Effectiveness for infinite variable words and the Dual Ramsey Theorem

Ramsey's Theorem deals with colorings of the k-element subsets of a fixed infinite set. The Dual Ramsey Theorem deals with colorings of the k-element partitions of a fixed infinite set. We review the statement of the Dual Ramsey Theorem and of its generalization to A-partitions. Here A is a finite alphabet. We discuss the problem of optimal recursion-theoretic bounds for the Dual Ramsey Theorem. We point out applications of the Dual Ramsey Theorem to initial segment constructions in degree theory. We state a result of Brackin concerning partitions which have no coarsenings of higher Turing degree. In an appendix we summarize the known results (and present two new results) concerning nearly recursive (i.e. bi-immune free) Turing degrees.

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

This paper investigates the determinacy and the complete determinacy of infinite games, following reverse mathematics program whose purpose is to find the set comprehension axioms that are necessary and sufficient for these statements in the frame of second order arithmetic. In some sense, this research clarifies how complex oracles we need to obtain the algorithms which give a winning strategies and which determine the winning positions for the players. It will be shown that, depending on the complexity of the rules of games, the complexity of the oracles changes drastically and that determinacy and complete determinacy statements are not always equivalent.

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak Konig's lemma) and prove that it is strictly weaker thanWKL (weak Konig's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive on open sets.

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay.We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

The uniform content of partial and linear orders

approach to finite Ramsey theory and a self-dual Ramsey theorem

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prove the Ordered Variable word for binary strings in ACA0.

On some variations of coloring problems of infinite words

In 2012 M. Sokić proved that the that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić.Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permutations has the dual Ramsey property as well. It was Leeb who pointed out in 1970 that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. In this paper we argue that this is even more the case when dealing with the dual Ramsey property.

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains.<br></p><p>The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element.<br></p><p>After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

We continue our research on the relative strength ofL-like combinatorial principles for successors of singular cardinals. In [3] we have shown that the existence of a λ+-special Aronszajn tree does not follow from that of a λ+-Souslin tree. It follows from [5], [4] and [6] that under G.C.H. □λ does imply the existence of a λ+-Souslin tree. In [2] we show that □λ does not follow from the existence of a λ+-special Aronszajn tree. Here we show that the existence of such a tree implies that of an ‘almost Souslin’ λ+-tree. It follows that the statement “All λ+-Aronszajn trees are special” implies that there are no λ+-Aronszajn trees.

A factorisation x = u 1 u 2 ⋯ $x = u_1 u_2 \cdots$ of an infinite word x $x$ on alphabet X $X$ is called ‘monochromatic’, for a given colouring of the finite words X ∗ $X^*$ on alphabet X $X$ , if each u i $u_i$ is the same colour. Wojcik and Zamboni proved that the word x $x$ is periodic if and only if for every finite colouring of X ∗ $X^*$ there is a monochromatic factorisation of x $x$ . On the other hand, it follows from Ramsey's theorem that, for any word x $x$ , for every finite colouring of X ∗ $X^*$ there is a suffix of x $x$ having a monochromatic factorisation. A factorisation x = u 1 u 2 ⋯ $x = u_1 u_2 \cdots$ is called ‘super-monochromatic’ if each word u k 1 u k 2 ⋯ u k n $u_{k_1} u_{k_2} \cdots u_{k_n}$ , where k 1 < ⋯ < k n $k_1 < \cdots < k_n$ , is the same colour. Our aim in this paper is to show that a word x $x$ is eventually periodic if and only if for every finite colouring of X ∗ $X^*$ there is a suffix of x $x$ having a super-monochromatic factorisation. Our main tool is a Ramsey result about alternating sums that may be of independent interest.

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.

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&lt;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.