peano, sets, foundations
Nov. 5th, 2019 12:15 pmWe had a rather confusing discussion with ![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
gonchar lxe sassa_nf
I think I was wrong here and there.
So, the issue was, can we declare that a "subset" of natural numbers is either finite or countable?
First, the question has no sense. Peano arithmetic has no notion of "finite", "countable", "set of".
What we can prove though, is that, e.g., for any natural number
Now, about the sizes.
We may dive into a set theory, e.g. ZFC. In this ZFC we can model Peano numbers, and we can, using Infinity Axiom and Separation Axiom Schema, produce "a set of all natural numbers". This set is infinite in a certain sense (there's more than one definition of finite and infinite). In any case, due to Choice Axiom (and others) and some theorems, a subset of ℕ is no bigger than ℕ. If it's bounded, it's finite; if it's not bounded, it can be counted anyway (thankslxe for the hint).
gonchar lxe sassa_nf I think I was wrong here and there.
So, the issue was, can we declare that a "subset" of natural numbers is either finite or countable?
First, the question has no sense. Peano arithmetic has no notion of "finite", "countable", "set of".
What we can prove though, is that, e.g., for any natural number
n there is a prime natural number p where p = n + k for some natural number k. Which can be interpreted roughly as "prime numbers are not limited", or "there is no natural number that can enumerate all prime numbers".Now, about the sizes.
We may dive into a set theory, e.g. ZFC. In this ZFC we can model Peano numbers, and we can, using Infinity Axiom and Separation Axiom Schema, produce "a set of all natural numbers". This set is infinite in a certain sense (there's more than one definition of finite and infinite). In any case, due to Choice Axiom (and others) and some theorems, a subset of ℕ is no bigger than ℕ. If it's bounded, it's finite; if it's not bounded, it can be counted anyway (thanks
lxe for the hint).потерял где-то нить, выдергиваю
Dec. 27th, 2014 08:39 amТут возникла такая тема, что натуральные числа в множествах определяются по индукции. Не знаю, как в множествах вообще, а в ZFC есть аксиома бесконечности:
∃X (∅∈X ∧ ∀y (y∈X → S(y)∈X))
where S(y) ≡ y ∪ {y}
Вот.
∃X (∅∈X ∧ ∀y (y∈X → S(y)∈X))
where S(y) ≡ y ∪ {y}
Вот.
