Date: Wed, 24 Mar 1999 10:25:04 -0700
From: Randall Holmes <holmes@catseye.idbsu.edu>
Subject: FOM: properties

Sazonov said:

Does anybody understand what is an _arbitrary_ property?  Say, 
what about the following property of natural numbers

P(n) <=> `n is a natural number which can be denoted by
     an English phrase consisting of < 1000 symbols'?

What about the (paradoxical) least n such that not P(n)?

Holmes answers:

In this case, the problem is with the use of "can be denoted by an
English phrase" in an English phrase, as formalization reveals.

Sazonov said:

We have too unsteady ground for discussing 2nd order logic in a 
"pure" form outside ZFC or the like 1st order formalism.  The 
only existing way to make any informal idea or notion (like 
`property') precise and mathematically rigorous consists in an 
explicit presenting some formal rules of reasoning on this 
subject.  We are doing this usually via suitable extension of 
1st order logic by non-logical axioms.

Holmes answers:

This is a widely held position with which I do not agree.  I do know
what a property or predicate is -- it is something which is true of
some things and not of others.  The problem in Sazonov's first example
is that the "property" he exhibits is not well-defined; if I write on
a blackboard the phrase "the successor of the smallest number not
written on this blackboard", the problem is with the failure of
reference of the phrase, not with the category "number" to which its
referent supposedly belongs.

I agree with Sazonov that second order logic cannot be effectively
used without setting up a (necessarily partial) formalization, which
will use first-order logic (and can be construed as a first-order
theory with nonlogical axioms).

By its nature, a completely "arbitrary" property is something which
one cannot actually exhibit.  The fact that given any formal scheme
for enumerating properties of natural numbers one can define a
property of natural numbers (by diagonalization) which cannot be
expressed in that scheme strongly suggests that there are "arbitrary"
properties of natural numbers, though.  The fact that there are
members of a kind which I can't become acquainted with does not imply
to me that I can't be aware of the existence of that kind (or quantify
over it).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes@math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes