FOM: Second-order validities

[Joe Shipman]

OK, maybe when you write down axioms and rules of inference for
second-order logic you are really indulging in a species of first-order
logic.  But at least it's a logically motivated species of first-order
logic that may be more powerful than the ordinary first-order predicate
calculus.  To recast an earlier query, do *logically motivated* axioms
and rules of inference for second-order logic allow one to prove all the
second-order validities that ZF does?  If yes, this is a very strong
argument for logicism; if not, let's have an example!

To repeat my other query from March 15th:  what are the best known lower
and upper bounds on the Lowenheim number for second-order logic?

-- JS