Re: Skolem's 'Paradox'

William of Ockham says...

>Thank you once again. But I can't see how you imagine this is an
>explanation.

And I can't see why you don't consider it an explanation.
The way (in math, anyway) to explain new terminology is
by relating it to old terminology that presumably you
already *do* understand. That's what I have done. I've
shown how to translate a sentence of the form "S is true
in model M" into an equivalent sentence that don't mention
the concept of truth in a model.

>I wanted to understand " true in model M". You explain
>it as meaning "S relativized to M is true". The rest of your
>explanation simply assumes the equivalence, without explaining it.
>
>What does "S relativized to M is true" mean?

That depends on the particular S, as I have explained.
If S is the sentence

"for all x, there is a y such that y > x"

Then "S relativized to M is true" means the following
sentence (call this sentence S_M):

"for all x, if x is an element of M, then there is a y
such that y is an element of M and y > x"

This sentence doesn't mention "true in a model" or "true"
or "relativized" or anything other than standard mathematical
and logical concepts: quantifiers, set membership, comparisons.
Assuming that you already understand those concepts,
then you understand sentence S_M. S_M means the same thing as
"S relativized to M is true" which means the same thing as
"S is true in model M".

If you *don't* understand the standard mathematical
and logical concepts, then you won't understand the
translated sentence, of course. But that difficulty
doesn't have anything to do with models.

So any time any logician says something about something being true in
some model, you just perform the translation to get a new sentence
that doesn't mention "true" or "model", and pretend that the logician
said that, instead.

If someone says that "S is true inside model M, but false outside it",
translate that into the sentence

S_M and not S

where S_M is constructed from M using the translation I provided.
That sentence is an ordinary sentence of mathematics that doesn't
mention "inside a model", "outside the model", or "true".

--
Daryl McCullough
Ithaca, NY

.

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