Reduction to a symmetric predicate

1956 ◽  
Vol 21 (1) ◽  
pp. 56-59
Author(s):  
Alan Cobham
Keyword(s):  
Set Theory ◽  
Quantification Theory ◽  
The Universe

It has been shown by Quine that an interpreted theory Θ, formulated in the notation of quantification theory, is translatable into a theory Θ′ in which the only primitive predicate is a dyadic F. In establishing this result Quine takes for the universe of Θ′ a set which comprehends the universe of Θ and has the property that if x and y are members then so is {x, y}. For this fragmentary set theory the distinctness of x from {x, y} and {{x}} is assumed. It will be shown here that, if a pair of somewhat more stringent restrictions are assumed for the set theory, then there is a symmetric dyadic predicate G definable within Θ′, i.e., in terms of F alone, in terms of which F is in turn definable. It follows from this extended result that the theory Θ is translatable into a theory in which the only primitive predicate is symmetric and dyadic.

Reduction to a dyadic predicate

1954 ◽  
Vol 19 (3) ◽  
pp. 180-182 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
Quantification Theory ◽  
Logical Type ◽  
Image Position ◽  
The Universe

Consider any interpreted theory Θ, formulated in the notation of quantification theory (or lower predicate calculus) with interpreted predicate letters. It will be proved that Θ is translatable into a theory, likewise formulated in the notation of quantification theory, in which there is only one predicate letter, and it a dyadic one.Let us assume a fragment of set theory, adequate to assure the existence, for all x and y without regard to logical type, of the set {x, y) whose members are x and y, and to assure the distinctness of x from {x, y} and {{x}}. ({x} is explained as {x, x}.) Let us construe the ordered pair x; y in Kuratowski's fashion, viz. as {{x}, {x, y}}, and then construe x;y;z as x;(y;z), and x;y;z;w as x;(y;z;w), and so on. Let us refer to w, w;w, w;w;w, etc. as 1w, 2w, 3w, etc.Suppose the predicates of Θ are ‘F1’, ‘F2’, …, finite or infinite in number, and respectively d1-adic, d2-adic, …. Now let Θ′ be a theory whose notation consists of that of quantification theory with just the single dyadic predicate ‘F’, interpreted thus:The universe of Θ′ is to comprise all objects of the universe of Θ and, in addition, {x, y) for every x and y in the universe of Θ′. (Of course the universe of Θ may happen already to comprise all this.)Now I shall show how the familiar notations ‘x = y’, ‘x = {y, z}’, etc., and ultimately the desired ‘’, ‘’, etc. themselves can all be defined within Θ′.

Category theory, introduction to

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Category Theory ◽  
Standard Set ◽  
Universe Of Sets ◽  
The Universe

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.

INNER MODEL THEORETIC GEOLOGY

2016 ◽  
Vol 81 (3) ◽  
pp. 972-996 ◽  
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Set Theory ◽  
Main Theme ◽  
Solid Core ◽  
Strong Cardinal ◽  
The Third ◽  
Inner Models ◽  
Inner Model ◽  
Basic Concepts ◽  
Woodin Cardinal ◽  
The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

Natural internal forcing schemata extending ZFC: Truth in the universe?

1994 ◽  
Vol 59 (2) ◽  
pp. 461-472
Author(s):  
Garvin Melles
Keyword(s):  
Set Theory ◽  
Mathematical Truth ◽  
Unified Theory ◽  
Rule Of Thumb ◽  
Guiding Principle ◽  
Survey Article ◽  
Inner Models ◽  
Independence Results ◽  
Universe Of Sets ◽  
The Universe

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

Power-like models of set theory

2001 ◽  
Vol 66 (4) ◽  
pp. 1766-1782 ◽  
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Mahlo Cardinal ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Elementary Submodel ◽  
Finite Language ◽  
The Universe ◽  
Models Of Set Theory ◽  
The Continuum

Abstract.A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ1-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2<0 = 0 then every consistent extension of ZF also has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ2-like model. Here we prove:Theorem A. If 0 has the tree property then the following are equivalent for any completion T of ZFC:(i) T has a 0-like model.(ii) Ф ⊆ T. where Ф is the recursive set of axioms {∃κ (κ is n-Mahlo and “Vκis a Σn-elementary submodel of the universe”): n ∈ ω}.(iii) T has a λ-like model for every uncountable cardinal λ.Theorem B. The following are equiconsistent over ZFC:(i) “There exists an ω-Mahlo cardinal”.(ii) “For every finite language , all ℵ2-like models of ZFC() satisfy the schemeФ().

