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lbThingrb2y50

Correct. Each iteration of the halting problem for oracle Turing machines (called the "Turing jump") takes you to a new level of relative undecidability, so in particular true arithmetic is strictly harder than the halting problem.

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[ Question ]

Does decidability of a theory imply completeness of the theory?

by Noosphere89

29th Jul 2023

1 min read

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I know right now that completeness implies decidability of a theory, but the question is essentially the converse of this:

Given an arbitrary formal theory that is somehow decidable by a Turing Machines, in the sense that for any sentence in the theory, you can decide it's truth or falsity by some means, can you show that it will inevitably be a complete theory?

If it isn't universally the case that decidability of a theory implies the completeness of a theory, are there any conditions that are required to have decidability of a theory be equivalent to completeness of a theory?

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Does decidability of a theory imply completeness of the theory?

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Algon

Jul 30, 2023

No. This answer to a related stack-overflow question explains why:

That is to say a first order logic without any predicate or function symbols is decidable. But you can have incomplete theories in such a signature.

[-]Noosphere892y40

Thank you for answering the question quickly, it was nice to know that being decidable wasn't implying completeness like completeness implied decidability.

2Algon2y

Completeness doesn't imply decidability though? Peano arithmetic is complete and undecidable. Maybe you meant completeness + semidecidability - > decidability?

6Rafael Harth2y

Peano arithmetic is not complete. I think the claim from my post is correct except that it misses having a computable set of axioms as a second requirement (which in the post more or less follows from context, though I should still probably add it explicitly) actually nvm I did say this in the post. (See e.g. https://math.stackexchange.com/questions/1280043/every-complete-axiomatizable-theory-is-decidable)

2Noosphere892y

Your post is below, so anyone else can follow this conversation if necessary. https://www.lesswrong.com/posts/Tb4mJWh3FYGRyid6C/understanding-goedel-s-incompleteness-theorem

2Algon2y

Sure, that's true. I should've said the theory of the standard model of Peano arithmetic. That's complete, undecidable and can't have a recursive axiom system.

2Noosphere892y

I'm going to ask a new question, different from the old question I asked: Given a halt oracle added to the Turing Machine, that is a magical oracle tape that can solve all problems Turing-reducible to the halting problem, can we nail down the standard interpretation/model of Peano Arithmetic?

9lbThingrb2y

The true first-order theory of the standard model of arithmetic has Turing degree 0ω. That is to say, with an oracle for true arithmetic, you could decide the halting problem, but also the halting problem for oracle Turing machines with a halting-problem-for-regular-Turing-machines oracle, and the halting problem for oracle Turing machines with a halting oracle for those oracle Turing machines, and so on for any finite number of iterations. Conversely, if you had an oracle that solves the halting problem for any of these finitely-iterated-halting-problem-oracle Turing machines, you could decide true arithmetic.

2Noosphere892y

So the answer is no, even a magical halt oracle cannot compute the true first order theory of the standard model of arithmetic, but there are machines in which you can compute the true first order theory of the standard model of arithmetic.

5lbThingrb2y

2Noosphere892y

Sort of, though I was thinking of this quote from Rafael Harth's post below: I was trying to think of an answer to how this could work, or alternatively how this couldn't work.

2Algon2y

Based off that passage, I don't see Rafael Harth's argument works either.

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Does decidability of a theory imply completeness of the theory?

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Jul 30, 2023

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