[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 Black

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