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", ...

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645 views

Do we need more than the periods? [closed]

Reading this question, and the Wikipedia page on reverse mathematics, I wonder whether one needs more than the subfield $\mathcal{P} \subset \mathbb{C}$ of periods for applied mathematics, or indeed ...
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2answers
1k views

Propositions equivalent to the completeness of the real numbers

Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't? ...
12
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1answer
564 views

Complementation of $\omega$-regular languages in reverse mathematics

Does anyone know where Büchi's theorem that $\omega$-regular languages are closed under complementation fits into the reverse-mathematics classification scheme? That is, is it equivalent over ...
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5answers
1k views

Consistency strength needed for applied mathematics

Given that we can never proof the consistency of a theory for the foundations of mathematics in a weaker system, one could seriously doubt whether any of the commonly used foundational frameworks (ZFC ...
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1answer
670 views

Soundness Theorem in reverse mathematics

STPL := soundness theorem for predicate logic (see this) When trying to figure out the strength of the STPL in reverse mathematics, I managed to convince myself of the following: a) ACA0 has a ...
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4answers
1k views

Is finitism an extreme form of constructivism?

I hope this question is not too soft for MO. The Wikipedia says about finitism that it is an extreme form of constructivism. See http://en.wikipedia.org/wiki/Finitism. I doubt that this is correct. ...
6
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1answer
286 views

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 $\Sigma^1_1$ ...
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Reverse mathematics strength of identically zero polynomials are the zero polynomial

According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
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1answer
221 views

Proving boundedness of continuous images of [0,1] in WKL0

I've been reading about reverse mathematics (mostly on wikipedia), and I had been thinking that I understood how to prove the equivalences to WKL0 and ACA0 mentioned in the its article. However, I ...
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4answers
884 views

Can transfinite induction be defined as axiom scheme in FOL on bin-tree structures?

Transfinite induction requires a second order induction hypothesis. So, that can not be defined as axiom scheme in FOL. However, if I look to Goodstein's theorem en the Hydra games, then they have to ...
6
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2answers
571 views

Weakest subsystems of second order arithmetic for mathematical logic

It is known that to prove completeness of first-order logic for countable languages WKL0 is enough. But, is it the weakest subsystem where one can prove it? What about the incompleteness theorems? Is ...
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2answers
850 views

Can Goodstein's theorem been proven with first order PA + Constructive Omega Rule?

I am trying to understand transfinite induction and Gentzen's theories. But I was wondering, if there is any connection with the Constructive Omega Rule (COR). With COR I mean that if you can proof: ...
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3answers
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What is the reverse mathematics of first-order logic and propositional logic?

Suppose one tries to formalize first-order logic. How much "strength" is required to do this? Strength can mean in various senses: The fragment of ZFC needed to codify first-order logic. Which ...
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3answers
750 views

Reducing ACA₀ proof to First Order PA

According to the Wikipedia ACA0 is a conservative extension of First Order logic + PA. http://en.wikipedia.org/wiki/Reverse_Mathematics First of all I have a few questions about the proof: a - What ...
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3answers
1k views

Reverse mathematics of (co)homology?

Background Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 ...