Re: Cantor's Argument Skolemized?

On 15 Mar, 19:59, hurbur...@xxxxxxx wrote:
On Mar 15, 7:28 pm, Fjodor <frode.bjor...@xxxxxxxxxxxx> wrote:

I think it would be of tremendous philosophical and mathematical
interest if we could construct a set theory without the transfinite
stuff, i.e. a set theory that could do all the practical mathematics
but either could not prove |P(N)| > |N| or could prove |P(N)| = |N|..

Occam's razor comes to mind.

In NF, Cantor's Theorem fails. Indeed we have V=P(V). however another
version of Cantor's Theorem can be saved: the set of unit subsets of A
equinumerous to A: such sets are called cantorian. Furthermore one
can't prove Cantor's theorem in NF. Indeed the condition {x: notxef
(x)} is not stratified.

This is right. But Cantor's argument to the effect that the power set
of N has higher cardinality than N goes through in NF, though in a
somewhat revised way. So NF is e.g. not consistent with the addition
of an axim stating that the universe is sountable. (I lways wondered
why Quine didn't say that there IS a function from N onto P(N) which
is not definable by a stratified condition.)
.

Relevant Pages

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