Skolem Again

P1. L-S tells us that orthographically identical theories may have
different models. By "orthographically identical", I mean that the
textbooks used to teach the theory, and which list the principal
theorems of the theory, are word for word the same.
P2. Theories that are learned in an identical way cannot have
different meanings.
P3. But theories with different models have different meanings.

Contradiction.

P1 is a consequence of a mathematical truth. I assume P2 is obvious
(and no one here has disagreed with it). Here is the argument for P3.

To learn a language is to learn what its model is. You understand the
meaning of the predicate "is a dolphin", when you understand -
concerning anything that is recognisably a dolphin - that the
expression "is a dolphin" applies to it. You understand the meaning of
"circle" when you understand that, concerning any X that is
identifiably circular, that "X is a circle" is true. Now suppose a
world identical to our own in every respect, except that all the
indefinable real numbers have been eliminated, leaving countably many.
Therefore, by hypothesis, the English sentence "there are countably
many things", in its standard English meaning, is true.

But the people in this world learned their language by learning what
its model was, i.e. they learned that "2" satisfies "1+1 = x", and that
"3" satisfies "1+2 = x", that "root 2" satisfies "x . x = 2" and so on.
Therefore, in the language they have learned, the sentence "there are
countably many things" is false. So it cannot have the same meaning as
the corresponding English sentence. (Since orthographically identical
sentences that have different truth values cannot have the same
meaning). Thus we could learn the same (i.e. orthographically
identical) language in such a way that the same sentence had an
entirely different meaning than its counterpart in the other language.

Yet the two languages would be learned in the same way, which
contradicts P2: languages learned in the same way should have the same
meaning.

.

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