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22
A classification of rapidly growing Ramsey functions
 PROC. AMER. MATH. SOC
, 2003
"... Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf i ..."
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Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=i  H −1 α (i) where  i h denotes the htimes iterated binary length of i and H−1 α denotes the inverse function of the αth member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.
On endextensions of models of : exp
 Math. Log. Quart
, 1996
"... Every model of I∆0 is the tally part of a model of the stringlanguage theory ThFO (a main feature of which consists in having induction on notation restricted to certain AC 0 sets). We show how to “smoothly ” introduce in ThFO the binary length function, whereby it is possible to make exponential ..."
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Every model of I∆0 is the tally part of a model of the stringlanguage theory ThFO (a main feature of which consists in having induction on notation restricted to certain AC 0 sets). We show how to “smoothly ” introduce in ThFO the binary length function, whereby it is possible to make exponential assumptions in models of ThFO. These considerations entail that every model of I∆0 + ¬exp is a proper initial segment of a model of ThFO and that a modicum of bounded collection is true in these models.
RANDOM REALS, THE RAINBOW RAMSEY THEOREM, AND ARITHMETIC CONSERVATION
, 2012
"... We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2RAN be the principle that for every real X there is a real R which is 2random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be impleme ..."
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We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2RAN be the principle that for every real X there is a real R which is 2random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the CsimaMileti fact that the existence of random reals has infinitarycombinatorial consequences we can conclude that 2RAN has nontrivial arithmetic consequences. In Section 4, we show that 2RAN is conservative over RCA0 + BΣ2 for Π1 1sentences. Thus, the set of firstorder consequences of 2RAN is strictly stronger than P − + I Σ1 and no stronger than P − + BΣ2.
The automorphism group of a countable recursively saturated structure
 Proceedings of the London Mathematical Society, Series 3 , 65:225244
, 1992
"... The automorphism groups of K0categorical structures have been studied extensively by both permutation group theorists and model theorists, and this collaboration has turned out to be very fruitful. (See, for example, [10,6,2].) The notion of a recursively saturated structure generalizes that of a ( ..."
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The automorphism groups of K0categorical structures have been studied extensively by both permutation group theorists and model theorists, and this collaboration has turned out to be very fruitful. (See, for example, [10,6,2].) The notion of a recursively saturated structure generalizes that of a (countable)
On embedding models of arithmetic of cardinality ℵ1 into reduced powers
 Fundamenta Mathematicae, Volume 176, Issue
"... In the early 1970’s S.Tennenbaum proved that all countable models of PA − + ∀1 − Th(N) are embeddable into the reduced product Nω/F, where F is the cofinite filter. In this paper we show that if M is a model of PA − + ∀1 − Th(N), and M  = ℵ1, then M is embeddable into Nω/D, where D is any regula ..."
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In the early 1970’s S.Tennenbaum proved that all countable models of PA − + ∀1 − Th(N) are embeddable into the reduced product Nω/F, where F is the cofinite filter. In this paper we show that if M is a model of PA − + ∀1 − Th(N), and M  = ℵ1, then M is embeddable into Nω/D, where D is any regular filter on ω.
On Wright’s inductive definition of coherence truth for arithmetic
 Analysis
"... As the first illustration of a potential satisfier for the ‘platitudes for truth ’ in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of ‘truth conceived as coherence ’ for arithmetic. This paper attempts to clarify certain a ..."
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As the first illustration of a potential satisfier for the ‘platitudes for truth ’ in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of ‘truth conceived as coherence ’ for arithmetic. This paper attempts to clarify certain aspects of Wright’s proposal. Take the standard firstorder language of arithmetic L. 1 Let B be some axiom system for arithmetic, which Wright calls the ‘coherence base’. With small notational modifications, Wright proposes the following inductive definition of the concept ‘coheres with B ’ (for Lsentences): (CAt) If ϕ is atomic, then ϕ coheres with B iff B � ϕ. (C¬) ¬ϕ coheres with B iff ϕ does not cohere with B. (C∧) ϕ ∧ ψ coheres with B iff ϕ and ψ cohere with B. (C∨) ϕ ∨ ψ coheres with B iff either ϕ or ψ coheres with B. (C→) ϕ → ψ coheres with B iff either ϕ does not or ψ does cohere with B. (C∀) ∀xϕ coheres with B iff, for each number n, ϕ(n) coheres with B. (C∃) ∃xϕ coheres with B iff, for some number n, ϕ(n) coheres with B. First, note that except for the basis clause (CAt) this is the same as the usual Tarskian inductive definition of truth for arithmetic: 2 (TAt) If ϕ has the form t = u, then ϕ is true iff val(t) = val(u). 3 1 The terms of L are defined recursively from a basis of variables, the constant 0 and the operation symbols s, + and ×. The numerals of L are written n, meaning 0 prefixed by n occurrences of the successor symbol s. The atomic formulas of L are equations of the form t = u (with t, u terms) and complex formulas of L are defined by recursion on complexity as usual. Below, Sent(L) is the set of Lsentences of L, AtSent(L) is the set of atomic Lsentences and Form(L) is the set of Lformulas. In the arithmetic formalization of semantics, we use SentL(x) and ClTmL(x) to mean arithmetic formulas expressing respectively that x is (the code of) a sentence of L or that x is (the code of) a closed term of
Infinite finitely generated fields are biinterpretable with N
"... Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theore ..."
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Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fields of curves, we show each infinite finitely generated field considered in the ring language is parametrically biïnterpretable with N. As a consequence, for any finitely generated field there is a firstorder sentence in the language of rings which is true in that field but false in every other finitely generated field, and, hence, Pop’s conjecture that elementarily equivalent finitely generated fields are isomorphic is true.
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of
 Institute for Mathematical Sciences, National University of Singapore, World Scientific
"... In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these ar ..."
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In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these areas. We begin with variations on König’s Lemma, and give an introduction to reverse mathematics and related parts of computability theory. We then focus on Ramsey’s Theorem as a case study in the computability theoretic and reverse mathematical analysis of combinatorial principles. We study Ramsey’s Theorem for Pairs (RT22) in detail, focusing on fundamental tools such as stability, cohesiveness, and Mathias forcing; and on combinatorial and model theoretic consequences of RT22. We also discuss the important theme of conservativity results. In the final section, we explore several topics that reveal various aspects of computable mathematics and reverse mathematics. An appendix contains a proof of Liu’s recent result that RT22 does not imply Weak König’s Lemma. There are exercises and open questions throughout the article. ∗Please send any corrections to
On the Independence of Goodstein's Theorem
"... In this undergraduate thesis the independence of Goodstein's Theorem from Peano arithmetic (PA) is proved, following the format of the rst proof, by Kirby and Paris. All the material necessary for its understanding is developed, beginning with the foundations of set theory, followed by ordin ..."
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In this undergraduate thesis the independence of Goodstein's Theorem from Peano arithmetic (PA) is proved, following the format of the rst proof, by Kirby and Paris. All the material necessary for its understanding is developed, beginning with the foundations of set theory, followed by ordinal numbers and a proof of Goodstein's Theorem, and concluded with basic model theory and the independence proof. Other relevant information, such as an outline for an alternative independence proof and an application to dynamical systems is also included. Contents 1.