We study the strength of RRT32, Rainbow Ramsey Theorem for colorings of triples, and prove that RCA0 + RRT32 implies neither WKL0 nor RRT42 source. To this end, we establish some recursion theoretic properties of cohesive sets and rainbows for colorings of pairs. We show that every sequence admits a cohesive set of non-PA Turing degree; and that every ∅′-recursive sequence admits a low3 cohesive set.

Answering a question in the reverse mathematics of combinatorial principles, we prove that the thin set theorem for pairs ) implies the diagonally noncomputable set principle over the base axiom system $\mathrm{RCA}_{0}$.

Previously, both Soare and Simpson considered sets without subsets of higher -degree. Cintioli and Silvestri, for a reducibility , define the concept of a -introimmune set. For the most common reducibilities , a set does not contain subsets of higher -degree if and only if it is -introimmune. In this paper we consider -introimmune and -introimmune sets and examine how structurally easy such sets can be. In other words we ask, What is the smallest class of the Kleene's Hierarchy containing (...) -introimmune sets for ? We answer the question by proving the existence of -introimmune sets in the class , bi--introimmune sets in , and bi--introimmune sets in. (shrink)

Ramsey’s theorem asserts that every k-coloring of $[\omega ]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable k-coloring of $[\omega ]^n$ whose solutions compute the halting set. On the other hand, for every computable k-coloring of $[\omega ]^2$ and every noncomputable set C, there is an infinite monochromatic set H such that $C \not \leq _T H$. The latter property is known as cone avoidance.In this article, we design a natural class of Ramsey-like theorems encompassing (...) many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every k-coloring of $[\omega ]^n$, of an infinite subdomain $H \subseteq \omega $ over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions. (shrink)

Ramsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave the same color. The strength of consequences of Ramsey’s theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey’s theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose (...) during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey’s theorem. (shrink)

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we study the infinite pigeonhole pr...

A set of integers A is computably encodable if every infinite set of integers has an infinite subset computing A. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this article, we extend this notion of computable encodability to subsets of the Baire space, and we characterize the Π10-encodable compact sets as those which admit a nonempty Σ11-subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly computably reducible to (...) Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch. (shrink)

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω (...) with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

The thin set theorem for n-tuples and k colors states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$, nor the free set theorem imply the Erdős–Moser theorem whenever k is sufficiently large. Given a problem $\mathsf {P}$, a computable instance of $\mathsf {P}$ is universal (...) iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance. (shrink)

In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 - (...) CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the (...) resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

We study the first-order consequences of Ramsey’s Theorem fork-colourings ofn-tuples, for fixed$n, k \ge 2$, over the relatively weak second-order arithmetic theory$\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of$\mathrm {RCA}^*_0$that does not satisfy$\Sigma ^0_1$induction,$\mathrm {RT}^n_k$is equivalent to its relativization to any proper$\Sigma ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe.We give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$for$n \ge (...) 3$. We show that they form a non-finitely axiomatizable subtheory of$\mathrm {PA}$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _{\ell +3}$fragment for$\ell \ge 1$lies between$\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$and$\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$form a subtheory of$\mathrm {I} \Sigma _2$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _4$fragment is strictly weaker than$\mathrm {B} \Sigma _2$but not contained in$\mathrm {I} \Sigma _1$.Additionally, we consider a principle$\Delta ^0_2$-$\mathrm {RT}^2_2$which is defined like$\mathrm {RT}^2_2$but with both the$2$-colourings and the solutions allowed to be$\Delta ^0_2$-sets rather than just sets. We show that the behaviour of$\Delta ^0_2$-$\mathrm {RT}^2_2$over$\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$is in many ways analogous to that of$\mathrm {RT}^2_2$over$\mathrm {RCA}^*_0$, and that$\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$-$\mathrm {RT}^2_2$is$\Pi _4$- but not$\Pi _5$-conservative over$\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of$\Pi _5$-conservativity is not provable in$\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)

It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible in (...) various senses. (shrink)

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show (...) that for all n ≥ 2, there exists a Δ 0 n+1 n-cohesive set. We improve this result for n = 2 by showing that there is a Π 0 2 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0 (n+1) and also characterize the jumps of the n-r-cohesive degrees. (shrink)

