[FOM] n-th order ZFC

[meskew at math.uci.edu]

> From: Robert Black <mongre at gmx.de>
> Date: July 10, 2011 12:18:12 AM GMT+08:00
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] n-th order ZFC
>
> 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


Why does second-order logic help the realist position?  One could just say
that the powerset operation is fully determinate, even though it cannot be
categorically described in first-order logic.  In second order logic you
have a logic with very restricted semantics, but one can also restrict
semantics in first-order logic by picking out a class of distinguished
models of a given theory, perhaps even a one-element class.  So a prior
commitment to realism about powersets seems to obviate any new
philsophical lessons of second-order logic.  Conversely, believing that
standard 2nd-order semantics make sense seems to require prior commitment
to realism about powersets.

Best,
Monroe