# [FOM] Paul Cohen was wrong

Monroe Eskew meskew at math.uci.edu
Mon Sep 12 12:11:32 EDT 2011

On Sun, Sep 11, 2011 at 8:17 PM, Daniel Mehkeri <dmehkeri at gmail.com> wrote:
>
> "Successor cardinals" seems to be an instance of positing an ordinal beyond
> the ones already considered. As such it seems to be in the spirit of the
> axiom of infinity, as well as of weakly inaccessible cardinals. Infinity and
> weak inaccessibility are definitely not special cases of powerset.
> Conversely, perhaps power set is not a consequence of various large cardinal
> axioms?

Without power sets, what mathematical reason would we have for
asserting that different sizes of infinity exist at all?  Cantor
established their existence using the assumption that the set of all
real numbers exists.  Although his argument was controversial in his
day, basing it on the assumption that R exists makes it much more a
proof from within ordinary mathematics than inventing a new system of
transfinite numbers and positing the existence of an ordinal beyond
those already considered.  If he had only done that, the world might
have been more inclined to side with Kronecker and Wittgenstein.

Of course power set is not a consequence of small large cardinals like
weakly Mahlo since, is \kappa is such a cardinal, you can force R to
be bigger than \kappa^+ while preserving Mahloness and then look at
H_{\kappa^+}.

However I maintain that powersets, successor cardinals, infinity, weak
inaccessibility, etc. are all motivated by the same principle.  Namely
that you can collect "all that you've got so far" into a set for
further manipulation.  Philosophically I see no difference between
"completed infinity," "completed R" and "completed \omega_1."

Monroe