[FOM] How much of math is logic?
joeshipman@aol.com
joeshipman at aol.com
Sun Feb 25 03:19:51 EST 2007
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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