Re: Moore on Skolem's Paradox

>> 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.

I had the following thought-experiment in mind. We discover a
different civilisation whose mathematics is the same as our own (how
otherwise?). They speak a language called *English, which, which is to
all intents and purposes identical to English, including mathematical
English. By "to all intents and purposes" I mean they learn
mathematics, and particularly the meaning of "real number" in a way
identical to the way used here. How would we "demonstrate" that the
*english expression for "the real numbers" has a different reference
from the corresponding expression in English?

Ockham:
> the difference is not demonstrable in any way,

Menzel:
> 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.

Yes, but one is not "given" the model for one's own language! It
presumes that in one language I am able to "refer" to a different bunch
of things - the "real" real numbers, if you like - than are refered
to in some another language - the "fake" real numbers. Even though
the two languages are taught and learned in identical ways. Again, how
can I suppose that the term "real numbers" refers to a different set of
things in *English, than it does in English, even though the two
languages are taught and learnt in identical ways? Skolem's Theorem
i.e. Lowenheim-Skolem, suggests that this is possible. This conflicts
with our common sense notion of what terms "refer to". We are to
suppose

(A) In our language, the "the real numbers" refers to the real numbers
(B) In language E* "the real numbers" does not refer to the real
numbers, but to some restricted sub-set.of the real numbers

Or is it that when we learn mathematics, we learn the meaning of
expressions referring to mathematical objects by some sort of Platonic
intuition, some direct reference to sets of Mathematical Objects?
Then, of course, my initial assumption that the *English learn
mathematics in a way identical to the English, is wrong. For, when the
teacher teaches the class the meaning of "the real numbers", their
grasp of the reference of that expression is different from ours.

.

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