Re: Non standard models of PA

Herman Jurjus wrote:
It seems to me that such explicit formulas do exist, only they are
probably very long. (What i have in mind is making a Henkin
construction, but with 'fixed' enumeration of sentences, indexes of
spare-constants, etc. in some explicit way. Dividing out the equivalence
relation at the end is handled by identifying every equivalence class
with its smallest member.)

That is indeed how the arithmetical completeness theorem is proved.

But there is one more worry:
Suppose we have such explicit formulas for sum and product.
If we have two models of ZFC, then in both models, our formulas lead to
some non-standard model of PA. But is there an a priori reason why these
two models must be isomorphic?

No. Why should that worry you?

If not, then is it really fair to say that these formulas provide a
concrete, explicit and unambiguous definition of a non-standard model?

Yes, in as much it's fair to say any mathematical definition of any
structure about which we can state problems undecidable using our
currently accepted principles defines a concrete, unambiguous and
explicit structure.

To put it differently, if someone says 'Tennenbaum's theorem shows that
there is no concrete, explicit and unambiguous definition of a
non-standard model of PA', then what counter arguments can be given?

In order to conclude that there is no concrete, explicit and
unambiguous definition of a non-standard model of PA from Tennenbaum's
theorem we'd need some assumption to the effect that only recursive
things are concrete, explicit and unambiguous -- a rather odd idea
having not much to do with how we actually conceive mathematics and do
mathematical reasoning.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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Relevant Pages

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    ... unambiguous definition of a non-standard model of PA from Tennenbaum's ... having not much to do with how we actually conceive mathematics and do ... mathematical reasoning. ...
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