Is 1 = 0.999... ? Really?

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.

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:

You can’t reason with a zealot.

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.

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?

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?

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.

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. :smiley:

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.

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.

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.

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.

I was.

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.

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.

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.

The magic number you speak of is a logical contradiction.

Obviously, you know absolutely nothing about how logic works.

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.

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

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.

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

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.)

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

There is NO such a number.

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

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”.)

It does not.

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.

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

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.

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

So is the largest number.

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

Well, I’m convinced.

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.

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.

:laughing-rolling: 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.)

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

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

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

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

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.

Yeah, Zeno’s Paradox.

Does too! :teasing-neener:

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.

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

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.

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.

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”.

Yes, they are. And your point is?

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 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.)

That’s not my interpretation.

(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.

“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”.

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”.

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.

Well, you could try, oh I don’t know, explaining what you mean by “number”?

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).

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.

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.

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

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.

Ok, well, I don’t believe in it.

[/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.

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).

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.

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

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

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”.

This is based on the erroneous assumption that every number has a number greater than it. This holds true for naturals, integers, rationals and standard reals but it does not hold true for all numbers. Indeed, it holds true for all numbers EXCEPT for one: the largest number.

Anything that is more or less than something else is a number. The largest number is a number because it’s something that is greater than every other number. The same applies to infinity: it’s a number because it’s something that is greater than every integer (or standard real, if you will.)

Why do I have to use the same exact words as other people do? Back when I was in school, it was highly desirable for students to use their own words for the simple reason that by doing so they prove they aren’t parrots.

“Greater than every integer” and “Greater than every standard real” are two different (and only slightly so) expressions of one and the same thing. There’s no difference. Nothing “new and exotic” about the former.

The Wiki says “Greater than every natural number”. Still not “Greater than every integer”, the “new and exotic” definition that I’m putting forward.

If something is greater than all integers, it’s also greater than all standard reals.

That’s not true.

Or, only somewhat tongue in cheek:

This is based on the erroneous assumption that every God has a God greater than it. This holds true for Christians, Muslims, Hindus, Shintos and standard Gods but it does not hold true for all Gods. Indeed, it holds true for all Gods EXCEPT for one: the largest God.

The one true God that you believe in. If, for example, you’re a Zoroastrianist, that would be Ahura Mazda.

In which case, you must have a serious problem with the expression (0.9 < 0.99 < 0.999 < \cdots < 1).

I have no idea how to respond to the statement that there is no such thing as the set of real numbers.

Maybe you should start with the definition of the term “discrete unit”?

Why are you saying that an integer such as (9) is a “discrete unit” whereas a real number such as (9.249238491) is not?

The fact is, absolutely anything can be a member of a set.

In other words, there is no set of all reals between (0) and (1) but there is a set of finite number of reals between (0) and (1)? And that’s because reals are not “discrete enough”?

Ah, there we go. Now was that so hard? You see what a little explanation can do (and an example to boot!)? I understand what you mean!

My confusion lay in the assumption that we were talking about the largest number (because that’s how this started), or the last number in the sequence of all integers–you know, at the infinitely big end–but you’re right, (\sum_{i=1}^{n} \frac{9}{10^i}) does represent an endless sequence of sums, and 1 does come after it. I stand corrected.

In this case, the “end” of the sequence has a double meaning and can lead to equivocation. In the one sense, the sequence (\sum_{i=1}^{n} \frac{9}{10^i}) has no end. You just keep adding (\frac{9}{10^i}) forever. On the other hand, it does have an end on the number line–that is to say, there is a definite limit which we can call the “end” (whether that’s 1 or < 1 is the central dispute of this thread, but let’s not worry about that now). In other words, there is no end term of the infinite sum but there is an end location on the number line–two different things.

It’s still problematic when we think of it as a sequence, however (at least for me), since the word “end” in that case has only one meaning: the last item in the sequence. And I am still troubled by the idea of the last item coming “after” a sub-sequence with no end. It might be argued that this isn’t a proper sequence. Maybe there are two sequences: a parent sequence and a child sequence. Maybe the parent sequence consists of only two items: P = ((\sum_{i=1}^{n} \frac{9}{10^i}), 1), where the child sequence is (\sum_{i=1}^{n} \frac{9}{10^i}): C = (0.9, 0.09, 0.009, …). In other words, the child sequence is treated as a single discrete item and placed in the first position of the parent sequence. This would work for me. This would permit that 1 still comes after all terms in (\sum_{i=1}^{n} \frac{9}{10^i}) without requiring that it be expressed as a “sketchy” sequence.

