Fair enough. It needs expansion. But I don't have the inclination to write an essay now. — Banno

No problemo.

A certain aspect of reality (quantity) is discovered all over the world
— TheMadFool

I think that's not what happens. Rather, a certain way of talking about the world is found in many places. There are, after all, languages without much by way of number. Would you say that the folk who speak them have failed to notice an aspect of reality, or would you say that they have no use for a particular process, a certain way of speaking? — Banno

Well, it wouldn't be wrong to say that, necessity is the mother of invention, if a people had no use for math, they wouldn't have ever adapted language to make it math-apt.

Your point then is...I draw a blank at this point.

Good, then you agree that the limit of the sequence is rigorously defined. — TonesInDeepFreeze

I agree to take your word for it, because I imagine that if I were more mathematically literate I would agree with you. In any case that was never something I was arguing against.

what we understand intuitively can be understood in ways inconsistent with that in mathematics — Janus

Informally inconsistent, yes.

Anyway, when we move beyond child-level thinking that there must be involved a "reaching" and instead we study rigorous mathematics, then we understand that .999... = 1.

That sounds fair enough...

Actually, this is implied after an infinite amount of operations on a series. Like a Riemann sum.

after an infinite amount of operations — Shawn

There is an infinite sequence and there is the limit of that infinite sequence. There is no supertask - no performing an infinite number of operations - used.

But, that's how it's usually phrased by others when thinking that you have to add an infinite amount of terms in a series, yes?

I've not seen it phrased that way, especially in a rigorous exposition in classical mathematics. Not even in freshman calculus. I don't know what writings in classical mathematics you have in mind.

There is an infinite sequence, with a finite description. And then there is a finite proof that the limit of that sequence is 1. No supertask.

It's just a Riemann sum, Tones, lol!

What exactly do you disagree with in what I just posted?

With that you were disregarding with instantiating infinity in a series sum that apparently converges to 1 for 0.9...(9)

What do you mean by 'instantiating infinity'. We instantiate the set of natural numbers in set theory, of course. And we instantiate the set of rational numbers in set theory, of course. The sequence is indexed by the set of natural numbers. And the range of the sequence is a subset of the set of rational numbers.

While I'm at it, you also don't have to "reach" 1 for 0.9...(9) for non-standard analysis and can instantiate a infinitesimal.

Of course non-standard has infinitesimals. (And non-standard analysis takes place itself in a larger environment of classical mathematics.) Though, I don't remember which, if any, infinitesimals are instantiated, since non-standard analysis, as I understand, non-constructivity is endemic to the proof of that infinitesimals exist. he proof of the existence . In any case, there is no notion of undefined notion of "reaching" needed for the classical method of a limit of a sequence such as discussed in this case.

What I mean is the typical notation for a Riemann sum of as n goes to infinity. So the implicit assumption is that you do reach a number, but as per calculus that amount of steps is reached once you instantiate an infinitesimal...

Specifically, we don't do that even in simple freshman calculus. I have never read an author say there is an "implicit reaching" (whatever that would actually mean, as you have not given a mathematical definition of 'series reaches').

And basic calculus does not use infinitesimals. It is in the context of ordinary classical mathematics that

SUM[n = 1 to inf] 9/10^n

is defined as

the limit of the sequence {<1 9/10> <2 99/100> <3 999/1000>...}.

And then with a finite proof we show that

the limit of the sequence {<1 9/10> <2 99/100> <3 999/1000>...} = 1.

No infinitesimals adduced and no undefined "reaching".

[Edit: I fixed the notation of the sequence.]

Well think about whether 1/x converges for the set of all irrationals, as x goes to infinity. It doesn't. So, you have to consider whether some value is reached or is the limit, no?

In addition I see it of more use to talk about how fast or slow a convergence occurs.

By the way, when I say that non-standard analysis is subsumed within classical mathematics, I mean non-standard analysis developed in ZFC. I'm not referring to IST axiomatically from scratch.

Of course, if the sequence does not converge then it's another ballgame. But we easily prove that the sequence we've been talking about does converge.

You seem to be offering up diversions - infinitesimals, nonstandard analysis, sequences that don't converge - when the point is actually as simple as I've shown.

I seem to think that the point being is to demonstrate the issue pictorially with a converging series. So, something is "reached", no?

People tend to think that 0.999 is a never ending sequence when in fact it's a converging series to 1.

You keep replying past the simple straightforward points I made. Especially again as you go past my point that you haven't mathematically defined "reached". The proof that .999... = 1 does not require using an undefined notion of "reached", so it's not my problem that you keep wanting to bring it back into discussion.

What? .999... is not a sequence. It's a number. That number is 1.

The sequence is {<1 9/10> <2 99/100> <3 999/1000>...}

and its limit is 1.

Once more:

SUM[n = 1 to inf] 9/10^n = lim of {<1 9/10> <2 99/100> <3 999/1000>...} = 1.

EDIT: Corrected formulation of the sequence.

What? .999... is not a sequence. It's a number. That number is 1. — TonesInDeepFreeze

How about:
[(10^n)-1]/(10^n) As n approaches infinity from 1?

You can see the infinitesimal hiding there.

Are you trolling me? — TonesInDeepFreeze

See https://en.wikipedia.org/wiki/Talk:0.999.../Arguments and the eleven archives attached.

It's not an argument worth entering in to; a troll's haven.

Are you trolling me? — TonesInDeepFreeze

Not at all. If people struggle with understanding that 0.999... is 1 then, 0.999... exists inside the sequence of [(10^n)-1]/(10^n) when starting from 1 to n approaching infinity.

If every thread on the principles of mathematics is allowed to degenerate into a thread about 0.999...<>1 it would become impossible to do any philosophy of maths.

So I will ask for Moderator support to discontinue this discussion on this thread.

By all means, folks, set up a thread of your own, or go back to the innumerable previous threads on the topic. Just go away.

What's happening?