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51
COUNTING SMALL SETS IN WEAK BOUNDED ARITHMETIC
"... ABSTRACT. We define a weak first order theory for $\mathrm{A}\mathrm{C}^{0} $ with an auxiliary axiom scheme which counts the cardinality of small sets defined by some $\mathrm{A}\mathrm{C}^{0} $ relation. The main result is that definable functions of this theory is exactly those which are $\mathrm ..."
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ABSTRACT. We define a weak first order theory for $\mathrm{A}\mathrm{C}^{0} $ with an auxiliary axiom scheme which counts the cardinality of small sets defined by some $\mathrm{A}\mathrm{C}^{0} $ relation. The main result is that definable functions of this theory is exactly those which
Ordinal Analyzes and Large Cardinals Abstract
, 2005
"... One of the major topics of proof theoretical research are ordinal analyzes of axiom system. Ordinal analyzes go back to Gentzen’s 1943 paper [1] in which he proved that the order–type of any elementarily definable ordering whose well–foundedness is provable from the axioms of Peano arithmetic has an ..."
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One of the major topics of proof theoretical research are ordinal analyzes of axiom system. Ordinal analyzes go back to Gentzen’s 1943 paper [1] in which he proved that the order–type of any elementarily definable ordering whose well–foundedness is provable from the axioms of Peano arithmetic has
LOGICAL DREAMS
, 2003
"... We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic. ..."
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Cited by 6 (0 self)
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We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic.
On the Relationship between Choice Schemes and Iterated Class Comprehension in Set Theory
"... The aim of this thesis is to show a few specific results about extensions of Von Neumann–Bernays–Gödel set theory NBG, by applying prooftheoretic techniques. We get the main results in a uniform way, by using cutelimination and asymmetric interpretations. The same technique was applied a few decad ..."
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decades ago, to analogous systems of second order arithmetic, by Cantini [1]. We consider natural extensions of NBG by a few axiom schemes, i.e., choice AC [Σ 1 1], dependent choice DC [Σ 1 1], full induction TI ∈[L 1], and iterated elementary comprehension (CA[Π 1 0])<c. And we are going to establish
A quantitative ergodic theory proof of Szemerédi’s theorem
, 2004
"... A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combin ..."
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Cited by 55 (15 self)
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, selfcontained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely
The continuum hypothesis
"... In this paper we prove the continuum hypothesis with categorical logic, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and that, conversely, the theorems ..."
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nition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the GödelBernaysvon Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal
Chapter 12 Rough Sets and Rough Logic: A KDD Perspective
"... Abstract Basic ideas of rough set theory were proposed by Zdzis law Pawlak [85, 86] in the early 1980’s. In the ensuing years, we have witnessed a systematic, world–wide growth of interest in rough sets and their applications. The main goal of rough set analysis is induction of approximations of con ..."
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Abstract Basic ideas of rough set theory were proposed by Zdzis law Pawlak [85, 86] in the early 1980’s. In the ensuing years, we have witnessed a systematic, world–wide growth of interest in rough sets and their applications. The main goal of rough set analysis is induction of approximations
Research Statement
"... My research concerns the search for and justification of new axioms in mathematics. The need for new axioms arises from the independence results. Let me explain. In reasoning about a given domain of mathematics (or, in fact, any domain) the question of justification is successively pushed back furt ..."
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system PA (Peano arithmetic) and, in the case of set theory, it led to the axiom system ZFC (ZermeloFraenkel set theory with the Axiom of Choice). Set theory is of particular interest since it is a sufficiently rich framework to incorporate all areas of mathematics (number theory, analysis, function
www.elsevier.com/locate/fss Finiteness notions in fuzzy sets
"... Finite sets are one of the most fundamental mathematical structures. In the absence of the axiom of choice there are many di"erent inequivalent de1nitions of 1nite even in classical logic. When we allow incomplete existence as in fuzzy sets the situation gets even more complicated. This paper g ..."
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Finite sets are one of the most fundamental mathematical structures. In the absence of the axiom of choice there are many di"erent inequivalent de1nitions of 1nite even in classical logic. When we allow incomplete existence as in fuzzy sets the situation gets even more complicated. This paper
For David Chalmers et al, eds., Metametaphysics, OUP forthcoming The Metaontology of Abstraction
"... §1 We can be pretty brisk with the basics. Paul Benacerraf famously wondered 1 how any satisfactory account of mathematical knowledge could combine a facevalue semantic construal of classical mathematical theories, such as arithmetic, analysis and settheory—one which takes seriously the apparent s ..."
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§1 We can be pretty brisk with the basics. Paul Benacerraf famously wondered 1 how any satisfactory account of mathematical knowledge could combine a facevalue semantic construal of classical mathematical theories, such as arithmetic, analysis and settheory—one which takes seriously the apparent
Results 11  20
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51