Is 1 = 0.999... ? Really?

Not really.

I addressed this “proof” around 20 pages ago and I can restate what’s wrong with it but I think it’s pointless since you don’t agree that we can do arithmetic with infinite quantities.

Basically, you don’t agree that adding a green apple to an infinite line of red apples increases the number of apples in the line. Instead, you prefer to contradict yourself by saying that the number of apples remains the same.

It has been claimed that it’s a contradiction in terms to say that an infinite sequence has a beginning and an end.

An infinite sequence has no end, so you cannot say that it has an end.

Well, you actually can, provided that the first occurrence of the word “end” and the second occurrence of the word “end” mean two different things (i.e. provided that they refer to two different ends.)

Don’t be fooled by homonyms.

(S = (e_1, e_2, e_3, \dotso, e_L)) is one such sequence. It’s an infinite sequence with a beginning and an end.

Note that a sequence with no repetitions is no more than a relation between the set of positions and the set of elements.

(S = (e_1, e_2, e_3, \dotso, e_L)) is a relation between the set of positions (P = {1, 2, 3, \dotso, \infty}) and the set of elements (E = {e_1, e_2, e_3, \dotso, e_L}).

Note that sets have no order. This means that, when visually representing a set, you can place its elements anywhere you want. This means that (P = {1, 2, 3, \dotso, \infty} = {\infty, 1, 2, 3, \dotso}). The same applies to (E). By moving the last element of the set to the beginning of the set, there are no longer any elements after the ellipsis, so there is less to complain about (:

Let’s represent the sequence as a set of pairs ((\text{position}, \text{element})). (S = (e_1, e_2, e_3, \dotso, e_L) = {(\infty, e_L), (1, e_1), (2, e_2), (3, e_3), \dotso}). And voila! There is nothing beyond the ellipsis anymore, so absolutely nothing to complain about (((:

“The last position in the sequence” refers to the largest number in the set of positions (P). Either there is such a number or there is not. In the case of our sequence, there is such a number and it is (\infty).

This is not the same as “The number of elements in the set of positions (P)”. This is an entirely different thing. In the case of our sequence, the number of positions is infinite (i.e. there is no end to the number of elements.) It’s also not the same as “The last element in the set of positions (P)”. No such thing exists, not because the set is infinite, but because sets have no order.

Aw, what a sad way to go out. We were so close to a break through. I don’t know why you didn’t want to get into hyperreals. I think that’s where you had your best shot and where I think you might have had a point. But I guess frustration got the better of you. Sayonara chico.

Talking about hyperreals is both unnecessary and pointless. How can you accept hyperreals if “you don’t agree that adding a green apple to an infinite line of red apples increases the number of apples in the line”?

I don’t accept hyperreals. But I’m willing to entertain them conditionally. Under the condition that you can have numbers greater than infinity (or numbers that are infinitely small), then hyperreals become not only a possibility but a necessity. We could then go on to debate the logic of hyperreals, argue about what can and can’t be said about them.

You need to address this argument:

Infinites can be measured?

I’ll address it. You’re assuming a “completed infinity”, that’s minus 1 or plus 1 adds or subtracts from it.

A COMPLETED infinity (contradiction, oxymoron!!)

Your response was that the argument is wrong because I’m assuming that adding an apple to an infinite set of apples makes the set larger.

Note that you did not say that I’m wrong because adding an apple to an infinite set of apples does not make the set larger (that would be a pretty bold statement.) No, you said that I’m assuming that adding an apple to an infinite set of apples makes the set larger. That’s not pointing out a flaw, that’s you not being to tell whether my conclusion logically follows or not. “I don’t see how it follows” is not pointing out a flaw. It’s merely an expression of ignorance. “It does not follow because of this and that”, on the other hand, is pointing out a flaw.

And the reason my conclusion follows is because by definition the operation of addition is the operation of increasing the quantity of things. What do you think the word “add” means?

You might want to argue that it is a contradiction in terms to say that the size of an infinite set has been increased. But this isn’t true because the word “infinite” does not mean “the largest number”. Indeed, if that’s what the word meant, then (\infty + 1 = \infty) would be just as wrong as (\infty + 1 > \infty). But that’s not what the word means. And that’s precisely what the word must mean in order for there to be a contradiction. To increase some number is to create a larger number, and if you’re increasing the largest number, then you’re creating a number greater than the largest number – which is a contradiction because by definition the largest number is a number greater than every other number i.e. there is no other number greater than it.

The word “infinite” is merely a number greater than every integer. And there isn’t one such number. There’s an infinity of them, the largest number being merely one of them.

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.