21/10/27 14:37:55.91 O7+c++yB.net
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[FOM] Higher-order arithmetic as an alternative to ZFC
Anthony Coulter
Wed Mar 30 10:03:55 EDT 2016
My official rationale is that second-order logic is simpler and more
natural than ZFC but it's still powerful enough to do most of your
interesting mathematics. (Many undergraduate textbooks have an
appendix with remedial set theory, but only rarely mention ZFC; all you
really need are the axioms of extensionality and separation, plus an
assumption that there exists some set containing all the objects you're
going to study in the textbook.) Occasionally you need to perform
induction on an unusually complex structure (that is, on a very large
ordinal) and when that happens, ZFC is still there and you can use it,
but now invoking super-powerful induction is like invoking the axiom
of choice---you have to do it explicitly and you're made to feel a
little guilty to encourage you to find a way to redo the proof without it.
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