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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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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
On ordertypes of models of arithmetic and a connection with arithmetic saturation
 Lobachevskii Journal of Mathematics
, 2004
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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.
CONSTRUCTING κLIKE MODELS OF ARITHMETIC
"... A model (M,�,…)isκlike if M has cardinality κ but, for all a�M, the cardinality of �x�M: x�a� is strictly less than κ. In this paper we shall give constructions of κlike models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The ..."
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A model (M,�,…)isκlike if M has cardinality κ but, for all a�M, the cardinality of �x�M: x�a� is strictly less than κ. In this paper we shall give constructions of κlike models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M�Π � �Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κlike; also hierarchical variants of this result for Π n �Th(PA); and (2) for every n�1, every singular κ and every M�BΣ n �exp��IΣ n there is a κlike model K elementarily equivalent to M.