I think what you proved essentially boils down to the fact that a consistent guessing oracle can be used to compute a completion of any consistent recursively axiomatizable theory. (In fact, it turns out that a consistent guessing oracle can be used to compute a model (in the sense of functions and relations on a set) of any consistent recursively axiomatizable theory; this follows from what you showed and the fact that an oracle for a complete theory can be used to compute a model of that theory.)
I disagree with
Philosophically, what I take from this is that, even if statements in a first-order theory such as Peano arithmetic appear to refer to high levels of the Arithmetic hierarchy, as far as proof theory is concerned, they may as well be referring to a fixed low level of hypercomputation, namely a consistent guessing oracle.
The translation from T to U is computable. The consistent guessing oracle only came in to find a completion of U, but it could also find a completion of T (in fact, a completion of U can be computably translated to a completion of T), so the consistent guessing oracle doesn’t really have anything to do with the relationship between T and U.
U axiomatizes a consistent guessing oracle producing a model of T. There is no consistent guessing oracle applied to U.
In the previous post I showed that a consistent guessing oracle can produce a model of T. What I show in this post is that the theory of this oracle can be embedded in propositional logic so as to enable provability preserving translations.
I see that when I commented yesterday, I was confused about how you had defined U. You’re right that you don’t need a consistent guessing oracle to get from U to a completion of U, since the axioms are all atomic propositions, and you can just set the remaining atomic propositions however you want. However, this introduces the problem that getting the axioms of U requires a halting oracle, not just a consistent guessing oracle, since to tell whether something is an axiom, you need to know whether there actually is a proof of a given thing in T.
The axioms of U are recursively enumerable. You run all M(i,j) in parallel and output a new axiom whenever one halts. That’s enough to computably check a proof if the proof specifies the indices of all axioms used in the recursive enumeration.
Yeah, sorry that was unclear; there’s no need for any form of hypercomputation to get an enumeration of the axioms of U. But you need a halting oracle to distinguish between the axioms and non-axioms. If you don’t care about distinguishing axioms from non-axioms, but you do want to get an assignment of truthvalues to the atomic formulas Q(i,j) that’s consistent with the axioms of U, then that is applying a consistent guessing oracle to U.
I think what you proved essentially boils down to the fact that a consistent guessing oracle can be used to compute a completion of any consistent recursively axiomatizable theory. (In fact, it turns out that a consistent guessing oracle can be used to compute a model (in the sense of functions and relations on a set) of any consistent recursively axiomatizable theory; this follows from what you showed and the fact that an oracle for a complete theory can be used to compute a model of that theory.)
I disagree with
The translation from T to U is computable. The consistent guessing oracle only came in to find a completion of U, but it could also find a completion of T (in fact, a completion of U can be computably translated to a completion of T), so the consistent guessing oracle doesn’t really have anything to do with the relationship between T and U.
U axiomatizes a consistent guessing oracle producing a model of T. There is no consistent guessing oracle applied to U.
In the previous post I showed that a consistent guessing oracle can produce a model of T. What I show in this post is that the theory of this oracle can be embedded in propositional logic so as to enable provability preserving translations.
I see that when I commented yesterday, I was confused about how you had defined U. You’re right that you don’t need a consistent guessing oracle to get from U to a completion of U, since the axioms are all atomic propositions, and you can just set the remaining atomic propositions however you want. However, this introduces the problem that getting the axioms of U requires a halting oracle, not just a consistent guessing oracle, since to tell whether something is an axiom, you need to know whether there actually is a proof of a given thing in T.
The axioms of U are recursively enumerable. You run all M(i,j) in parallel and output a new axiom whenever one halts. That’s enough to computably check a proof if the proof specifies the indices of all axioms used in the recursive enumeration.
Yeah, sorry that was unclear; there’s no need for any form of hypercomputation to get an enumeration of the axioms of U. But you need a halting oracle to distinguish between the axioms and non-axioms. If you don’t care about distinguishing axioms from non-axioms, but you do want to get an assignment of truthvalues to the atomic formulas Q(i,j) that’s consistent with the axioms of U, then that is applying a consistent guessing oracle to U.