Re: The set of necessary FISONs

On 03.03.2025 18:01, FromTheRafters wrote:

WM formulated the question :

On 03.03.2025 12:58, joes wrote:

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

>

Ah no, you're shifting the quantifier there.

That is allowed if Zermelo can do it.

He can't.

>
But it has been accepted that with all defined elements the set is defined.

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).

>
Zermelo has no other deduction. The set is constructed by its elements.

>

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.

>
Of course not. It is the set of all elements.

 Discrete elements.

Yes, of course: { }, {{ }}, {{{ }}}, ... This is the sequence of numbers. According to Zermelo its existence guarantees
the set of numbers Z₀ = {{ }, {{ }}, {{{ }}}, ...}.
Regards, WM