Re: Skolem's 'Paradox'


William of Ockham wrote:
|I understand that the Loewenheim-Skolem theorem says that if a theory
|has a model, then it has a finite or countable model. Any such theory
|can be interpreted as "about" a bunch of things that are at most
|countably infinite.

The phrase "can be interpreted as" covers a host of problems
with the reasoning in the paradox.

If we understand "can be interpreted" broadly, then the
claim is correct; we can (re)interpret the theory that way.
But one should remember that in such a broad sense, anything
can be reinterpreted as meaning something other than what it
normally would be taken to mean. There's a tendancy for
discussions of the paradox to slide into language like, "the
theory is unable to prevent" this reinterpretation. The point
to be clear on is, in what sense do we mean to consider an
interpretation as "prevented" by the theory? Words are just
words; they can never reach out and grab you, so as to prevent
you from taking them in some other way. If you are hell-bent
on interpreting the word "cow" to mean "chicken", nothing is
going to stop you.

The other, narrower and more subtle, sense of "can be
interpreted" has to do with what interpretations can be
*reasonably* held.

In order for words to convey a meaning, they need to be
uttered in a context where the listener is able to recognize
that meaning, and distinguish it from other conceivable
interpretations. Skolem's paradox reasons as though the only
basis for doing that is first-order logic. This tendancy can
be called "first-order-itis", since it gives a special role
to first-order logic without ever actually saying why first-
order logic deserves such a special role.

I claim that it's possible for you to understand what I mean
when I say, "set of natural numbers". Skolem's paradox shows
that if I'm right, then we can rule out the idea that the
way you can do this is by creating a translation of what
I've written, plus context, into a first-order theory, then
go hunting for any old model of it, with no further
restriction.

Of course, if you think about it, there never was any reason
to expect that such a practice would serve to convey the
meaning of mathematical terminology to you. I think of the
era your quotes come from as one of "flaming" optimism about
foundations. It's easy to say in hindsight that it shouldn't
come as a surprise that there don't exist complete or
categorical axiomatizations of every domain of discourse in
mathematics (so that the truth of a statement about the domain
could be defined as provability from the axioms for the
domain, and the objects in the domain could be identified as
the elements of the one and only model of the axioms). But
it seems to be necessary to point it out.

It's surely true, after all, that the way you know what is
meant by the terms derives *somehow* from experience with
people using them. It's understandable that in mathematics
one would expect axioms for a certain set of terms to be the
key statements about those terms, hence perhaps the ones that
a person could turn to, to determine what the terms mean.
And first-order logic is a familiar framework in which to put
a set of axioms. As pretty a picture as it would make if things
could work like that, though, it just doesn't work that way.
Not even for such terms as "natural number" or "first-order
logic" does first-order-itis work out.

Keith Ramsay

.

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