Re: Moore on Skolem's Paradox

On 26 Sep 2005 11:17:39 -0700, William of Ockham <d3uckner@xxxxxxxxxxxxxx> said:
> What's paradoxical about that, you say? Perhaps this. Common sense
> suggests that
>
> (*) The meanings of two terms are different iff their extensions are
> demonstrably different.

I don't think commonsense dictates the "only if" direction, but
whatever.

> If by contrast the difference in extension is not in principle
> demonstrable, then the two terms (surely) have the same meaning. Yet
> this is what Skolem's theorem seems to deny.

What is "Skolem's theorem"? Do you mean the claim that there is no
absolute sense of cardinality?

> We are to suppose that certain terms in two apparently identical
> languages (like American and English) have different extensions (i.e.
> they apply to different mathematical objects),

What two languages do you mean? It's just the one language of set
theory that is at issue here. Or by a "language" do you mean the
language of set theory plus a given interpretation? If so, it seems
we're back simply to the mundane fact that open formulas of set theory
-- any of several that pick out the uncountable sets in the standard
interpretation, for example -- have different extensions in different
interpretations.

> but that the difference is not demonstrable in any way,

Why not? Given a countable model, I can demonstrate that the extension
of "x is uncountable" is different in that model than it is in an
intended model.

> because demonstration in mathematics is a method of proof, and all
> methods of proof will give the same result in both languages.

Sorry, I'm prett lost.

> That is too much to swallow. It is not a paradox, rather it is a
> conflict (rather like the notion of a set having the same number as
> its proper subset) between our ordinary intuitions, and what
> mathematics says.

Where, exactly is the conflict with ordinary intuition? Our best theory
of sets tells us that there are uncountably many things. Beyond this,
Skolem's "paradox" relies upon technical results where ordinary
intuitions have no purchase -- unless of course the technical results
are mischaracterized in such a way as to make them sound deeper than
they are -- which is pretty much the sense I'm getting for your
reconstruction here (which, frankly, I don't even recognize as Skolem's
"paradox" anymore).

.

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