Re: Continuum hypothesis

aatu.koskensilta@xxxxxxxxx writes:

The result the original poster probably has in mind is that both ZFC
+CH and ZFC+~CH are conservative over ZFC for arithmetical statements,
i.e. statements in which the quantifiers range over the hereditarily
finite sets (these are equivalent to statements in the first-order
language of arithmetic, of course). The conservativity extends to
Pi^1_4 sentences (in the arithmetical hierarchy), if I recall
correctly.

A few additional comments. The conservativity results hold for GCH and
not merely CH. In addition, Woodin has shown that if there are
sufficiently large large large (sic) cardinals no statement of
second-order arithmetic can be disturbed by forcing. In a sense, then,
large large cardinals large enough yield a theory of V_omega+1 that is
"complete".

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.