Term extraction and Ramsey's theorem for pairs

Kreuzer, Alexander; Kohlenbach, Ulrich

doi:10.2178/jsl/1344862165

Cited by 10 publications

(13 citation statements)

References 35 publications

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“…Using Theorem one can extend the conservation and program extraction results obtained in and to

{AA}_{weak}

and obtain the following theorem. Theorem The theory

{sans-serifWKL}_{0} + Σ_{2}^{0} - IA + {sans-serifAA}_{sans-serifweak} i s Π_{1}^{1}

‐conservative over

{sans-serifRCA}_{0} + I Σ_{2}^{0}

. From a proof of a sentence of the form

\forall f \in N^{N} \exists x \in double-struckN A_{qf} (f, x)

in the system

{sans-serifWKL}_{0}^{ω} + Π_{1}^{0} - CP + {sans-serifAA}_{sans-serifweak}

one can extract a primitive recursive term t realizing x , i.e., a term t such that

\forall f A_{qf} (f, t (f))

holds. In particular,

{AA}_{weak}

Π_{2}^{0}

‐conservative over

sans-serifPRA

. Proof For 1), cf.…”

Section: Uniform Equicontinuity Versus Equicontinuitymentioning

confidence: 89%

From Bolzano‐Weierstraß to Arzelà‐Ascoli

Kreuzer

2014

Mathematical Logic Qtrly

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Abstract. We show how one can obtain solutions to the Arzelà-Ascoli theorem using suitable applications of the Bolzano-Weierstraß principle. With this, we can apply the results from [9] and obtain a classification of the strength of instances of the Arzelà-Ascoli theorem and a variant of it.Let AA be the statement that each equicontinuous sequence of functions fn : [0, 1] → [0, 1] contains a subsequence that converges uniformly with the rate 2 −k and let AA weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate.We show that AA is instance-wise equivalent over RCA 0 to the BolzanoWeierstraß principle BW and that AA weak is instance-wise equivalent over WKL 0 to BW weak , and thus to the strong cohesive principle (StCOH). Moreover, we show that over RCA 0 the principles AA weak , BW weak + WKL and StCOH + WKL are equivalent. We will give two different formalizations of this theorem, show how these can be reduced to suitable instances of the Bolzano-Weierstraß principle and, using this, obtain a classification of them in the sense of reverse mathematics and computable analysis. Bolzano-WeierstraßIn [9] we investigated the strength of the following two variants of the BolzanoWeierstraß principle:• The (strong) Bolzano-Weierstraß principle (BW) is the statement that each bounded sequence of real numbers contains a subsequence converging at the rate 2 −k . (This is the usual formulation in reverse mathematics. The rate 2 −k stems from the fact that real numbers are coded as sequences that converge at this rate. However, 2 −k is just an arbitrarily chosen rate. In fact, one can easily convert a sequence converging at a given rate into a sequence converging at any other given rate.)2010 Mathematics Subject Classification. Primary 03F60; Secondary 03D80, 03B30.

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“…Using Theorem one can extend the conservation and program extraction results obtained in and to

{AA}_{weak}

and obtain the following theorem. Theorem The theory

{sans-serifWKL}_{0} + Σ_{2}^{0} - IA + {sans-serifAA}_{sans-serifweak} i s Π_{1}^{1}

‐conservative over

{sans-serifRCA}_{0} + I Σ_{2}^{0}

. From a proof of a sentence of the form

\forall f \in N^{N} \exists x \in double-struckN A_{qf} (f, x)

in the system

{sans-serifWKL}_{0}^{ω} + Π_{1}^{0} - CP + {sans-serifAA}_{sans-serifweak}

one can extract a primitive recursive term t realizing x , i.e., a term t such that

\forall f A_{qf} (f, t (f))

holds. In particular,

{AA}_{weak}

Π_{2}^{0}

‐conservative over

sans-serifPRA

. Proof For 1), cf.…”

Section: Uniform Equicontinuity Versus Equicontinuitymentioning

confidence: 89%

From Bolzano‐Weierstraß to Arzelà‐Ascoli

Kreuzer

2014

Mathematical Logic Qtrly

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“…Proof. Using Fact 2.3, Corollary 3.3, the lower bound from Corollary 3.7, the upper bound from Corollary 4.15 and the fact that parallelization is a closure operator we obtain the following reduction chain for n ≥ 2, k ≥ 2, N: 5 This approach has been used in [34, Proposition 6.6.1]; for a proof theoretic analysis of Ramsey's theorem this method has been applied in [28,29,1], and in reverse mathematics it has been used to prove that Ramsey's theorem is provable over ACA 0 ; see [19] for the Erdős-Rado method in general. 6 This property has been called "join-irreducible" in previous publications, but formally it is a strengthening of join-irreducibility in the lattice theoretic sense.…”

Section: Jumps Increasing Cardinality and Color And Upper Boundsmentioning

confidence: 99%

On the Uniform Computational Content of Ramsey’s Theorem

Brattka¹,

Rakotoniaina²

2017

J. symb. log.

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We study the uniform computational content of Ramsey's theorem in the Weihrauch lattice. Our central results provide information on how Ramsey's theorem behaves under product, parallelization and jumps. From these results we can derive a number of important properties of Ramsey's theorem. For one, the parallelization of Ramsey's theorem for cardinality n ≥ 1 and an arbitrary finite number of colors k ≥ 2 is equivalent to the n-th jump of weak Kőnig's lemma. In particular, Ramsey's theorem for cardinality n ≥ 1 is Σ 0 n+2 -measurable in the effective Borel hierarchy, but not Σ 0 n+1 -measurable. Secondly, we obtain interesting lower bounds, for instance the n-th jump of weak Kőnig's lemma is Weihrauch reducible to (the stable version of) Ramsey's theorem of cardinality n+2 for n ≥ 2. We prove that with strictly increasing numbers of colors Ramsey's theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey's theorem that are indistinguishable from a non-uniform perspective play an important role. For instance, the colored version of Ramsey's theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey's theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey's theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey's theorem, and we show that there are some significant differences with the nonuniform situation in reverse mathematics.

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“…The functionals of T 1 are said to be primitive recursive functionals in the sense of Kleene, in contrast to the functionals of T , which are are said to be primitive recursive functionals in the sense of Gödel (see [8,2,10]). The results of Howard [7] show the following: (See also [17,Section 10], which relates Howard's results explicitly to the fragments T n .) Theorem 2.4 implies that if M is a primitive recursive functional in the sense of Kleene, then M ′ is of level T 2 (which is to say, roughly Ackermannian).…”

Section: A Combinatorial Factmentioning

confidence: 97%

A metastable dominated convergence theorem

Avigad¹,

Dean²,

Rute³

2012

JLA

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The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( fn) converges to the integral of the pointwise limit. Tao [20] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space. We prove a slight strengthening of Tao's theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov's theorem, and introduce a new mode of convergence related to these notions.

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Term extraction and Ramsey's theorem for pairs

Cited by 10 publications

References 35 publications

From Bolzano‐Weierstraß to Arzelà‐Ascoli

From Bolzano‐Weierstraß to Arzelà‐Ascoli

On the Uniform Computational Content of Ramsey’s Theorem

A metastable dominated convergence theorem

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