Moore on Skolem's Paradox

I have just found a paper by A.W. Moore, where it is argued that
Skolem's Paradox really is a paradox after all. I include an extract
below. I'm not saying I agree with his argument, I include it only to
refute the arguments that inability to see that it is not a paradox is
proof of the need to read a logic book. Moore is professor of logic at
Cambridge University and has written and lectured extensively on set
theory. I assume he has read all the necessary books. Perhaps he did
not understand them? Very well, but then I am hardly likely to
understand them either, and the exhortations are needless.

Ockham

PS I have contributed a short article to Wikipedia here

http://en.wikipedia.org/wiki/Skolem%27s_paradox

with an account of Moore's version.

And here is the extract:

-----------------------------------------------------
MOORE ON SKOLEM'S PARADOX
-------------------------------------------------------

The article from which this extract is taken appeared as Moore, A.W.
"Set Theory, Skolem's Paradox and the Tractatus", Analysis 1985, 45.


.... The only possible conclusion [given the Lowenheim-Skolem Theorem]
seems to be that notions such as countablility and uncountability are
inherently relative. ... 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.

The crux of the matter is this. If there is an implicit relativization
in our claim that P(w) is uncountable (the claim which is established
by Cantor's argument), then it ought to be possible to make it explicit
(just as it is possible to make explicit any relativization in the
claim that a physical object is moving). But it is possible to do so
this only insofar as it is possible to construe our discourse about
sets as discourse about a particular collection of objects, the
collection to which such claims must be relativized. And this in turn
is not possible unless we endorse the fundamental error that there is a
set which contains all the sets we intend to talk about. 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.

Admittedly, there are interpretations of the language of set theory
under which all the "right" sentences come out true and in which w is
correctly represented as such even though the set represented as its
power set is countable. Any such interpretation can be thought of as a
limited point of view; there are correlations to which we have access
and which are not in its domain. But there are also further subsets of
w to which we have access, as Cantor's argument testifies. What we
ourselves take to be P(w) never appears to be anything but uncountable.
The relativist, convinced that our own point of view is in turn
limited, urges us to acknowledge the possibility that what we ourselves
take to be P(w) is not - as viewed from some even higher vantage-point
from which it may yet be countable. But how are we to make sense of
this? Certainly not by trying to view P(w) from two different points of
view at once; that would be incoherent. Nor by trying to view it simply
from this point of view; thay would make the possibility
unintelligible. But if it were possible to view it from an absolute
standpoint, then relativism itself would lose its rationale and there
could be no objection to saying that P(w) contained all of w's subsets
and that it was unconditionally uncountable. So if we do deny the
absolute uncountability of P(w), then what exactly are we denying and
where, so to speak, are we denying it? (The mere fact that there are
legitimate concepts of countability and uncountability which do involve
relativisation to certain domains is beside the point. The relativist
wants to insist that there are no absolute concepts of countability and
uncountability - that it makes no sense to describe P(w) as
unconditionally uncountable). We, in mounting a general investigation
into what sets are like, can only aspire to know whether or not P(w) is
countable, as it were here. It is not. But we can have no grasp on any
distinction between what is true here and what is true simpliciter. So
P(w) is uncountable simpliciter.

Yet the very use of the word "here" appears to vindicate the
relativist. There remains a real predicament. That which cannot
legitimately be stated (relativism) appears, for all that, to impress
itself upon us as soon as we step outside mathematical practice and
reflect on what is revealed therein. This predicament is directly
analagous to that which Wittgenstein faces in the Tractatus. There is
no particular point of view in the world which can be spoken of as
here: our point of view is a limit of the world. That is, there is no
particular set in the hierarchy of sets which can be spoken of as the
intended range of the quantifiers: they are intended to range over the
whole hierarchy (though not even this can properly be said). And here
an intriguing possibility arises. If we are prepared to extend the
analogy with the Tractatus, then it will become apparent that, despite
the fact that relativism defies any coherent statement, the debate
between the relativist and the non-relativist is in a very deep sense
irresoluble. For what the relativist means is quite correct; only it
cannot be said, but makes itself manifest (5.62).

.

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