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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What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the ...
24
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4answers
2k views

In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?

I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems ...
24
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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 ...
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2answers
1k views

Prospects for reverse mathematics in Homotopy Type Theory

Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include Subsystems of Second Order Arithmetic ...
13
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5answers
2k 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 ...
13
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2answers
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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? ...
13
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1answer
544 views

Higher recursion theory and reverse mathematics: What is to $\Pi^1_1-CA_0$ as $RCA_0$ is to $ACA_0$?

There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" - indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in ...
12
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1answer
762 views

Reverse mathematics of Hilbert's Theorem 90

What is known, and what is published, on the reverse mathematics of the nest of results called Hilbert's Theorem 90?
12
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2answers
625 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 ...
11
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3answers
778 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 ...
11
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1answer
348 views

Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$

Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free. In them he ...
11
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0answers
479 views

What is known about the reverse mathematics of algebraic number fields?

I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...
11
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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 ...
10
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5answers
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Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prerequisites? Good introductory texts?

I'm still studying maths at undergraduate level, but intend to continue exploring topics in pure maths after I have graduated, so am thinking already about what directions I'd like to persue now, (as ...
10
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4answers
2k 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. ...
9
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2answers
463 views

Reverse mathematics below RCA

I'm sure this is a fairly basic question, but I can't seem to find a solid answer: My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to ...
9
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2answers
429 views

Z_2 versus second-order PA

These days, Peano Arithmetic ($PA$) refers to the first-order version of the axioms, where induction is only over formulas referring to natural number variables. Peano's original version of the ...
9
votes
1answer
449 views

New research on coding in reverse mathematics?

Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey ...
9
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1answer
537 views

Weakest choice principle required for Robertson-Seymour Graph Minor Theorem?

The main Robertson-Seymour Theorem states that finite graphs form a well-quasi-ordering under the graph minor relation. In other words, in every infinite set of finite graphs, there exist two graphs ...
9
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0answers
385 views

“Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
8
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3answers
617 views

truth vs. provability for ordered fields

In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
8
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2answers
856 views

Sperner's lemma and Tucker's lemma

In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proof that Tucker's ...
8
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1answer
208 views

Strength of $\Delta_1^0$ subset of $2^\mathbb{N}$ as finite union of specific basic open sets.

This question is to find the Reverse Mathematical strength of writing $\Delta_1^0$ (clopen) subset of $2^\mathbb{N}$ as a finite union $\bigcup_{\sigma \in F} [|\sigma|]$ where $F \subset ...
7
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7answers
384 views

Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$

This question came out of this other MO question of mine. My question is Is there a formal comparison between $\mathsf{RCA}_0$ and $\mathsf{BISH}$ (Bishop style constructive mathematics as used ...
7
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2answers
392 views

Proof complexity of two directions of equivalency?

This question is not precise, but I believe has a precise formulation. Consider a mathematical theorem which gives an equivalency between two conditions. As an extreme example: \begin{theorem} A ...
7
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3answers
2k views

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 ...
7
votes
3answers
441 views

Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
7
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1answer
441 views

What is the status of Cantor-Schroder-Bernstein in Reverse Math?

I'd like to know which of the set theories in SOSOA prove what versions of Cantor-Schroder-Bernstein? For my own purposes I can use arbitrarily high quantifier complexity, but I wonder how little ...
7
votes
1answer
295 views

The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...
7
votes
1answer
214 views

Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question. For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...
7
votes
0answers
163 views

Strength of claims about extensions of partial preorders and orders to linear ones

Consider these two axioms: Every partial order extends to a linear order. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: ...
7
votes
0answers
313 views

What is known of the reverse math of Riemann-Roch?

I hope this is not too trivial, but I think this may be well known to someone (not me).
7
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0answers
328 views

Fragments of Morse—Kelley set theory

Morse—Kelley set theory (hereafter MK) is the impredicative counterpart of von Neumann—Bernays—Gödel set theory (NBG), where formulas containing class quantifiers are permitted in the comprehension ...
6
votes
2answers
601 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 ...
6
votes
2answers
728 views

reverse mathematics strength of “Lipschitz functions are somewhere differentiable”

What is the reverse mathematics strength of "For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ? ...
6
votes
1answer
407 views

Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...
6
votes
1answer
92 views

Does $WKL_0$ provide more comprehension than $RCA_0$?

$WKL_0$ extends $RCA_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL_0$ Prove that there are sets which are not proven to exist by the ...
6
votes
1answer
131 views

Is 0' of PA degree relative to a non-low set?

Definitions: A set $X$ is of PA degree relative to a set $Y$ if every infinite $Y$-computable binary tree has an infinite $X$-computable path. A set $X$ is low if $X'$ is computable from ...
6
votes
1answer
192 views

Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code

It is well known that every Borel set has the property of Baire. That is, for every Borel set $B$, there is an open set $U$ and a sequence of dense open sets $D_n$ such that for every $x\in \cap_n ...
6
votes
1answer
295 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 $\Sigma^1_1$ ...
6
votes
1answer
270 views

Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?

I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
5
votes
1answer
377 views

First order consequence of a combinatorial principle

(Base theory $RCA_0$)The principle says there exists a function g such that g dominates any X-recursive function for any X in the model. i.e. For any $f\le_T X$, $\exists b\in M$ such that ...
5
votes
1answer
331 views

What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?

I am researching a logical system that is limited to $\Pi^0_2$ sentences and I am busy to prove that FOL + PA is a conservative extension of that system. Meaning that with $\Sigma^0_n$ sentences (that ...
5
votes
2answers
435 views

Did Gödel prove that the Ramified Theory of Types collapses at $\omega_1$?

Second-Order Arithmetic is considered impredicative, because the comprehension scheme allows formulas with bound second-order variables that range over all sets of natural numbers, including the set ...
5
votes
1answer
140 views

Attribution of an equivalence of the existence of omega-models of RCA0

There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of ...
5
votes
1answer
520 views

Why is adopting Russell's Axiom of Reducibility as strong as eliminating the Ramified Hierarchy?

In order to respond to concerns of impredicativity, Bertrand Russell developed a system of ramified second-order logic, which is like regular second-order logic except the comprehension schema is ...
5
votes
1answer
288 views

Constructive compactness for countable models?

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's ...
4
votes
4answers
415 views

Strength of some claims about finitely additive measures on infinite sets?

Assume ZF. Consider the claim: (1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$. Then (1) is ...
4
votes
3answers
776 views

The Reverse Mathematics of writing a set as a union?

To be more precise, a countable collection of sets $(S_n)_{n \in \mathbb{N}}$ is encoded as the row of some given set $S$, i.e. $S_n = S^{[n]}$. Futhermore, for any function from $\mathbb{N} ...
4
votes
2answers
139 views

What can be achieved by liberalizing induction for $RCA_0$?

$RCA_0$ has $\Delta_0$-comprehension and $\Sigma_1$ induction. Let $X\Sigma_{n}$ be $RCA_0$ plus $\Sigma_n$-induction and let $X\Sigma_{\omega}$-induction be $RCA_0$ plus the full induction schema. ...