On 7/30/2023 11:22 AM, WM wrote:
>
markus...@gmail.com schrieb am Sonntag,
> 30. Juli 2023 um 16:11:57 UTC+2:
>> I'm not unable to give
>> a formal definition of what a set is.
>> A set is any object that satisfies
>> the axioms of whenever set theory
>> you are dealing with.
>> A ZFC-set is an object that satisfies
>> the ZFC axioms.
>
> What is an object that satisfies the axioms?
We don't need to say what it is.
It is enough that
it is an object that satisfies the axioms.
The axioms are true claims about
what the axioms refer to.
True claims can be augmented, using only
visibly not-first-false claims.
Augmented in that way, only-not-first-falsely,
each descriptive and augmenting claim
is not-first-false.
Description plus augmentation are
a finite sequence of claims.
In a finite sequence of claims,
if each claim is not-first-false,
then each claim is not-false.
The descriptive claims are not-false,
of course.
The only-not-first-false augmenting claims
are also not-false,
and we know they are not-false
without knowing what they refer to,
but by their being only-not-first-false
in that sequence.
> What is an object that satisfies the axioms?
We don't need to say what it is.
That explains the counter-intuitive power
we finite beings have to know about
infinitely-many, almost all of which
we cannot even in principle interact with.
We know the augmenting claims are true
by examining them for not-first-falsity,
not by examining what they refer to.
It doesn't matter what they refer to.
It doesn't matter if we can't,
even in principle, examine what they refer to.
It us by _that claim-sequence_ that we know.
> The concept of infinity from Greeks to Gauss
> differs from actual infinity.
Your actual-infinityᵂᴹ requires the existence of
things which do not equal themselves.
You reject actual-infinityᵂᴹ or you should.
But that doesn't reject _our work_
Actual-infinityᵂᴹ is not _our work_