FOM: CH and 2nd-order validity

[I Jane]

Roger Jones asked (Fri, 13 Oct 2000):
Is it known whether CH is independent of 2nd-order validity?

This may answer your question:
You can write down a pure (i.e with no non-logical symbols) second-order
sentence S which is valid iff CH holds. S can be a formalization of:
If X admits a complete separable dense linear ordering, then every
uncountable subset of X is bijectable with X.

You can also write down a sentence T which is valid iff CH does not hold. T
can be:
If X admits a complete separable dense linear ordering, then some
uncountable subset of X is not bijectable with X.

(S and T work because any complete separable dense linear ordering is
isomorphic to the ordering of the real numbers.)

Of course, the equivalences

 S is universally valid iff CH
 T is universally valid iff not CH

are expressible and provable in ZF. So, there is at least one sense in
which your question is not a theological one.

Ignacio Jane'

Departament de Logica
Facultat de Filosofia
Universitat de Barcelona