And how does one tell? Don’t you think there ought to be something in the notation that indicates the existence of implicit terms? Take the notation representing the continuity between the reals and the infinitely large hyperreals for example:

1, 2, 3, … , R-2, R-1, R, R+1, R+2, …

Here, the … represents implicit terms on both sides: 4, 5, 6, … on the left, … R-5, R-4, R-3, on the right.

Is it the fact that it’s obviously a sequence on both sides (i.e. the hyperreals are listed out as a sequence rather than a single number)?

Hard to tell when I don’t even know what you’re talking about. But now that you’ve elucidated what you’re talking about, we can put this line of argument to rest.

Fine, if you mean something incredibly insightful and profound by “the largest number” and my brain is just too stupid to grasp it, I can’t possibly know whether you’re right or wrong. But given your lack of explanation for what kind of number the “largest number” is, I fall back on the ordinary concept of “number” and say that I know no such number in that sense exists. And chances are pretty good you don’t have a special meaning of “number” in mind and just misunderstand how numbers work (even if we include hyperreals in the mix, you can’t have a largest number).

That’s an example of a sequence that has a beginning (first element, which is (a_1 = 0.9)) and an end (last element, which is (b = a_L = 1)) but no second last element, no third last element and so on.

Note that there is no (0.\dot9) in (0.9 < 0.99 < 0.999 < \cdots). The expression only contains finite decimals. So we both agree that (0.9 < 0.99 < 0.999 < \cdots < 1) is true.

What does “the last item in the sequence” mean?

I’ve explained this myself on multiple occasions. Basically, what it means is “the item occupying the position that has the highest index in the sequence” or “the item that comes after all other items in the sequence, which means, there is no item in the sequence that comes after it”. Thus, the last item in the sequence ((0.9, 0.99, 0.999, \dotso, 1)) is (1) since it comes after all other items in the sequence. The fact that it’s located an infinite number of positions away from all other items does not change that fact.

In which case it is necessary to explain what the word “sequence” means to you, what makes a sequence proper and what makes it improper, as well as the relevance of all of that.

It’s fine either way.

The first three elements in ((a_1, a_2, a_3, \dotso, a_L)) reveal the pattern. If I wanted to say the sequence is bi-infinite, I would have written something like ((a_1, a_2, a_3, \dotso, a_{L-2}, a_{L-1}, a_L)) instead.

There is nothing about naturals, integers, rationals, standard and non-standard reals that says “Look, there can’t be a number greater every other number.”

There is precedent for this, in that division works for all numbers except (0). We could similarly say that addition is undefined with respect to (L). I suspect that changing the definition of addition in that way would have other repercussions, but I’m not sure. I think all operations would have to be undefined with respect to (L).

If I might reply to one of Magnus’ objections that I don’t think was fully addressed.

If I understand your objection to the mainline proof, you think that this is false:
(10 * .999… = 9 + .999…)

This might be a good place to address this, because the question of what (.999…) means seems to be live right now.

How do you feel about a recursive definition: (.999…) is a number that satisfies two conditions: the first decimal place is a (9), and for the (n)th decimal place, if the (n-1)th decimal place is a (9), then the (n)th decimal place is also a (9).

Are those conditions sufficient to uniquely define (.999…)? I’d guess Magnus and Gib will disagree.

I think “convergence theory” is a load of crap.

“Convergence theory” is the theory that infinite sums equal (not approximately equal) to a whole number.!!

Magnus

Yes.

Well, I think it’s best to define “sequence” in order to answer that question. To me, a sequence is a series of objects with a defininte order. These objects can obviously be absract but they must be discrete (i.e. the objects have clearly defined identities, no ambiguity about where one object ends and another begins, no blurring of identities, etc.). For example, sections of a river would be difficult to describe as a sequence as there is no clear border between one section and another, and if you wanted to define each section as a collection of (H_2O) molecules, the molecules would blend in with and over take the molecules of the other sections, thereby making it unclear where one section ends and another begins (to the point where it can be unclear whether one section is always “before” another). On the other hand, carts in a train can much more easily be described as a sequence as they are definitely discrete and always maintain a specific order.

