FOM: CH and 2nd-order validity
I Jane
jane at mat.ub.es
Sat Oct 14 07:46:40 EDT 2000
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
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