[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 

Harvey Friedman

More information about the FOM mailing list