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> So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property.

Mathematicians have proposed several different constructions of the reals. If you were familiar with, say, constructing the rationals from the integers or constructing the reals from the rationals, you would know that this is not a problem. One of the more straightforward constructions of rationals define a rational number as an equivalence class of tuples of integers, corresponding to the numerator and the denominator. We often use one of these tuples to represent a rational number, e.g. 2/3 or 4/6. And since they are in the same equivalence class, either alone is sufficient to represent the same rational number. This is despite the fact that the tuple 2/3 and the tuple 4/6 are different mathematical objects.

As for the definition of 0.999..., it's not what you think it is. There is a specific definition for infinitely repeating digits that is different from representations without repeating digits.

That is, the positive real number that 0.x... represents for all digits x is defined as L, where L is the supremum of the set containing 0, 0.x, 0.xx, 0.xxx, 0.xxxx, ...

A supremum L in E of a set S subsetof E is defined as follows: L is the smallest number in E such that L is greater than or equal to all numbers in S. I shall not prove here that L is unique, but it is. With respect to the reals and the set containing 0, 0.x, 0.xx, 0.xxx, the supremum of that set in the reals is 1.

My knowledge of the construction of the reals comes from Chapter 1 of Water Rudin's Principles of Mathematical Analysis, ISBN 0-07-085613-3, which you may be interested to read, but beware that it's considered a difficult read for beginners.

I completely agree with you, and understand all that.

> L is the smallest number in E such that L is greater than or equal to all numbers in S.

And note that this doesn't imply that L is in S!

So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based on condensing the names of the elements in S, like "0.x...", or 0.x with a bar over the x, and making that notation denote something not in S, namely 1.

This doesn't appear incompatible with the limit-based construction.

Are there instances in which this supremum approach toward continued decimals disagrees with the limit approach, in establishing the value?

Yes, that is exactly what that notation means, just as 0.1111... is not any of the numbers 0.1, 0.11,...

Supremum of an increasing sequence of reals is equal to the limit of the sequence in all cases.

I think I was too hasty in replying to your post and misread and misunderstood the point you were trying to make. No, I don't think there's a difference.

They're coming out of the woodwork...

Edit: sorry, didn't mean to ad hominem, but I'm going to leave my original comment there all the same. To make my post a bit more constructive, OP, what exactly is your definition of "constructible"? Because it doesn't relate to any mathematical concept I'm familiar with. Other than maybe "finitary".

https://en.wikipedia.org/wiki/Constructivism_(mathematics)

Do you have remarks not regarding the semantics of "constructable"?

But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.

First we have to agree on what 0.999 is, then we can call it constructible or not.

Numbers formed by repeated 9's appended to 0.9 are certainly Turing computable. Which has the meaning that we have a terminating algorithm which, given a natural number N, will compute the N-th digit of the infinite sequence 0.999...

This is the same way that pi is computable. Given an N, we can compute the N-th digit of pi in a finite number of steps.

We don't say that 3.141... = pi! Unless, by convention, we agree that this "..." syntax has the semantics of (for instance) denoting the limiting numeric value of the non-terminating algorithm for producing the digits of pi ad infinitum. That is to say, "insert here a process for calculating the remaining digits of pi, and take this whole expression then to be the limiting value to which that converges".

But that's exactly what the ellipsis means. What other interpretation is there? I might as well argue that 1 = 1 only if we agree on the convention that = means "equals".

https://en.wikipedia.org/wiki/Finitism

Correction: Induction is how we can understand that a property holds for all finite values.

It seems that you object to the commonly accepted meaning of "...".

So you're saying that "for all positive integers n : property(n)" means that property(k) might not hold for some k that is not a "finite value"?

The only problem is that this not-finite value does not occur among the integers; thus the inductive procession doesn't actually have that value k as its target. It does not lie in its path, so to speak.

> It seems that you object to the commonly accepted meaning of "...".

I formulated it as a limit and stated that it denotes 1 in the case of 0.999...

How do I object?

You might be interested in reading about transfinite induction, which is an interesting and legitimate mathematical proof technique. However your 'induction' does not succeed in doing this. Transfinite induction requires two induction steps, you've only provided one of them. Not that you would actually use transfinite induction for anything resembling this, that's what calculus and analysis is for.

For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit! After all, each one is finite so by induction the limit is finite too. Except of course no one would expect this induction to apply to the limit as well as to the individual steps.

I'm not surprised that something called "induction" doesn't succeed in achieving what "transfinite induction" can do, otherwise the latter couldn't exist as a separate technique with its own name. :)

> For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit!

How so? Induction is in fact the basis for the common proof that there is no highest integer: for any integer k, we can add 1 to find a larger integer. Every integer k has the property P(k) := "k is not the highest integer".

The problem with this argument is it assumes you can get to infinity 9s by adding them one-by-one.

It's like adding sides to a regular polygon to get to a circle. No matter how many sides you add, you still have a polygon. You can only say that a circle is a polygon with infinity sides, but you'll never get there adding them one by one.

The argument is that you cannot get to infinity by adding one by one.

(If the argument assumed otherwise, it would only be temporarily, for the sake of setting up a reductio ad absurdum.)

You can't use induction to get to an infinite amount of nines, so induction doesn't work in this case because you never get there.

The number of nines is not in the set of natural numbers, yet you're trying to use induction by covering every natural number.

You cannot simultaneously believe that "you never get there" (i.e. an infinite number of nines is not constructible) and believe that the syntax 0.999... denotes an actual infinite string of 9's which has a straightforward value (that happens to be precisely 1).

That simply isn't the basis for how 0.999... is regarded as 1.

Rather, 1 is the supremum ("least upper bound", LUB) of the countably infinite set { 0.9, 0.99, 0.999, ... } as a subset of the reals. We define that 0.9... denotes that supremum: i.e. that the ellipses suffixed to 0.9 denote the expansion of 0.9 into the set { 0.9, 0.99, 0.999, ... } followed by determining the LUB of subset among the reals.

No "infinite string of 9's whose length isn't a natural number" nonsense is involved.

Cassius Clay differs from Mohamed Ali on the name property therefore they are not identical?

Good grief ...

If we have an entity X and we use the function name(X) to inquire about its name property and obtain "Mohamed Ali". And then if elsewhere we use name(X) and it evaluates to "Cassius Clay", then name() is not a mathematical function.

You have to find a way to reason about an object and its properties so that they are functions. If it can have multiple names, you have to model that appropriately.

Or a name is not an attribute and .999... and 1 name the same thing.

I would humbly submit that if your code is showing Cassius Clay is not the same person as Mohammed Ali, it is your code that would be in error rather than the universe.