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Results tagged with reverse-math
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user 8991
The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms, typically of set existence, needed to prove them; originated in its modern form in the 1970s by H. Friedman and S. G. Simpson (see R.A. Shore, "Reverse Mathematics: The Playground of Logic", 2010).
7
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Strength of Transfinite Induction on the Difference Hierarchy
I'm wondering if a particular theory of second order arithmetic has been studied or is known to be equivalent to some other theory.
Consider the formulas generated by $\Pi^1_1$ and $\Sigma^1_1$ formu …
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3
votes
Accepted
Strength of Transfinite Induction on the Difference Hierarchy
I hope it's not too tacky to answer my own question now that I've had a few days and train rides to think about it.
The answer is that the proof theoretic ordinal of $\Delta-TI_0$ is the Howard-Bachm …
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7
votes
Accepted
Am I counting quantifiers correctly?
To state the general rule: when counting quantifier changes, you ignore bounded quantifiers that come after all unbounded ones. But even a bounded quantifier, if it comes before an unbounded one, m …
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7
votes
Accepted
What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?
I'm not clear whether you're asking if there are any interesting non-$\Pi_2$ theorems in the literature, or any proofs of $\Pi_2$ theorems with interesting non-$\Pi_2$ intermediate steps which cannot …
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9
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Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prer...
Historically, reverse math is very closely tied to ideas in proof theory, but as Andreas points out, over the last decade or so, the connection to recursion theory has been very strong. For the found …
2
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Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$
I believe BISH includes the Fan Theorem, which, being the contrapositive of weak König's lemma, is not provable in $RCA_0$.
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6
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What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
The difficulty is in what it means to go up the ramified hierarchy. When talking about theories, you can't write down a computable or c.e. theory which perfectly captures the whole ramified hierarchy …
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5
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What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
This is a second answer (think of it as part two of the other, already long, answer), in response to the clarification of the question by Keshav Srinivasan in the comments on that question.
First, no …
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