[FOM] n-th order ZFC

[Roger Bishop Jones]

On Monday 11 Jul 2011 00:35, meskew at math.uci.edu wrote:

> Why does second-order logic help the realist position? 

I myself feel that introduction of metaphysical terminology 
here confuses rather than clarifies the issues, which are 
semantic and orthogonal to the metaphysics.

> 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.

It is possible (as I mentioned in an earlier post) to 
stipulate a more discriminating semantics for a first order 
language of sets and get a language of similar 
expressiveness to second order set theory,
However, this would be hard work, because it is not common 
to work with first order set theory in that way.  You would 
have to spend time explaining what you were doing.
It is simpler to talk about second order set theory, at 
least people know what you are talking about, even if they 
may not like it.

There is also a degree of artificiality involved, if one 
adopts a semantics which reflects second order set theory 
exactly.
The natural thing to do is to stipulate to the full power 
set, but this does not give an exact match with second order 
set theory because of the differences between the first order 
replacement schema and the full second order replacement 
axiom.
You would have to stipulate to models whose rank is an 
inaccessible cardinal, and in explaining this you would 
probably then have to refer to second order set theory.

In my view the important thing is not the drawing of 
philosophical questions, but the ability to express 
mathematical problems exactly which would otherwise require 
complications.
For example, when we consider CH in  second order set theory 
we have a definite problem on our hands.
When an author considers CH in a first order context it is 
often without any clear disambiguation of the problem.
I dip occasionally into the literature surrounding CH and it 
is the norm even in informal overviews that I am unable to 
discover what problem is being addressed.
Sometimes it is quite clear that the problem has not been 
pinned down, since V=L is admitted as one possible 
resolution of the problem, when to be strict it is rather 
one possible way of disambiguating the question (with the 
advantage that the problem is then already solved).

> Conversely, believing that standard 2nd-order semantics
> make sense seems to require prior commitment to realism
> about powersets.

It does not require any metaphysical position at all.
Acceptance that "all possible subsets" is meaningful, like 
acceptance that the concept of natural number is meaningful, 
is well within the normal standards of mathematics, and is 
independent of metaphysical ontology.
(it is true that philosophers do sometimes refer to the 
acceptance of the objectivity of truth in some domain as 
realism, but then some philosophers are unable to accept 
that one can believe in the objective truth of arithmetic 
without also believing in the existence of numbers.
On that latter point I can assure them as a matter of 
empirical fact that it is possible).

Roger Jones