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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 ..."
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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.
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 ..."
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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.
The atomic model theorem
"... 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 ..."
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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 Π 1 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.
Domination, forcing, array nonrecursiveness and relative recursive enumerability
, 2009
"... We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof ..."
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We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof of a recent result of AmbosSpies, Ding, Wang and Yu [2009] that every degree above any not in GL2 is recursively enumerable in a 1generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.
Reverse Mathematics: The Playground of Logic
, 2010
"... The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are ..."
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The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are really needed to develop classical mathematics
RAMSEY’S THEOREM FOR PAIRS AND PROVABLY RECURSIVE FUNCTIONS
"... Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 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 ..."
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Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 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 Σ0 1induction (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 ∀∃theorems. There are two components this work. The first component is a general prooftheoretic result, due to the second author ([13, 14]), 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 RT2 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 RT2 2 that are primitive recursive in f. 1.
THE STRENGTH OF THE RAINBOW RAMSEY THEOREM
, 2009
"... The Rainbow Ramsey Theorem is essentially an “antiRamsey” 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 mathe ..."
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The Rainbow Ramsey Theorem is essentially an “antiRamsey” 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 2random bounds an ωmodel 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.
STABILITY AND POSETS
"... Abstract. Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite Π 0 1 chains and antichains, in infinite computab ..."
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Abstract. Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite Π 0 1 chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite Π 0 1 chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite Π 0 1 antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA0 to WSCAC, the corresponding principle for weakly stable posets. 1.
THE MAXIMAL LINEAR EXTENSION THEOREM IN SECOND ORDER ARITHMETIC
"... Abstract. We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0. 1. ..."
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Abstract. We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0. 1.
Π 1 1CONSERVATION OF COMBINATORIAL PRINCIPLES WEAKER THAN RAMSEY’S THEOREM FOR PAIRS
"... Abstract. We study combinatorial principles weaker than Ramsey’s theorem for pairs over the system RCA0 (Recursive Comprehension Axiom) with Σ0 2bounding. It is shown that the principles of Cohesiveness (COH), Ascending and Descending Sequence (ADS), and Chain/Antichain (CAC) are all Π1 1conservat ..."
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Abstract. We study combinatorial principles weaker than Ramsey’s theorem for pairs over the system RCA0 (Recursive Comprehension Axiom) with Σ0 2bounding. It is shown that the principles of Cohesiveness (COH), Ascending and Descending Sequence (ADS), and Chain/Antichain (CAC) are all Π1 1conservative over Σ0 2bounding. In particular, none of these principles proves Σ0 2induction. Key words. Reverse mathematics, Π1 1conservation, RCA0, Σ0 2