# FOM: Questions on higher-order logic

JoeShipman@aol.com JoeShipman at aol.com
Thu Aug 31 20:38:45 EDT 2000

```Here are some precisely posed questions that might help make the current
discussion more clearly focused.  In the following, "standard semantics" is
assumed, and I am working in first-order ZFC, using the definition of
"second-order validity in standard semantics" given in the first chapter of
Manzano's book.

1)  For which ordinals alpha is  the the truth set for V(alpha) Turing
reducible to the set of second-order validities?

2) Is the set of second-order validities reducible to the truth set of
V(alpha) for any  alpha?  This seems unlikely at first, because GCH, a
statement about arbitrarily high ranks of sets, is equivalent to the validity
of a particular second-order sentence.  On the other hand, we know that if
GCH holds high enough up (a supercompact cardinal) then it holds universally,
so maybe it's not so unlikely.

3) Is the set of validities for 3rd-order-logic or for type theory stronger
under Turing reducibility than the set of validities for SOL?

4) Let X be the set of sentences phi of SOL such that ZFC proves that phi is
a second-order validity.  Is there a "reasonable" deductive calculus for SOL
whose set of derivable validities is not contained in X?  (Here "reasonable"
means that simply replacing ZFC by ZFC+Y for some set-theoretic Y in the
above won't do.  Unlike the other four questions, this one is necessarily
imprecise.)

5) Let X be the set of second-order validities.  Let Y be the set of
statements "phi is a second-order validity" for phi in X.  Let Z be the
closure of Y under logical implication.  Are any theorems of ZFC outside of
Z?  (Such a theorem would refute a form of logicism.)

-- Joe Shipman

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