[FOM] Turing machines and set theory

[Bob Hadley]

Hello,

I'd be grateful for references to papers or books that elaborate on the
following ideas.   The statement provided below was, in large part, given to
me by a colleague who is knowledgeable about recursion theory, but he was
not able to supply a reference.

Suppose we have a Turing machine, T, that accepts a single input set
and then recursively enumerates the theorems derivable, in first-order
predicate logic, within the arithmetic formal system known as Q.
Then it will be possible to formulate a special set of axioms,
using the language and enough of the axioms of standard set theory,
to capture the functionality of this Turing machine.  That is, the special
axioms expressing the recursive functionality of the Turing machine will be
added to standard axioms of set theory, and the totality of these would be
expressed in the language of set theory.

In addition to this, it is possible to formulate a sentence of set
theory which says, in effect, that "all Q-sentences enumerated by T are
true".

Is there a single paper or text which essentially provides proof of these
claims?

Many thanks,

Bob Hadley

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Robert F. Hadley (Bob)              Phone: 604-291-4488
  Professor                           email: hadley at cs.sfu.ca
  School of Computing Science         
  and Cognitive Science Program

  Simon Fraser University
  Burnaby, B.C. V5A 1S6
  Canada

  Web page:   www.cs.sfu.ca/~hadley/
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