Is 1 = 0.999... ? Really?

You replied to the post yourself ! Honestly! This is getting absurd!

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What exactly does that prove?

Just what I said it does. Infinite series don’t converge.

To answer your question.

This started out with you asking:

“You’d have to explain why you’re limiting yourself to integers.”

To which I said: “Try it without integers. It doesn’t work.”

I was showing you it doesn’t work. Of course, what you really meant was the non-integer (\infty).

I don’t get the distinction between these two cases. Sounds like the exact same case just worded differently. In the one case you’re saying x can equal (\infty), in the other that it can’t.

This logic:

(x = 0.\dot9)
(10x = 9.\dot9)
(10x = 9 + 0.\dot9)
(10x = 9 + x)
(9x = 9)
(x = 1)

A sum cannot stop after an infinite number of terms because if it could it would be finite so the concept of completed infinity is entirely fallacious
And so your first sentence and third sentence contradict each other because if (1) increases by (1) and does not end then logically the sum cannot stop

I0x = 9.999…
I0x = 9 + .999…
I0x = 9 + x
I0x = 9 + I
I0x = I0
x = I

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.