[FOM]: Independece without Forcing
Harvey Friedman
friedman at math.ohio-state.edu
Sat Jul 12 11:21:34 EDT 2003
Reply to Eisworth 7/11/03 /5:24PM.
I forgot to respond to the material from Eisworth 7/11/03 /5:24PM
after my signature in my posting 2:15AM 7/12/03.
>
>I asked earlier about obtaining independence results without forcing
>and asked about getting "mathematical statements" independent of
>ZFC + V=L.
>
>When I originally posed the question, I had in mind speculating about
>the existence of "mathematical" statements P such that ZFC + V=L + P
>is consistent but not in a "standard" (read transitive?) version of L.
>
I don't quite know what you mean by this.
When you write ZFC + V = L, there is only one notion of L that makes
sense. For other notions of L, V = L is easily refutable in ZFC.
Perhaps you are asking for a "mathematical statement" P such that for
small initial segments A of L, the statement
P holds in A
is itself independent of ZFC + V = L?
Obviously, this will be the case if P is an arithmetic independence
result. It is also the case for certain statements in Boolean
Relation Theory. It is even the case for other Borel independence
results.
Harvey Friedman
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