FOM: Questions on higher-order logic
JoeShipman at aol.com
Wed Sep 6 10:48:13 EDT 2000
This answer is indeed much more informative. It means that using axioms of
the form "phi is a 2nd-order validity" within the framework of the 1st-order
language of set theory doesn't get you very far, basically because such
statements are limited in logical complexity.
"An uncountable cardinal exists" is certainly important in "ordinary
mathematics", for example to obtain standard counterexamples in general
topology (see, for example, Munkres's introductory text "Topology", which
relies heavily on the "minimal uncountable well-ordered set").
This doesn't quite do what I had hoped, which is refute "2nd-order logicism"
by showing that non-logical axioms are necessary for an important
mathematical theorem, since assuming "phi is a 2nd-order validity" within the
framework of 1st-order set theory is not the same thing as assuming "phi"
itself within the framework of 2nd-order logic. Someone favoring an
alternative foundation of mathematics via 2nd-order logic might therefore not
find the argument "You can't show an uncountable cardinal exists" persuasive.
I find this situation a little confusing. By working in SOL with standard
semantics one can develop much of ordinary mathematics in a more natural way
than via the usual foundation in first-order ZFC (or ZC, since the
Replacement Axiom is not needed in the standard developments). But
apparently within set theory one needs more information than which sentences
are second-order validities, there have to be axioms specifically about sets,
such as the powerset axiom.
-- Joe Shipman
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