[FOM] Paul Cohen was wrong

[Daniel Mehkeri]

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.

Daniel Mehkeri