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Squares, Scales and Stationary Reflection
"... Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by ZermeloFraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many are ..."
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Cited by 25 (10 self)
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Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by ZermeloFraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven eective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(}) and square() discovered by Jensen. Simultaneously, attempts have been made to nd suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of c...
Bounded Hyperset Theory and Weblike Data Bases
 Computational Logic and Proof Theory, 5th Kurt Gödel Colloquium, KGC’97, Springer LNCS
, 1997
"... this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike Data Bases (WDB) and on the corresponding querying to WDB. Let us stress that it is not only about databases with an access via Web. The database itself should be organized in the same ..."
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Cited by 18 (5 self)
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this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike Data Bases (WDB) and on the corresponding querying to WDB. Let us stress that it is not only about databases with an access via Web. The database itself should be organized in the same way as Web. I.e. it must consist of hyperlinked pages distributed among the computers participating either in global network like Internet or in some local, isolated from the outside world specic network based essentially on the same principles, except globality, and called also Intranet [15].
Square in core models
 Bull. Symbolic Logic
, 2001
"... Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1small construction is an iterate of K, 4) (joint with W. J. Mitchel ..."
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Cited by 18 (3 self)
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Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1small construction is an iterate of K, 4) (joint with W. J. Mitchell) K�κ is universal for mice of height ≤ κ whenever κ ≥ℵ2,5)ifthereisaκ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and � <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ωErdös cardinal suffices to develop the basic theory of K. 1.
Combinatorial principles in the core model for one Woodin cardinal
 ANNALS OF PURE AND APPLIED LOGIC
, 1995
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Indexed Squares
 Journal of Symbolic Logic
"... . We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1 ..."
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Cited by 17 (8 self)
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. We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1extendibility is incompatible with square; we prove this is close to optimal by showing that 1extendibility is compatible with square. 1. Introduction In this paper we study some variations on Jensen's celebrated combinatorial principle (variously pronounced as \square kappa" or \box kappa"). is a principle which is helpful in constructing objects of cardinality + ; for example Jensen showed that if holds then there is a special + Aronszajn tree, and every stationary subset of + contains a nonreecting stationary subset. Jensen proved [Je1] that if V = L then holds for every uncountable cardinal ( ! is a trivial theorem in ZFC). In combination with Jen...
Turing computations on ordinals
 Bulletin of Symbolic Logic 11 (2005
, 2005
"... We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel ..."
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Cited by 12 (4 self)
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We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel’s constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L. 1 Introduction. A standard Turing computation may be visualized as a timelike sequence of elementary readwritemove operations carried out by one or more “heads ” on “tapes”. The sequence of actions is determined by the initial tape contents and by a finite Turing program. The specific choice of alphabet, operations
The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
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Cited by 12 (1 self)
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SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
On Bounded Set Theory
"... We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation ..."
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Cited by 10 (1 self)
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We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation between BST, possibly with AntiFoundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.