Each member of the sequence can be represented by a unique and unambiguous symbol, and the sequence itself can be represented by a linear arrangement of symbols (ex. (a_1), (a_2), (a_3), …) demonstrating the order each one comes in. This linear arrangement necessitates that there are no gaps in the sequence. So (a_1), (a_2), , (a_3), … is just bad notation, and more or less means the same as (a_1), (a_2), (a_3), … (or if you want to say the gap really is a member of the sequence, it should be represented with a symbol–like 0 or ‘null’–thereby making it at least an abstract object).

The last item in the sequence would just mean the symbol in the sequence taking the last position. So if you wrote out the sequence from left or right, it would be the rightmost item. In the sequence of the first 10 integers after 0:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

… 10 is the last item in the sequence.

I don’t dismiss the possibility of infinite sequences, so I’m not saying sequences must have an end. If we were to extend the above sequence to include all integers after 0, we would express it as:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

…and it wouldn’t have an end.

Infinite sequences can be infinite in both directions, as in the case of the sequence of all integers (not just the ones coming after 0), and in that case it would have neither a beginning nor an end.

Any time you talk about an item that comes after the end of an endless sequence, I have a problem with that. I hope you see the problem. If the sequence is endless, it doesn’t have an end. But if the last item comes after the end of the endless sequence, we are essentially saying it comes after something that doesn’t exist. Please tell me you see the problem here.

To talk about 1 coming after the end of the sequence (0.9, 0.99, 0.999, …) isn’t a problem because we’re not quite saying it’s the last item in the sequence, but that it comes after the end of the region on the number line covered by 0.9, 0.99, 0.999… which isn’t infinite. It actually has an end. But when you say 1 is the last item in the sequence, you’re no longer talking about a point on the number line but a position in the sequence. You’re saying the sequence is infinite (and therefore doesn’t have an end), yet 1 shows up at the end. ← There is a clear contradiction there which is why I have a problem with it (and why I brought up the two layered sequence model of parent/child).

Picking up from my definition above, what makes this kind of sequence improper (maybe) is just the problem noted above. It suggests something contradictory. It says the last item of the sequence comes after the end of an endless sub-sequence. The relevance is that this decides whether we can call (0.9, 0.99, 0.999, … , 1) a sequence or not.

Good. Then we have at least one way to think about it that we can agree upon.

That’s what I thought. But now here’s a challenge: what if we wanted to add 2 and 3 to the sequence above:

(0.9, 0.99, 0.999, … , 1, 2, 3)

Then it looks as though we are saying there is a number that comes just before 1.

If you were to ask me, I’d say we need a new notation. Maybe a bar:

(0.9, 0.99, 0.999, …| , 1, 2, 3)

The bar after the … indicates that there is no continuim between the endless sub-sequence represented by … and the sub-sequence immediately following it (no item immediately preceding 1).

This is the first time I’ve seen you use the term “non-standard reals”. Are these the kinds of strange numbers you have in mind? Can there be a largest non-standard real?

I’m not sure if you meant to lump them together with all other number types in this statement, but it sounds like you all of a sudden are saying there can be a largest number greater than all naturals, integers, rationals, and standard reals. So let me get this straight: you’re saying there is no largest natural, no largest integer, no largest rational, and no largest standard real, but there is a largest non-standard real which is greater than all natural, integers, rationals, and standard reals. Is that correct? Are you willing to explain to me what a non-standard real is?

Carleas

I’m not sure what Magnus’s objection to this step was, but I seem to recall something about a 0 appearing at the end of the string of 9s. Magnus is of the camp that believes you can have a last position of an endless sequence, which I would guess implies he thinks there’s a last 9. If you multiply (0.\dot9) by 10, therefore, there ends up being a 0 at the end of the sequence. This means the number you get is not (9 + 0.\dot9), and therefore the step that follows (where we replace (0.\dot9) with X) is unjustified.

I’m putting words in his mouth, obviously, but if this is correct, he might not agree with your definition since it insists that for every 9, there is a 9 that follows, and he would say this is true except for the infinitieth 9.