[FOM] Why would one prefer ZFC to ZC?
Monroe Eskew
meskew at math.uci.edu
Thu Jan 28 18:16:35 EST 2010
> (2) It should be possible to view the "universe" as a model in the
> ordinary sense of first-order logic;
The question is: Possible by what means? It is not possible to prove
V{\omega+\omega} exists in ZC. You may simply assume it without
accepting full replacement. But why? To me it seems whatever
intuition makes you think that V{\omega+\omega} exists would also make
you think replacement is true. (Not in terms of logical necessity but
concepts.)
Likewise there are a variety of relatively mild assumptions, with some
intuitive backing, that give a natural transitive model of ZFC. For
example inaccessibles can be seen as justified by some closure
principle. These assumptions go beyond ZFC but likewise \omega*2 goes
beyond ZC.
> (3) It should be possible to prove beautiful, generally accepted
> mathematical results using the axiom system -- the more the better.
Then this should push you towards replacement, unless there is
something holding you back. What might it be?
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