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A ∆ 2 2 WellOrder of the Reals And Incompactness of L(Q MM)
"... 403 revision:19981215 modified:19981217 Preface A forcing poset of size 22ℵ1 which adds no new reals is described and shown to provide a ∆2 2 definable wellorder of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this wellorder ..."
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order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees. The paper also deals with the MagidorMalitz logic: it is consistent that this logic is highly non compact. This paper deals with three issues: the question of definable wellorders of the reals
and
, 2001
"... Let HC ′ denote the set of sets of hereditary cardinality less than 2ω. We consider reflection principles for HC ′ in analogy with the Levy reflection principle for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x1,..., xn) is a property which is provably persist ..."
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Let HC ′ denote the set of sets of hereditary cardinality less than 2ω. We consider reflection principles for HC ′ in analogy with the Levy reflection principle for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x1,..., xn) is a property which is provably
The downward transfer of elementary satisfiability of partition logics
 Mathematical Logic Quarterly
"... a set into several disjoint nonempty subsets, so that the elements in each partition subset are homogeneous or indistinguishable with respect to some given properties. H.D.Ebbinghaus first (in 1991) distilled from this phenomenon, which is a monadic second order property in nature, a special kind ..."
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to Malitz quantifiers [6] if we restrict them with some infinite cardinality requirements. However the latter one appeared earlier and their backgrounds are also different. There are several types of partition quantifiers, such as 2partition or multipartition, monadic or nonmonadic type. When augmenting
Absoluteness, truth, and quotients
"... The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘settheoretic infinite ’ that concerns me here. Once the existence of an infinite set is accept ..."
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The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘settheoretic infinite ’ that concerns me here. Once the existence of an infinite set is accepted, the axioms of set theory imply the existence of a transfinite hierarchy of larger and larger orders of infinity. I shall review some wellknown facts about the influence of these axioms of infinity ([28]) to the everyday mathematical practice and point out to some, as of yet not understood, phenomena at the level of the thirdorder arithmetic. Technical details from both set theory and operator algebras are kept at the bare minimum. In the Appendix I include definitions of arithmetical and analytical hierarchies in order to make this paper more accessible to nonlogicians. In this paper I am taking a position intermediate between pluralism and nonpluralism (as defined in [34]) with an eye for applications outside of set theory. Acknowledgments This paper is partly based on my talks at the ‘Truth and Infinity ’ workshop (IMS, 2011) and the ‘Connes Embedding Problem ’ workshop (Ottawa, 2008). I would like to thank the organizers of both meetings. Another driving force for this paper—and much of my work—originated in conversations with functional analysts, too numerous to list here, over the past several
Mathematisches Forschungsinstitut Oberwolfach Report No. 55/2005 Set Theory
, 2005
"... Abstract. This meeting covered all important aspects of modern Set Theory, including large cardinal theory, combinatorial set theory, descriptive set theory, connections with algebra and analysis, forcing axioms and inner model theory. The presence of an unusually large number (19) of young research ..."
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researchers made the meeting especially dynamic. Mathematics Subject Classification (2000): 03Exx. Introduction by the Organisers This meeting was organised by SyDavid Friedman (University of Vienna), Menachem Magidor (Hebrew University, Jerusalem) and Hugh Woodin (University of California, Berkeley
ABSOLUTENESS FOR UNIVERSALLY BAIRE SETS AND THE UNCOUNTABLE I
, 2006
"... Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as t ..."
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Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as the Borel Conjecture, Whitehead’s problem, and automatic continuity for Banach Algebras, were proved independent, perhaps leaving an impression that most nontrivial statements about infinite sets can be neither proved nor refuted in ZFC. 1 Moreover, some classical statements imply the consistency of ZFC and stronger theories, and by Gödel’s incompleteness theorems the consistency of these statements with ZFC can be proved only by using strong axioms of infinity, socalled large cardinal axioms. A classical example is Banach’s ‘Lebesgue measure has a σadditive extension to all sets of reals. ’ While it is fairly easy to find a model in which this is false and there is no known ZFCproof of its negation, proving the consistency of this statement requires assuming the existence of a measurable cardinal ([32]). A remarkable result was proved by Shoenfield ([31]): every statement of the form
Laver and Set Theory
, 2014
"... orist of remarkable breadth and depth, and his tragic death from Parkinson’s disease a month shy of his 70th birthday occasions a commemorative and celebratory account of his mathematical work, work of an individual stamp having considerable significance, worth, and impact. Laver established substa ..."
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orist of remarkable breadth and depth, and his tragic death from Parkinson’s disease a month shy of his 70th birthday occasions a commemorative and celebratory account of his mathematical work, work of an individual stamp having considerable significance, worth, and impact. Laver established substantial results over a broad range in set theory from those having the gravitas of resolving classical conjectures through those about an algebra of elementary embeddings that opened up a new subject. There would be crisp observations as well, like the one, toward the end of his life, that the ground model is actually definable in any generic extension. Not only have many of his results as facts become central and pivotal for set theory, but they have often featured penetrating methods or conceptualizations with potentialities that were quickly recognized and exploited in the development of the subject as a field of mathematics. In what follows, we discuss Laver’s work in chronological order, bringing out the historical contexts, the mathematical significance, and the impact on set theory. Because of his breadth, this account can also be construed as a