Re: Skolem's 'Paradox'

William of Ockham wrote: 
What is this paradox?  I have yet to find any explanation of it that
makes sense.

My crude understanding is that an interpretation of a language is all
the objects which the language is able to talk about. An interpretation
is a "model" of a theory, iff the interpretation makes true all axioms
(and hence, all theorems) of the theory.

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.

Fine.  Then suppose there is a theorem in language L that says "an
uncountable set exists".  But then the model, the things the language
is about, are countable.  So an uncountable set does not exist.
Paradox - apparently.

Here's an explanation of the paradox which I have adapted from a
well-known writer.

"The mapping f [from elements to sets of elements] does exist, yet it
exists outside the model!. Do you think that it "must" be located in
the model? Why? If you are living (as an "internal observer") within
the model, the set of sets of elements seems uncountable to you
(because you cannot find a 1-1 function from sets of elements into
elements in your world). Still, for me (an an "external observer") your
uncountable set is countable - in my world I have a 1-1 function from
sets of elements into elements!"

This makes no sense to me at all.  If "an uncountable set exists" is
true, then "there is no mapping from elements to sets of elements" is
surely true.  So how can one exist at all, whether "outside the model"
or not?

The stuff about observers makes no sense to me.  If it merely "seems"
that there is no mapping, then the sentence "there is no mapping" is
false, although "it seems there is no mapping" is true.  And what does
"in my world I have a mapping" mean?

Apparently early writers did think it was a paradox (or at least
problematical).  Von Neumann said "At present we can do no more than
note that we have one more reason here to entertain reservations about
set theory and that for the time being no way of rehabilitating this
theory is known."  Zermelo declared the Skolem paradox a hoax.  In 1937
he wrote a small note entitled "Relativism in Set Theory and the
So-Called Theorem of Skolem")in which he gives a refutation of
"Skolem's paradox", i.e., the fact that Zermelo-Fraenkel set
theory--guaranteeing the existence of uncountably many sets--has a
countable model.

Frankel said "Neither have the books yet been closed on the antinomy,
nor has agreement on its significance and possible solution yet been
reached. "

Skolem said "I believed that it was so clear that axiomatization in
terms of sets was not a satisfactory ultimate foundation of mathematics
that mathematicians would, for the most part, not be very much
concerned with it. But in recent times I have seen to my surprise that
so many mathematicians think that these axioms of set theory provide
the ideal foundation for mathematics; therefore it seemed to me that
the time had come for a critique.


Do you happen to have references for these quotes?
They're quite remarkable, to put it mildly.

--
Cheers,
Herman Jurjus






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