[FOM] Why would one prefer ZFC to ZC?
Jeremy Bem
jeremy1 at gmail.com
Fri Jan 29 19:50:14 EST 2010
Monroe Eskew wrote:
>> (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.)
How about countable union?
Meanwhile, V simply cannot be viewed as a model in the ordinary sense.
> 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.
I am speaking about V itself. I am showing that the left hand side of
the analogy ZC : V_{omega+omega} :: ZFC : V is fundamentally better
behaved than the right.
>> (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.
Correct.
> What might it be?
For the moment, the argument above!
-Jeremy
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