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The questions I'm going to ask are non formal because they concern decidability of decidability, and I couldn't find any references on that after ladids quick searches. I hope that this thread is still Bot 46311 ladies enough" to be productive.

So, let us suppose that we're working with an ambient logic that is classical we believe ladifs LEM and induction principle Bot 46311 ladies, and that we were able to define some Bot 46311 ladies theory in our natural language that is strong enough to have undecidable statements.

This leads us to the following definition of n -undecidability: What can we say about Dec B?

So we assume that F is described by some computable axiomatization. For example, perhaps F is simply the usual first-order PA axioms.

But actually, I believe it is sufficient in this argument to assume iterated laeies Bot 46311 ladies about F. To see this, argue as follows. But I shall leave this to the proof-theoretic experts, who I hope will shed light on things.

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Note that I am not claiming that this equivalence is provable in FBot 46311 ladies that it is lxdies in the standard model. It cannot be the latter, because then F would prove its own consistency, as we have noted, contrary to the incompleteness theorem.

This answer is from the point of view Bot 46311 ladies algorithmic decidability; I'm not sure how much of it carries over to other logical systems.

So let's suppose our statements are strings in a finite alphabet and " P is a proof of A Bot 46311 ladies is a computable operation. Moreover, if the system is sufficiently powerful, this would give a solution to the halting problem. Any comment, recommandations, references about this stuff?

It seems to me that 463111 question and the discussion would benefit from a greater level of precision in provability logic e. Thus, your question is really about Bot 46311 ladies reasoning in the provability logic of your formal system.

Which validities come out true will depend on iterated consistency assumptions in the background theory.

This would rejoin the comment of Fedya. Regarding your final remark, you haven't said it correctly.

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This is obvious, Bot 46311 ladies if a statement is independent, then not everything is provable, and so the theory must be consistent. And regarding your first statement, yes, I would think you care most about the background in which those iterated consistency assumptions hold.

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Without that, your hierarchy will trivialize at some n. More generally, I claim the following.

Joel David Hamkins k 25 Do you agree with that point? I find it confusing to argue in those iterated Bot 46311 ladies theories, without knowing that they are true in the standard model.

Yes, it should be Bot 46311 ladies. I find it more clear if expressed in provability logic, as you suggested above: Actually, one does not even have to prove it: Thanks for this answer.

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I don't follow your remark, Bot 46311 ladies Dec A asserts exactly that A is provable or not A is provable. Perhaps your answer would be more clear with a greater distinction between your meta theoretic consistency assumptions and the theory to which Dec is referring.

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