FOM: Henkin semantics

[John Mayberry]

Two minor points
1.) Strictly speaking 2nd order logic with Henkin semantics is not the 
same as the many-sorted 1st order theory that Steve Simpson has in 
mind. In Henkin semantics the second order variables are constrained to 
range over suitable subsets of their natural domains in full 2nd order 
semantics. The theories are, however, equivalent in the sense of being 
mutually interpretable.
2.)There is nothing to prevent our taking the very same axioms and 
rules that are complete in the Henkin semantics as axioms and rules 
in full 2nd order logic with standard semantics. It's just that they 
won't be complete for that semantics. I think Joe Shipman has raised a 
very interesting question in asking for a natural extension of those 
axioms that would allow us to prove any formula which can be proved 
valid in ZFC. 

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John Mayberry
J.P.Mayberry at bristol.ac.uk
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