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Simple cardinal characteristics of the continuum, in: Set theory of the reals
 Israel Math. Conf. Proc. 6, BarIlan Univ., Ramat
, 1993
"... Abstract. We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (Σ0 2 and, under suitable restrictions, Π0 2) are shown to have pleasant properties, related to Baire category. ..."
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Abstract. We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (Σ0 2 and, under suitable restrictions, Π0 2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) Π0 2characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah’s theorem that the dominating number is less than or equal to the independence number. 1.
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
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On ordertypes of models of arithmetic and a connection with arithmetic saturation
 Lobachevskii Journal of Mathematics
, 2004
"... Abstract. First, we study a question we encountered while exploring ordertypes of models of arithmetic. We prove that if M � PA is resplendent and the lower cofinality of M �N is uncountable then (M, <) is expandable to a model of any consistent theory T ⊇ PA whose set of Gödel numbers is arithmeti ..."
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Abstract. First, we study a question we encountered while exploring ordertypes of models of arithmetic. We prove that if M � PA is resplendent and the lower cofinality of M �N is uncountable then (M, <) is expandable to a model of any consistent theory T ⊇ PA whose set of Gödel numbers is arithmetic. This leads to the following characterization of Scott sets closed under jump: a Scott set X is closed under jump if and only if X is the set of all sets of natural numbers definable in some recursively saturated model M � PA with lcf(M � N)> ω. The paper concludes with a generalization of theorems of Kossak, Kotlarski and Kaye on automorphisms moving all nondefinable points: a countable model M � PA is arithmetically saturated if and only if there is an automorphism h: M → M moving every nondefinable point and such that for all x ∈ M, N < x < Cl ∅ � N, we have h(x)> x. 2000 Mathematical Subject Classification. 03H15, 03C62, 08A35. Key words and phrases. models of Peano arithmetic, linearly ordered sets,
The automorphism group of an arithmetically saturated model of Peano arithmetic
 J. London Math. Soc
, 1995
"... One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA ..."
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One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA is characterized up to isomorphism by two invariants: its firstorder theory Th(^) and its standard system SSy {Jt). At present, there seems to be no indication of how to recover any information about Th {Jt) from Aut {Jt) with the exception of whether or not Th {Jt) is True Arithmetic. We define the notion of arithmetically saturated in Definition 1.7; however, a model Jt of PA is arithmetically saturated if and only if it is recursively saturated and the standard cut is a strong cut. The following is our main theorem. THEOREM. Suppose that Jtx and Jt2 are countable, arithmetically saturated models of PA such that Aut(^) s Aut {Jt2). Then SSy {Jtx) = SSy {Jt2). In the Theorem, it suffices to assume that Jt2 is just recursively saturated. For, as shown by Lascar [8], if Jtx and Jt2 are countable, recursively saturated models of PA
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.