Re: Moore on Skolem's Paradox

(I sent this twice as I'm not sure what "reply to author" actually
does)

Ockham
>A consequence of the theorem as I understand it, is that, if there is a
>countable model for set theory as expressed in some language *English,
>the reference of "the real numbers" as used in *English is different
>from its reference in actual English.

> Hmm, wait ... maybe you meant "model for
> [set theory as expressed in some language]".

Hurrah - I'll ignore your earlier comments. But you say

> To which someone will probably reply that an
> _interpretation_ of a formal language supplies
> referents for the terms in the language. That's
> correct, given a pair consisting of a formal
> language plus an interpretation of it the terms
> do have referents, those referents being elements
> of the "universe" of. the interpretation But that
> does not mean that terms in a formal language
> _per se_ have referents, which is what we need
> to make sense of your assertion.

Of course, no terms of any language have referents per se. The
expressions themselves are just noises. What we call "learning a
language" is assigning a reference to all of the terms that we learn.
So we learn that "hamburgers" mean hamburgers, "Socrates" means
Socrates, "two"means the number two, and so on. An interpretation (as
I understand) is the information which we must have about a sentence in
order to understand it.

So by "set theory as expressed in some language *English" I mean the
language+ the interpretation given to it as a part of learning the
language itself. In that case, the reference for "the real numbers",
given the expression with its meaning (reference) in that language, has
to be different from the reference of "the real numbers" as used in
actual English (assuming "English" includes the meaning of all the
English words as normally learnt and used, as is the convention in
philosophy).

Is that a problem? Well, I don't see that L-S rules out the
possibility of a world where mathematics was learned in exactly the
same way as this world, but where the interpretation involved a
countable number of objects only (perhaps because only countably many
objects were available).

Nothing about the learning process seems to rule this possibility out.
In what sense are the referents of mathematically referring expressions
like "2" accessible to us? How do we learn what "2" refers to? Godel
thought we have a mental faculty of direct intuition for numbers. But
this (as Graham Priest has argued) seems to fall foul of the Private
Language Argument in one way or another. If the fixing of reference is
performed solely by subjective mental acts, then there is nothing to
prevent each of us fixing reference in a quite different way, which is
another way of saying that (in respect of our public language),
reference is not fixed at all. The language of set theory is, after
all, a public and shared language. So the the criteria for
reference-fixing must equally be public.

.

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