Re: The set of necessary FISONs

Am Thu, 13 Feb 2025 12:59:56 +0100 schrieb WM:

On 12.02.2025 20:39, Jim Burns wrote:

On 2/12/2025 4:19 AM, WM wrote:

 

One is Zermelo's proof by induction that there is an infiite set Z.

 
https://en.wikipedia.org/wiki/Zermelo_set_theory
The proof of Z you refer to is the bare assertion that inductiveᶻ Z
exists, AXIOM VII "Infinity".
  Z exists such that Z∋{} ∧ ∀a∈Z∋{a}

From the assumption UF = ℕ we have obtained U{ } = { } = ℕ.

 

We have obtained that,
for each FISON F′ ∈ {F} and 𝒜 ∈ 𝒫{F}
if ⋃𝒜 = ℕ then _not.necessarily_ F′ ∈ 𝒜

 
All of us who know induction know that by induction we have obtained
that all FISONs can be removed without changing the result.

The result has certainly changed from a nonempty set, whatever
you think the union of inf. many FISONs is.

From the assumption UF = ℕ we have obtained U{ } = { } = ℕ.

 
 From the assumption ⋃{F} = ℕ
we have obtained ⋂𝒫ᵁᐧᙿᴺ{F} = {}
and also 𝒫ᵁᐧᙿᴺ{F} ≠ {}

 
Learn induction. Mathematical induction is a method for proving that a
statement is true for every natural number {P(0),P(1),P(2),P(3),\dots }
all hold. No FISON remains.

With P = „You can leave out A(n) and all preceding FISONs” you get an
infinite number of such sentences, where each leaves infinitely many
FISONs in the union.


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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.