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51
Mechanizing set theory: Cardinal arithmetic and the axiom of choice
 Journal of Automated Reasoning
, 1996
"... Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this resu ..."
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Cited by 16 (9 self)
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Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving
Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice
, 1995
"... A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [23], number theory [22], group theory [25],calculus [9], etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published in the ..."
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A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [23], number theory [22], group theory [25],calculus [9], etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published
Cardinal Arithmetic in Weak Theories
, 2008
"... In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an emp ..."
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In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence
Consequences of arithmetic for set theory
 Journal of Symbolic Logic
, 1994
"... 488 revision:19930827 modified:20020716 In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider seq 1 ..."
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Cited by 3 (3 self)
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488 revision:19930827 modified:20020716 In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider seq
Cardinal arithmetic for skeptics
 Bull. Amer. Math. Soc. New Series
, 1992
"... When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency ” rather than “truth ” may be felt to give the subject an air of unreality. ..."
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Cited by 14 (4 self)
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. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of 2 ℵ0, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than
FORCING AXIOMS, SUPERCOMPACT CARDINALS, SINGULAR CARDINAL COMBINATORICS
, 2007
"... The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using idea ..."
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Cited by 1 (0 self)
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to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs. The singular cardinal problem. Cardinal arithmetic is a central subject in modern set theory and one of the key problems in this domain is to evaluate the gimel function κ ↦ → κ cof(κ) for a singular cardinal κ
Sets with Cardinality Constraints in Satisfiability Modulo Theories
"... Abstract. Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifierfree BAPA (QFBAPA). In contrast to ..."
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Cited by 6 (1 self)
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Abstract. Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifierfree BAPA (QFBAPA). In contrast
Independence and Large Cardinals
, 2010
"... The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intr ..."
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and their interrelations. Section 1 surveys the classic independence results in arithmetic and set theory. Section 2 introduces the interpretability hierarchy and describes some of its basic features. Section 3 introduces the notion of a large cardinal axiom and discusses some of the central examples. Section 4 brings
Large Cardinals and Determinacy
, 2011
"... The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measura ..."
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Cited by 1 (0 self)
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—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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Cited by 23 (2 self)
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for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number
Results 1  10
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51