Inner models for set theory – Part III

1953 ◽  
Vol 18 (2) ◽  
pp. 145-167 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Original System ◽  
Axiom System ◽  
Complete Model ◽  
Universal Class ◽  
Inner Models ◽  
Consistency Results ◽  
Image Position ◽  
The Universe ◽  
New Hypothesis

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in (1) this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system.

scholarly journals A Detailed Study of the Distributed Rough Set Based Locality Sensitive Hashing Feature Selection Technique

2021 ◽  
Vol 182 (2) ◽  
pp. 111-179
Author(s):  
Zaineb Chelly Dagdia ◽  
Christine Zarges
Keyword(s):  
Feature Selection ◽  
Big Data ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Search Space ◽  
Locality Sensitive Hashing ◽  
Data Dependency ◽  
Distributed Environment ◽  
The Universe

In the context of big data, granular computing has recently been implemented by some mathematical tools, especially Rough Set Theory (RST). As a key topic of rough set theory, feature selection has been investigated to adapt the related granular concepts of RST to deal with large amounts of data, leading to the development of the distributed RST version. However, despite of its scalability, the distributed RST version faces a key challenge tied to the partitioning of the feature search space in the distributed environment while guaranteeing data dependency. Therefore, in this manuscript, we propose a new distributed RST version based on Locality Sensitive Hashing (LSH), named LSH-dRST, for big data feature selection. LSH-dRST uses LSH to match similar features into the same bucket and maps the generated buckets into partitions to enable the splitting of the universe in a more efficient way. More precisely, in this paper, we perform a detailed analysis of the performance of LSH-dRST by comparing it to the standard distributed RST version, which is based on a random partitioning of the universe. We demonstrate that our LSH-dRST is scalable when dealing with large amounts of data. We also demonstrate that LSH-dRST ensures the partitioning of the high dimensional feature search space in a more reliable way; hence better preserving data dependency in the distributed environment and ensuring a lower computational cost.

HODL(ℝ) is a Core Model Below Θ

1995 ◽  
Vol 1 (1) ◽  
pp. 75-84 ◽  
Author(s):  
John R. Steel
Keyword(s):  
Set Theory ◽  
Transitive Closure ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
The Universe ◽  
Do So

In this paper we shall answer some questions in the set theory of L(ℝ), the universe of all sets constructible from the reals. In order to do so, we shall assume ADL(ℝ), the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L(ℝ) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L(ℝ), and for reasons we cannot discuss here, ZFC + ADL(ℝ) yields the most interesting “completion” of the ZFC-theory of L(ℝ).ADL(ℝ) implies that L(ℝ) satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L(ℝ). Nevertheless, there is a natural inner model of L(ℝ), namely HODL(ℝ), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets x such that every member of the transitive closure of x is definable over the universe from ordinal parameters (i.e., “OD”). The superscript “L(ℝ)” indicates, here and below, that the notion in question is to be interpreted in L(R).) HODL(ℝ) is reasonably close to the full L(ℝ), in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL(ℝ): what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)

scholarly journals Olbers’ paradox bijective solution

2020 ◽  
Author(s):  
Amrit S. Sorli
Keyword(s):  
Set Theory ◽  
The Earth ◽  
Universal Space ◽  
Finite Distance ◽  
Bijective Function ◽  
The Universe

The bijective research of the Olbers’ paradox based on the bijective function of set theory confirms that the paradox is fictitious. The set X is the universe and set Y is the model of the universe. Every element in the set X has exactly one correspondent model in the set Y. Elements in both sets are defined on the basis of the elementary perception. NASA has measured that the universal space has Euclidean shape, which means it is infinite. The luminosity of stars that are on a finite distance from the Earth is not strong enough to make the night a day.

A partial model for Quine's “New foundations”

1954 ◽  
Vol 19 (3) ◽  
pp. 197-200 ◽  
Author(s):  
Václav Edvard Beneš
Keyword(s):  
Direct Product ◽  
Set Theory ◽  
Natural Numbers ◽  
Unordered Pair ◽  
Partial Model ◽  
New Foundations ◽  
The Universe ◽  
Ordered Pairs

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model.2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples.We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.

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