On 30 Dec 2013, at 08:25, LizR wrote:

I admit I have difficulty understanding how Bruno's UD "runs" inside arithmetic

Don't push me too much as I really want to explain this to you :)

It is not completely obvious, especially if we want be 100% rigorous.

There are not so much textbook which do that entirely correctly. But here are three best one:

Boolos and Jeffrey (and Burgess for late edition).

Epstein and Carnielli   (out of stock!)


Matiyasevitch shows explicitly how to emulate any Turing machine with diophantine polynomials.

Oh, well, there is also the old good Stephen Kleene 1952 book, and many by Smullyan (although like Gödel they do that in PA or equivalent, and not in RA, which ask for more verification. Matiyasevitc shows that for diophantine equation, which means that it makes the RA universal quantifier not needed, and so gives the stronger result.

The main deep idea is already in Gödel 1931.

May be the shortest path is to explain the phi_i and use Kleene predicate to explain that equalities involving the phi_i are made arithmetical by the use of Kleene's predicate, but this needs the Gödel coding, which is long to describe, and even longer to prove that it does correctly the job.

I am thinking how to explain this without going in the technical details.



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