 9th becomes the 8th, and so on.

And how exactly does that relate to what I'm saying?

"The largest number" does not mean "A natural, integer, rational or a standard real greater than every other number". It means "A number greater than every other number".

Ok, well, I don't believe in it.

Saying that you do not understand a position is not an argument against that position. It's merely an expression of ignorance. You have to understand an argument before you can say it's wrong. Indeed, most of what you're doing in this thread amounts to "I don't understand it, therefore it's wrong".