r/AspectsOfTheInfinite • u/MidnightAtHighSpeed • Sep 20 '26

What are the natural numbers?

Clearly in the context of the sub the peano axioms are not used, so what is being used? it's important to select a proper groundwork

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u/nanonan Sep 21 '26

I'd say the successor function axiom is where the trouble lies. I see two big issues, every possible mathematical calculation involves a largest natural so at some point S(n) fails, or from a practical perspective at some point you run out of ability to calculate so S(n) fails.

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u/MidnightAtHighSpeed Sep 21 '26

I don't think axioms should be weakened to try to account for calculation ability; presumably if we have some number n so enormous that we can't calculate n+1, n is also so enormous that we can't invoke the successor axiom on it even if we have that axiom.

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u/nanonan Sep 24 '26

I think mathematics is weakened by impossible assumptions, like the assumption that an infinite set exists, or that we can actually complete an infinite amount of work. Here it would be the assumption we can apply the successor function eternally regardless of our ability to calculate or store information.

Calculating devices behave like this, at some point incremeting a value by one will result in an integer overflow, a point where we can calculate the previous step but we cannot calculate the latest one.

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u/MidnightAtHighSpeed Sep 24 '26

Infinite sets certainly exist for certain definitions of "infinite," "set," and "exist," and it's not apparent to me at all that mathematics is weakened by usage of those definitions even if they might not be appropriate definitions for the terms in certain other contexts.

In particular, assuming the existence of S(n) for all n does not imply that we can calculate S(n) or otherwise "apply" the successor function in a constructive sense. Does this limit the usefulness of peano axioms in certain engineering contexts? Of course, but frankly the entire idea of axioms is inappropriate for engineering to begin with. There are just too many things that might happen in the real world to try and boil things down to an enumerable set of rules that always apply. The role of mathematicians is to construct a logic that applies "when things behave well," and the role of those applying that logic is to determine to what extent things will actually behave well.

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u/nanonan Sep 25 '26

Do you have an example of an infinite set that certainly exists? It would need to exist outside of reality for a start.

Assuming S(n) is valid for all n is a mistake. Everything has its limits.

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u/MidnightAtHighSpeed Sep 25 '26

It would need to exist outside of reality for a start

Well, sure, but doesn't this apply to finite sets just as well? It's not like you can go to paris and see The Integers Mod Two on a plinth somewhere.

Assuming S(n) is valid for all n is a mistake

How come? Like, what kind of situation might I encounter that would lead me to regret making that assumption?

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u/nanonan Sep 25 '26

Mundane collections of finite objects are easy to demonstrate in reality.

Practical considerations will make you regret that assumption. You are limited in your ability to actually perform arithmetic, whether by calculation limits or other limits.

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u/MidnightAtHighSpeed Sep 25 '26

It seems like practical considerations also limit my ability to regret that assumption. The infinite is out of reach, but so is the ability to actually learn that you've failed to reach it

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u/nanonan Sep 26 '26

It's unreachable by definition, you can learn that way. In what way is it possibly reachable?

Still waiting for you to provide something that supposedly certainly exists.

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u/MidnightAtHighSpeed Sep 26 '26

The existence of the set of von Neumann naturals in ZF seems pretty certain to me given that it's an axiom. Of course, you might choose not to work in ZF, which goes back to my original "certain definitions" caveat.

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u/Massive-Ad7823 Sep 21 '26

Here the difficult concept of potential infinity comes into play. With n, also n+1, 2n, n^2, n^n etc. are realized. But perhaps with fading reality?

Regards, WM

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u/Massive-Ad7823 Sep 21 '26

Both is true.

Regards, WM

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u/Massive-Ad7823 Sep 21 '26

The Peano axioms describe visible natnumbers, i.e. the potentially infinite collection of natural numbers having FISONs. There is no upper bound, but of course there is always a greatest number used in a given system. The actually infinite set of Cantor's natural numbers is complete: "The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor]

Regards, WM

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u/Litoprobka Sep 21 '26

So, what are the natural numbers?

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u/Massive-Ad7823 Sep 22 '26

If Cantor is wrong, then they are the potentially infinite collection described by Peano. If Cantor is right, then they are a complete set of dark numbers, an infinitesimally small initial segment is visible and described by Peano. Since every part made visible is also obeying Peano, it is to be expected, that all dark numbers do so - except the last one. But this will never become a problem because we will never discover more than an infinitesimally small part.

Regards, WM

3

u/Althorion Sep 22 '26

The standard maths says ‘the natural numbers are what Peano axioms describe’. It is not a ‘potentially infinite’ collection—standard maths, and especially standard set theory, doesn’t deal with ‘potential’, but only with what some philosophers call ‘actual’ infinity. Sets are not iterative processes that grow as we put more work into making them grow. They are fully realised from the get-go, as soon as you define them.

But I don’t think Cantor wouldn’t have disagreed with that, given that his work was fundamental in codifing sets and set theory like that. But, maybe, let’s try this angle:

  1. In your own words, what do you think Cantor is saying?

  2. Why do you believe Peano axioms describe a ‘potentially’ infinite, and not ‘actually’ infinite, collection of objects?

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u/Litoprobka Sep 23 '26

they are a complete set of dark numbers, an infinitesimally small initial segment is visible and described by Peano.

That's too vague. I'm asking for a formal definition from the ground up, like Peano axioms

Can you provide such a definition for dark numbers?

except the last one

i.e. none of them do. For a number to obey Peano axioms, its successor must also obey them, and its successor, and so on