[FOM] How much of math is logic?

[joeshipman@aol.com]

Logicism seems to be considered passe nowadays, but I have not found 
the arguments against it convincing.

Treating "logical validity" as an undefined term which I would like to 
understand better, I request examples of the folllowing:

1) A theorem of Peano Arithmetic which is not equivalent to a logical 
validity

2) A theorem of ZF without the axiom of Infinity which is not 
equivalent to a logical validity

3) An existing open question (outside of set theory) which is not 
equivalent to a sentence of second-order logic (with standard semantics)

4) An argument against second-order logic with standard semantics that 
does not simply amount to "a logic should have a well-behaved proof 
theory or a complete deductive calculus" or "since second-order logical 
validity depends on which set-theoretical assumptions are true it is 
not really logical validity".


I am interested in defendiing these two propositions:

i) Any arithmetical sentence provable in ZF is a logical consequence of 
the axiom of infinity.
ii) An oracle for second-order validity decides any mathematical 
question of interest to non-logicians.

-- JS
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