[FOM] n-th order ZFC
Robert Black
mongre at gmx.de
Sat Jul 9 12:18:12 EDT 2011
Am 09.07.11 06:09, schrieb meskew at math.uci.edu:
> Can you name any mathematical problem that can be solved by appealing to
> the categoricity of second order set theory?
Obviously not, since it's routine to turn any proof in a second-order
theory into one into a two-sorted first-order theory with appropriate
comprehension axioms. (Second-order ZF slides seamlessly into
first-order Kelly-Morse, doesn't it?) But that's not the point. Those of
us who think we understand second-order quantification in a determinate
way (i.e. that there is a determinate totality of *all possible* subsets
of a given infinite set) can conclude from this that 'the natural
numbers', 'the real numbers', 'the sets of rank below the first
inaccessible' and so on are all (up to isomorphism) determinate
structures and that questions about them have determinate answers
independently of whether or not those answers are decided by ZFC (or any
other particular system). That's a very interesting philosophical claim
which has consequences about what mathematicians are doing, even though
using second-order logic doesn't in any way help them to do it.
Robert
--
Robert Black
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