[FOM] n-th order ZFC

[Robert Black]

Am 11.07.11 01:35, schrieb meskew at math.uci.edu:
> 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.
>
I agree entirely: the crucial point is the determinacy of the power-set 
operation (or some equivalent in terms of Fregean concepts or plural 
quantification). But once you believe that, second-order logic is an 
obviously neat way of expressing categorical theories. It's worth noting 
that second-order logic is a very natural system - it's not a 
coincidence that when Frege created modern logic in the 
_Begriffssschrift_ he went straight to second-order logic. Of course he 
needed to do that to get the ancestral construction, but also 
(disregarding what we now know about incompleteness) once you've got 
individual constants and variables and predicate constants it's really 
rather odd not to allow predicate variables.

Robert

-- 
Robert Black