We examine the reverse mathematical strength of a variation of Hindman’s Theorem (HT) constructed by essentially combining HT with the Thin Set Theorem to obtain a principle that we call thin-HT. This principle states that every coloring c:N→N has an infinite set S⊆N whose finite sums are thin for c, meaning that there is an i with c(s)≠i for all nonempty sums s of finitely many distinct elements of S. We show that there is a computable instance of thin-HT such (...) that every solution computes ∅′, as is the case with HT, as shown by Blass, Hirst, and Simpson (1987). In analyzing this proof, we deduce that thin-HT implies ACA0 over RCA0+IΣ20. On the other hand, using Rumyantsev and Shen’s computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to ∅′. Hence there is a computable instance of this restriction of thin-HT with no Σ20 solution. (shrink)

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see (...) text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman. (shrink)

In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically saturated models of Peano (...) Arithmetic such that Aut(M1)≅ Aut(M2).ThenSSy(M1) = SSy(M2).We show that ifMis a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1,M2be countable arithmetically saturated models of Peano Arithmetic such thatAut(M1) ≅ Aut(M2).Then for every n<ωHere RT2nis Infinite Ramsey's Theorem stating that every 2-coloring of [ω]nhas an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic. (shrink)

The present volume is an introduction to the use of tools from computability theory and reverse mathematics to study combinatorial principles, in particular Ramsey's theorem and special cases such as Ramsey's theorem for pairs. It would serve as an excellent textbook for graduate students who have completed a course on computability theory.

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of (...) pairs admits a pair of low₂ infinite homogeneous sets whose degrees form a minimal pair. (shrink)

We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for ${\Sigma^0_2}$ formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs.

A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a noncomputable set A and a computable instance of a problem ${\mathsf {P}}$, to find a solution relative to which A (...) is still noncomputable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of one cone, of $\omega $ cones, of one hyperimmunity or of one non- $\Sigma ^{0}_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non- $\Sigma ^{0}_1$ definitions coincide. On the other hand, none of these notions coincide in a nonrelativized setting. (shrink)

We present and analyze \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{ACA}_0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$$\end{document}, whereas Mathias forcing does not. We also show that the needed reals for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing. (shrink)

The Rainbow Ramsey Theorem is essentially an "anti-Ramsey" theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA₀ of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey's Theorem for pairs over RCA₀. The separation involves techniques (...) from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin's theorem that the hyperimmune degrees have measure one. (shrink)

A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to $\mathit{ACA}_{0}$ over $\mathit{RCA}_{0}$. A characterization theorem for finite-level partial orderings with Ramsey properties has been proven by the (...) second author. We show, over $\mathit{RCA}_{0}$, that one direction of the equivalence given by this theorem is equivalent to $\mathit{ACA}_{0}$, and the other is provable in $\mathit{ATR}_{0}$. We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over $\mathit{RCA}_{0}$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to $\mathit{ACA}_{0}$. (shrink)

We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$ . Four natural formulations are presented, and their relative strengths are compared. In the analysis of the pigeonhole principle for $\omega^{2}$ , we uncover two weak variants of Ramsey’s theorem for pairs.

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is (...) not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkndenote Ramsey's theorem fork–colorings ofn–element sets, and let RT<∞ndenote (∀k)RTkn. Our main result on computability is: For anyn≥ 2 and any computable (recursive)k–coloring of then–element sets of natural numbers, there is an infinite homogeneous setXwithX″ ≤T0(n). LetIΣnandBΣndenote the Σninduction and bounding schemes, respectively. Adapting the casen= 2 of the above result (whereXis low2) to models of arithmetic enables us to show that RCA0+IΣ2+ (...) RT22is conservative over RCA0+IΣ2for Π11statements and that RCA0+IΣ3+ RT<∞2is Π11-conservative over RCA0+IΣ3. It follows that RCA0+ RT22does not implyBΣ3. In contrast, J. Hirst showed that RCA0+ RT<∞2does implyBΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2is strictly stronger than RT22overRCA0. (shrink)

We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n -generics and weak n -generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n -generic with n≥2n≥2 then it satisfies the jump property G≡TG′⊕∅G≡TG′⊕∅. We prove that every such G has generalized high Turing degree, and so cannot have even (...) Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real. (shrink)

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 cardinalityn≥ 1 and an arbitrary finite number of colorsk≥ 2 is equivalent to then-th jump of weak Kőnig’s lemma. In particular, Ramsey’s theorem for cardinalityn≥ 1 is${\bf{\Sigma }}_{n + 2}^0$-measurable in (...) the effective Borel hierarchy, but not${\bf{\Sigma }}_{n + 1}^0$-measurable. Secondly, we obtain interesting lower bounds, for instance then-th jump of weak Kőnig’s lemma is Weihrauch reducible to Ramsey’s theorem of cardinalityn+ 2 forn≥ 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 nonuniform 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 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. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...) of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)