Re: The set of necessary FISONs

Am Mon, 03 Mar 2025 09:48:16 +0100 schrieb WM:

On 02.03.2025 20:31, joes wrote:

Am Sun, 02 Mar 2025 18:51:05 +0100 schrieb WM:

On 02.03.2025 13:22, Richard Damon wrote:

On 3/1/25 1:09 PM, WM wrote:

On 01.03.2025 17:25, Richard Damon wrote:

On 3/1/25 9:31 AM, WM wrote:

On 28.02.2025 20:45, Richard Damon wrote:

On 2/28/25 11:44 AM, WM wrote:

>

No, Zermelo uses induction. I did not say that he uses the term
induction.

No, what you describe is NOT "induction".

Correct your qualifiers. Or look it up.
F(1) ∈ F and F(n) ∈ F ==> F(n+1) ∈ F describes the infinite
inductive set F of FISONs.

You have a source that uses the term "Induction" for the recursive
iteration that builds the set?

∀P(P(0) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)))
Wikipedia

Induction is the nucleus: P(0) /\ ∀k(P(k) ==> P(k+1)

>
And what SET did that build? P was a statement about a relationship.

P gives a relationship, for instance "is element of F". The universal
quantifier ∀ proves that all FISONs belong to the set that can be
removed.

Ah no, you're shifting the quantifier there.

That is allowed if Zermelo csn do it.

He can't. The *set of* all FISONs can't be removed.

There are many sets of FISONs that can be removed, one set per natural,
containing all FISONS of numbers less than that natural (also more sets
of non-contiguous FISONs). But those are all finite.

Of course all FISONs are finite. All FISONs of Zermelo's set are finite.
By induction Zermelo produces the set. Quantifier shift, obviously
allowed by Zermelo. There is no other way to construct an infinite set.

A quantifier shift is never a valid deduction (even though the resultant
sentence may be true otherwise).

The set of sets of FISONs that can be removed together does not contain
the set of all FISONs (although it does contain the infinite sets of
the odd or even FISONs).

In exactly the same way as Z₀ is constructed by its elements, the set of
removable FISONs is constructed by its elements.

And in the same way Z_0 doesn't contain the set of all elements.

--
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.