[FOM] n-th order ZFC

Roger Bishop Jones rbj at rbjones.com
Tue Jul 12 16:18:21 EDT 2011

```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

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

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

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