HC of an admissible set

1979 ◽  
Vol 44 (1) ◽  
pp. 95-102
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Admissible Set ◽  
The Real ◽  
Admissible Sets

AbstractIf A is an admissible set, let HC(A) = {x∣x ∈ A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the “real parts” of admissible sets and the analytical consequences of admissible set theory.

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

The pure part of HYP(ℳ)

1977 ◽  
Vol 42 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Mark Nadel ◽  
Jonathan Stavi
Keyword(s):  
Admissible Set ◽  
Admissible Sets ◽  
Negative Results

AbstractLet ℳ be a structure for a language ℒ on a set M of urelements. HYP(ℳ) is the least admissible set above ℳ. In §1 we show that pp(HYP(ℳ)) [= the collection of pure sets in HYP(ℳ)] is determined in a simple way by the ordinal α = ° (HYP(ℳ)) and the ℒxω theory of ℳ up to quantifier rank α. In §2 we consider the question of which pure countable admissible sets are of the form pp(HYP(ℳ)) for some ℳ and show that all sets Lα (α admissible) are of this form. Other positive and negative results on this question are obtained.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

scholarly journals Structure of a decision support subsystem in real estate management

2013 ◽  
Vol 13 (1) ◽  
pp. 56-75 ◽  
Author(s):  
Małgorzata Renigier-Biłozor
Keyword(s):  
Decision Making ◽  
Decision Support ◽  
Real Estate ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
The Real ◽  
The Real Estate ◽  
Estate Management ◽  
Real Estate Management

Abstract This study proposes a decision support subsystem in real estate management. Owing to the complex and multi-layered character of the discussed problem, only selected aspects of real estate management are discussed in this paper. The described system will play the role of a relatively simple and effective “assistant” which is expected to maximize the effectiveness of a decision and shorten decision-making time. The author has made an attempt to develop a subsystem as an adviser to subjects operating in the real estate management. This system was developed accounting for and combining the classical economic and real estate market theories with the implementation of non-classical methods in the data mining category in an effort to increase its effectiveness. The rough set theory has been proposed as a tool that supports analytical processes. Fuzzy logic best reproduces expert knowledge, and it is one of the most effective tools for solving “vaguely defined” problems. The given work is an attempt to prove the hypothesis that: the reduction of uncertainty in the real estate management decision-making process is possible by the development of the advisory system based on the rough set theory. The main aim of this work is to increase the efficiency and efficacy of entities operating in the real estate management, thus influencing the effectivness of the entity and management.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

scholarly journals Wellordering proofs for metapredicative Mahlo

2002 ◽  
Vol 67 (1) ◽  
pp. 260-278 ◽  
Author(s):  
Thomas Strahm
Keyword(s):  
Set Theory ◽  
Upper Bounds ◽  
Transfinite Induction ◽  
Admissible Set ◽  
Two Systems ◽  
Universe Of Discourse

AbstractIn this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed sharp.

scholarly journals Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

2001 ◽  
Vol 66 (2) ◽  
pp. 935-958 ◽  
Author(s):  
Gerhard Jäger ◽  
Thomas Strahm
Keyword(s):  
Set Theory ◽  
Upper Bounds ◽  
Admissible Set

AbstractIn this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.

Anti-admissible sets

1999 ◽  
Vol 64 (2) ◽  
pp. 407-435
Author(s):  
Jacob Lurie
Keyword(s):  
Canonical Extension ◽  
Axiom System ◽  
Admissible Set ◽  
Standard View ◽  
Modal Language ◽  
Admissible Sets ◽  
Interesting Alternative

AbstractAczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the “circular logic” of [3], This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical “extension” to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU−, that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between “non-wellfounded sets over the admissible set A” and the fragment LA of the modal language L∞.

Sets and Numbers

2021 ◽  
pp. 3-27
Author(s):  
James Davidson
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Euclidean Distance ◽  
Final Section ◽  
Real Analysis ◽  
Real Numbers ◽  
The Real ◽  
Basic Ideas ◽  
Distance Sets

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

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