Re: Cardinality vs Subsets



On Feb 22, 1:45 pm, Transfer Principle <lwal...@xxxxxxxxx> wrote:
On Feb 22, 4:05 am, Math1723 <anonym1...@xxxxxxx> wrote:

On Feb 22, 1:52 am, Transfer Principle <lwal...@xxxxxxxxx> wrote:
I fully accept that Cantor's theorem is a theorem of ZFC. But not
everyone wants to be, or should be, bound by what ZFC proves. So
those who don't can choose a set theory, such as NFU, in which
Cantor isn't a theorem (assuming consistency), or search for a
new theory in which the negation of Cantor is provable.
Speaking of which, have you determined which of the nine specific ZFC
axioms [snip link] that you have a problem with?

For the vast majority of posters who criticize ZFC, the axiom
that they have a problem with is the Axiom of Infinity. It's
Infinity that leads directly to the results that such posters
are trying to avoid.

(Perhaps occasionally, a poster who accepts infinite sets but
not uncountable sets might be satisfied by dropping Powerset
rather than Infinity, but this is rare.)

Edward Belaga would allow countable, but not uncountable, ordinals:
http://www-irma.u-strasbg.fr/~belaga/
http://www-irma.u-strasbg.fr/~belaga/4*MathInfinity.pdf
http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.3207v3.pdf
http://www-irma.u-strasbg.fr/~belaga/HalfwayUpToTheMathematicalInfinityIII_SPUC09slides090324.pptx

"Are we really living in the world where the Power-Set Axiom is a
universal mathematical truth?" II. Proc. of the 17th Congress of the
Canadian Society of History and Philosophy of Mathematics, eds. H.
Grant et al., Kingston, May (1991)
.