Is 1 = 0.999... ? Really?

Completed infinity is not an oxymoron. You’re merely confused by homonyms.

And so far, you’ve done nothing to show that it’s an oxymoron.

Try this: define what infinity is and then define what completed infinity is.

Magnus!!! For the nth-millionth time!!! Infinity is not a number!!!

Sure, infinity is that which begins but never ends.

Completed (anything) is that which ends. (That which you can add or subtract from). (Finite)

That’s a simple-minded understanding of these terms, no doubt responsible for your confusion.

Try tackling this post of mine.

Note that (S = (e_1, e_2, e_3, \dotso, e_L)) is an example of so-called completed infinity.

Show me the contradiction.

Properly speaking, infinity is a number greater than every integer.

When you say that a set has an infinite number of elements, what you’re saying is that the number of its elements is greater than every integer.

Note that sets have no beginning and no ends defined. There is no first element, no last element, no beginning and no end of any sort. And yet, they can be said to be infinite. What this tells us is that the concept of infinity has little to do with notions such as beginnings and ends. It really is just a number greater than every integer.

You can even take an infinite set and define where it beings (i.e. which one of its elements is the first element) and where it ends (i.e. which one of its elements is the last element) thereby turning it into some sort of sequence that nonetheless remains infinite (since it still has an infinite number of members.)

I can take a set of natural numbers (N = {1, 2, 3, \dotso}) and turn it into a sequence with a beginning and an end like so (N = (2, 3, 4, \dotso, 1) = {(\infty - 1, 1), (1, 2), (2, 3), (3, 4), \dotso}). The resulting sequence, despite having a beginning and end, is still an infinite sequence.

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.

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!

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

And yet one ball is bound to be picked.

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.

Yes.

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

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

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.

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.

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.)

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?

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.

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.)

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

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”.

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.

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.

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.

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 can’t add 1 to the largest number.

It’s not a new and exotic definition.

“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.

There’s no need to do so.

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.

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.

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.”

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.

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.

You are such an avoider.

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.

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?

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…

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.

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.

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”.

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).

  1. 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.

  1. 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…

  1. 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.

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

Avoiding avoidance =/= avoidance (:

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.

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.

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.

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.”

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.)

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

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.

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”.

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

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.

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.)

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.