Results 1  10
of
50
Combinatorial Principles Weaker than Ramsey's Theorem for Pairs
"... We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order ha ..."
Abstract

Cited by 26 (7 self)
 Add to MetaCart
We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (ChainAntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs (RT
Foundations of BQO Theory and Subsystems of SecondOrder Arithmetic
 Pennsylvania State University
, 1993
"... Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open pro ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open problems about the axiomatic strength of both elementary and advanced results. A quasiordering (i.e. a reflexive and transitive binary relation) is awqo (well quasiordering) if it contains no infinite descending chains and no infinite sets of pairwise incomparable elements. This concept is very natural, and has been introduced several times, as documented in [19]. The usual working definition of wqo is obtained from the one given above with an application of Ramsey’s theorem: a quasiordering on the setQ is wqo if for every sequence {xn  n ∈ N}of elements ofQ there existm < n such thatxm xn. The notion of bqo (better quasiordering) is a strengthening of wqo which was introduced byNashWilliams in the 1960’s in a sequence of papers culminating in [30] and [31]. This notion has proved to be very useful in showing that specific quasiorderings are indeed wqo. Moreover the property of being bqo is preserved
Partition Theorems and Computability Theory
 Bull. Symbolic Logic
, 2004
"... The computabilitytheoretic and reverse mathematical aspects of various combinatorial principles, such as König’s Lemma and Ramsey’s Theorem, have received a great deal of attention and are active areas of research. We carry on this study of effective combinatorics by analyzing various partition the ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
The computabilitytheoretic and reverse mathematical aspects of various combinatorial principles, such as König’s Lemma and Ramsey’s Theorem, have received a great deal of attention and are active areas of research. We carry on this study of effective combinatorics by analyzing various partition theorems (such as Ramsey’s Theorem) with the aim of understanding the complexity of solutions to computable instances in terms of the Turing degrees and the arithmetical hierarchy. Our main focus is the study of the effective content of two partition theorems allowing infinitely many colors: the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our results on the complexity of solutions rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma, and these connections will be emphasized. We also study Ramsey degrees, i.e. those Turing degrees which are able to compute homogeneous sets for every computable 2coloring of pairs of natural numbers, in an attempt to further understand the effective content of Ramsey’s Theorem for exponent 2. We establish some new results about these degrees, and obtain as a corollary the nonexistence of a “universal ” computable 2coloring of pairs of natural numbers.
The canonical Ramsey theorem and computability theory
"... Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable in ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma. 1
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... 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 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
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 Hilbertstyle 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 modeltheoretic arguments.
The atomic model theorem and type omitting
 Trans. Amer. Math. Soc
"... We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π1 1conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ωmodel consisting of the recursive sets.
On a conjecture of Dobrinen and Simpson concerning almost everywhere domination
, 2005
"... Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the L ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article,
THE STRENGTH OF SOME COMBINATORIAL PRINCIPLES RELATED TO RAMSEY’S THEOREM FOR PAIRS
"... We study the reverse mathematics and computabilitytheoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2coloring of pa ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
We study the reverse mathematics and computabilitytheoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT 2 2 does not imply RT 2 2.