[FOM] Finite Set Theory

[Nik Weaver]

Harvey Friedman describes how he explains what set theory is
to non-mathematicians, and asks

> Now, where would FOM subscribers have issues with this development?

I have issues with step 1:

> 1. A set is a "bunch of things" arranged in any order whatsoever,
> with no repetition. The only thing that matters about a set is
> what is in it, and what is not in it. It doesn't matter how one
> describes what is in it. There may be many ways to describe the
> same set.

You have told *what matters about a set* in completely clear terms,
but you haven't said what a set *is*.  There is a hidden assertion
that there exist abstract objects with the behavior you describe.

Read Hartley Slater's post
http://www.cs.nyu.edu/pipermail/fom/2006-February/009887.html
for a thorough analysis of the various kinds of "set talk"
that appear in natural language which shows that the putative
abstract objects they are thought to be about are a grammatical
mirage.

Implicit in your post is the question "if set theory is nonsense,
why is it so easy for ordinary people to understand how it works?"

My answer to that is that the behavior you've described is very
familiar to ordinary people.  It's just how nouns behave, or maybe
I should say noun phrases.  The supposed "set of all mammals"
functions in exactly the same way as the word "mammal", namely,
to identify which things are mammals and which are not.  So the
behavior of your fictional sets is unmysterious and that's
probably why people have an easy time of picking it up (at least
for finite sets).

This suggestion implies a possible route to rehabilitating set
theory, by identifying sets with linguistic descriptions in some
way.  There could be other ways of doing this too.  I find it
unbelievable that if this were done carefully the result wouldn't
be predicative.

One more comment.  I've said this before, but perhaps it bears
stressing.  Any attempt to justify set theory by saying *what
sets are* (e.g., a "bunch of things") seems not only sure to
fail in the sense of being, literally, nonsense.  It is also
clear that were any such attempt to succeed, it would succeed
all too well and would in fact justify naive set theory, which
is inconsistent.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver