Re: Moore on Skolem's Paradox

Ullrich
> Do you specify in the article that the author does
> not know exactly what Skolem's theorem states?

I have looked at the paper again. In an earlier section, which I did
not quote, Moore writes

"One truth about sets, which can be established by Cantor's diagonal
argument and couched in this first-order language [whose sole
non-logical constant is the two-place predicate 'is a member of'], is
that there is no one:one correlation between w, the set of natural
numbers, and its own power set; that is, uncountably many sets belong
to P(w). Let S be the sentence which expresses this truth. It is a
consequence of the Loewenheim-Skolem theorem, in the version whose
proof requires the Axiom of Choice, that any true sentence from this
language, under the intended interpretation, will still be true under
an interpretation which results from the intended interpretation by the
elimination of all but countably many of its sets. In particular, this
is true of S. Not that this is paradoxical; nor does it constitute our
difficulty."

You said the author does not know exactly what Skolem's theorem states.
Is what he says above incorrect, then? He later says, in the version
which I did quote, what the "problem" alluded to above really is. I
shall quote it again.

" Our description of P(w) as uncountable, even though correct, must be
understood relative to our own current point of view. From another
point of view this very set may be countable. But I want to argue that
such relativism, compelling though it is, is subject to the by now
familiar predicament that any statement of it, if it is to be
intelligible at all, will have to be understood within a framework that
casts it as a straighforward error. ****It is this which I take to be
Skolem's paradox****."

That is to say, when it is claimed that P(w) is not _unconditionally_
uncountable, we have no way of understanding this except as the
demonstrably false claim that it is not uncountable at all.

On your remark about philosophers in the other thread (which I won't
reply directly, for reason of the mistake referred to), I wonder if
mathematicians actually read any philosophers? I mean, what appears in
peer-reviewed journals, as opposed to popularisations such as
Hofstadter, Rucker et nauseous alia?

.