Re: The set of necessary FISONs

Am Thu, 06 Feb 2025 20:32:53 +0100 schrieb WM:

On 06.02.2025 19:54, Jim Burns wrote:

On 2/6/2025 11:55 AM, WM wrote:

On 06.02.2025 15:57, Jim Burns wrote:

 

The key is that  ∀ᴺ¹n: ∃ᴺ¹j′: n<j′

The key is that the set ℕ is created by induction.

The set ℕ₁ is described as having induction valid for it.

Then it is the collection ℕ_def of definable numbers.

 
Sets missing natural numbers and sets with extra, non.inducible,
un.natural numbers are not ℕ₁

Then it is the set ℕ of all natural numbers.
You contradict yourself.

No. N is exactly the set of all, and only the natural numbers.
It does not matter that you call it N_def. Your extension of
that set is unclear, since you have not provided any axioms.

Which is curious, when one considers that ℕ₁ appears nowhere in it.

The axiom of induction holds for all predicates P which satisfy
induction.

Tautologically.

If the set M is described as the smallest set satisfying F(1) ∈ M and
F(n) ∈ M ==> F(n+1) ∈ M then M contains all FISONs which can be
subtracted from U(Fn)) without changing the assumed result ℕ.

Wrong. M cannot be finite.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.