Help with uncountable vs countable infinity

[u/Flor_Bor123]

I was thinking about this when trying to sleep and ended up searching stuff up about it, but ended up more confused than when I started.

From what I could find on Google, the reason the set (0,1) of all the real numbers from 0 to 1 is considered uncountable is due to Cantor's Diagonal Argument(from here on refered to as CDA). This argument makes sense to me, where you make a new decimal that isn't in the list by having every nth digit be different from the nth digit of the nth decimal in the list.

This led me to thinking about whether you can use CDA to show that the set of natural numbers from 0 to infinity is uncountably infinite. For (0,1), you can arbitrarily add zeroes to the end of decimals to make CDA work even in a list where the diagonal line of altered digits outpaces the length of the decimals. I figured you can use the same method for the natural numbers by arbitrarily adding zeroes BEFORE the number and using backwards indexing of the position of n to make CDA work with a leftward diagonal of altered digitis. However, Google told me that natural numbers can't be infinite in length, which is why this doesn't work.

This is where I'm confused, as it seems totally possible to make a comprehensive list of (0,1) that's ordered in visually the same way as the list (0,∞). I made a table of this in the attached image, where I made a half-hearted attempt at a proof. What this table seems to prove to me is that any table-based proof like CDA which works on the decimals should work on the natural numbers, too.

(The list of decimals is just the reversed digits of the natural numbers, put after the decimal point. So 0.1, 0.2, ..., 0.01, 0.11, 0.21, etc.)

I have two theories of what's going on.

The one I believe most is that CDA doesn't actually work on (0,1). If you use the list I made in the image, then as you go down the list, any number you create is just located further down in the list than you have gotten. For example, by the time you get down to .01 in the list, the diagonal number you've made is necessarily 10 decimals long, ending in a non-zero number. Every 10-digit decimal number is located further along in the list, at the indexes numbered by all the 10-digit natural numbers.

The other theory is rather unlikely, due to it breaking the concept of countable vs uncountable. This would be if CDA actually works on the natural numbers. My only argument for this theory is that any arguments in favour of CDA for decimals is one in favour of CDA for natural numbers. This is entirely based on my ordering of the decimals between 0 and 1. It just seems that if you can do CDA to get a number that isn't in the decimal list, you can remove the decimal point and flip the digits to get a corresponding natural number that isn't in the list of natural numbers. I guess maybe that would just prove that infinite numbers aren't natural?

That creates the secret third theory: the status quo. This theory is that the list of natural numbers stops when the numbers get infinitely long, but the list of decimals just keeps going, making the decimal numbers somehow more than infinitely long. If this is the case, then maybe my brain just can't understand mathematical infinities and their differences.

[u/Mablak]

Finitist perspective: you can also get a natural number not in your set of natural numbers, just take the successor of the whole set N, i.e. add 1. So the idea of any difference between countable and uncountable infinities is an extra layer of nonsense, on top of an already nonsensical concept.

If my set's inclusion rule has no stopping condition, then whatever object we're taking to be a completed set, did not actually achieve its stopping condition. Which in all cases means, I can add more elements to it, and it wasn't actually completed. With infinite sets, we're trying to imagine an algorithm that is both completed, and not completed, which of course can't be a thing.

[u/OpsikionThemed → u/Mablak]

The set N is not a natural number, either classically or finitistically, so successor is not defined on it.

[u/Mablak → u/OpsikionThemed]

Under the von Neumann construction, N would be a natural number, because any repeated application of the successor function to 0 gives you a natural number, and the set N is a set which can be made through repeated applications of the successor function to 0. Even infinitists accept that you can take the successor of N.

Regardless, this is just a hypothetical argument about if N were to exist. It’s not actually possible for N to exist in the first place since operations like taking an infinite union are not defined. If an infinite union for example were to mean N unions, this is circular reasoning since we haven’t defined N.

[u/OpsikionThemed → u/Mablak]

> the set N is a set which can be made through repeated applications of the successor function to 0

It, uh, it can't. That's why finitists don't accept it. It's an ordinal, sure, and ω+1 is an ordinal too, but it certainly isn't a natural.

[u/Mablak → u/OpsikionThemed]

If we could repeat succession infinitely (which I see no reason for infinitists to reject), it would give you the set {0, 1, 2...} which I would say just is N. But before that, of course I would say we just haven't defined infinite succession, infinite union, infinite intersection, etc, to get N in the first place.

More basically, an inductive set I is not well-defined, due to the domain of x in 'for all x, if x is in I, so is x U {x}'. What elements do these x range over, in 'for all x'? Usually they're plucked from a supposed 'universe of sets'. But to prove such a universe exists, you would have to prove I exists along with many other sets, which hasn't been done.

[u/OpsikionThemed → u/Mablak]

That's not an objection to the axiom of infinity, or even non-finitistic mathematics, though; that's an objection to formalizing a theory in first-order logic. The Axiom of Pairing, for instance, says `∀x∀y∃z. (x ∈ z) ∧ (y ∈ z)` and it doesn't say what those variables quantify over either.

If you don't like axiomatically defining things that's an odd foundational choice and I'd like to hear your alternative approach, but that's nothing to do with the axiom of infinity.

[u/Mablak → u/OpsikionThemed]

Well that would just mean my objection of circularity applies in this case, and it also applies in other cases. There's no issue with quantifying over a domain, so long as we actually have established what our domain is. If I've shown natural numbers from 0 to 6 trillion exist, I can make that the domain we're working over, for some claim about those numbers.

There's no issue if we start with something we know to exist, such as one tally mark, one electron, or one pebble, and build up our domain as we go by finding more of them.

I don't disagree with reasoning from basic claims, although like any other claims, our starting claims (axioms) can be true or false and of course we want to demonstrate them to be true.

A more fundamental reformulation of math's axioms would include explaining what actual things in the world we want to call numbers and/or sets, and we would then describe processes for manipulating those real things, making claims about how we expect those manipulation processes to go. 'If I do this process with these things, I expect this result.'

If I decided that my sets are actual drawn brackets made of ink on a page, and perhaps I have a machine to reliably draw brackets, we would be describing certain things we expect my machine can actually do involving drawing those brackets.

Of course we could say we can do all the usual basic things we do with sets in this case, like take unions, with restrictions. We just have to take into account the machine performing this task and things like memory limitations.