[FOM] The necessity of forcing

joeshipman@aol.com joeshipman at aol.com
Thu May 10 22:43:15 EDT 2007

-----Original Message-----
From: eilya497 at 013.net
The only ways of proving A is not a theorem of ZFC that I am aware of
are 1) proving that ~A can be forced and 2) proving that ~A is theorem
of ZF + V=L. The latter cannot work for both A and ~A, so I am inclined
to think your claim is true.

Of course there is 3) showing A implies Con(ZF) but I had ruled that 

If the only method that does not involve consistency strength and does 
not use forcing was "follows from V=L", that would be a good argument 
for V=L.

But I can define a slightly stronger axiom than V=L which adds no 
consistency strength and does not require forcing: "V=M", which is 
equivalent to "V=L and ZFC has no standard models". This axiom has as a 
consequence that there are no inaccessibles. It is obviously 
equiconsistent with ZF+V=L, because if there are standard models there 
is a minimal one, in which "V=M" is true.

Is there any way to strengthen this further? Can we make an axiom 
scheme out of my claim, along the lines of "anything which can be shown 
consistent with ZFC without forcing is true"? The problem is saying 
precisely what it means to "not use forcing".

-- JS
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