[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
out.
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
________________________________________________________________________
AOL now offers free email to everyone. Find out more about what's free
from AOL at AOL.com.
More information about the FOM
mailing list