[FOM] Paul Cohen was wrong
dmehkeri at gmail.com
Sun Sep 11 23:17:03 EDT 2011
Monroe Eskew writes:
> There is no "powerful new principle" that separates powersets from
> cardinal successors. The same principle underlies both, and
> restricting our attention to fragments of ZFC, the same axiom
> underlies both. You can introduce new axioms weaker than full
> powerset to tease them apart.
You could just as well say dependent choice is just a special case of
choice. But they are not obviously in the same spirit. (Especially if
you are a constructivist.)
"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?
I am reminded of something I read once to the effect that ZFC was the
child of an unhappy marriage between power set and well-ordering. I
can't find this quote.
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