[FOM]: Independece without Forcing

[Harvey Friedman]

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