scholarly journals A system of axiomatic set theory - Part VII

1954 ◽  
Vol 19 (2) ◽  
pp. 81-96 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Basic Set ◽  
Set Up ◽  
Axiomatic Set Theory

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

scholarly journals A system of axiomatic set theory. Part V. General set theory (continued)

1943 ◽  
Vol 8 (4) ◽  
pp. 89-106 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Transfinite Induction ◽  
Recursive Definition ◽  
Axiomatic Set Theory

We have still to consider the extension of the methods of number theory to infinite ordinals—or to transfinite numbers as they may also, as usual, be called.The means for establishing number theory are, as we know, recursive definition, complete induction, and the “principle of the least number.” The last of these applies to arbitrary ordinals as well as to finite ordinals, since every nonempty class of ordinals has a lowest element. Hence immediately results also the following generalization of complete induction, called transfinite induction: If A is a class of ordinals such that (1) ΟηA, and (2) αηA → α′ηA, and (3) for every limiting number l, (x)(xεl → xηA) → lηA, then every ordinal belongs to A.

A system of axiomatic set theory—Part VI

1948 ◽  
Vol 13 (2) ◽  
pp. 65-79 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory ◽  
Present Section

Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.

The General Idea Behind Gödel’s Proof

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
The Other ◽  
General Idea ◽  
Original Proof ◽  
Extensive Class ◽  
The One ◽  
Axiomatic Set Theory ◽  
Gödel’S Theorem ◽  
Mathematical Systems

In the next several chapters we will be studying incompleteness proofs for various axiomatizations of arithmetic. Gödel, 1931, carried out his original proof for axiomatic set theory, but the method is equally applicable to axiomatic number theory. The incompleteness of axiomatic number theory is actually a stronger result since it easily yields the incompleteness of axiomatic set theory. Gödel begins his memorable paper with the following startling words. . . . “The development of mathematics in the direction of greater precision has led to large areas of it being formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems to date are, on the one hand, the Principia Mathematica of Whitehead and Russell and, on the other hand, the Zermelo-Fraenkel system of axiomatic set theory. Both systems are so extensive that all methods of proof used in mathematics today can be formalized in them—i.e. can be reduced to a few axioms and rules of inference. It would seem reasonable, therefore, to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions which can be formulated in the system concerned. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems, there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms.” . . . Gödel then goes on to explain that the situation does not depend on the special nature of the two systems under consideration but holds for an extensive class of mathematical systems. Just what is this “extensive class” of mathematical systems? Various interpretations of this phrase have been given, and Gödel’s theorem has accordingly been generalized in several ways. We will consider many such generalizations in the course of this volume. Curiously enough, one of the generalizations that is most direct and most easily accessible to the general reader is also the one that appears to be the least well known.

A system of axiomatic set theory—Part I

1937 ◽  
Vol 2 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Set Theory ◽  
Axiom System ◽  
Von Neumann ◽  
Logical Calculus ◽  
First Order ◽  
Usual Manner ◽  
Considerable Simplification ◽  
Individual Variables ◽  
Set Up ◽  
Axiomatic Set Theory

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell).

MODELING THE SOUTH ATLANTIC OCEAN FROM MEDIUM TO HIGH RESOLUTION

2014 ◽  
Vol 31 (2) ◽  
Author(s):  
Mariela Gabioux ◽  
Vladimir Santos da Costa ◽  
Joao Marcos Azevedo Correia de Souza ◽  
Bruna Faria de Oliveira ◽  
Afonso De Moraes Paiva
Keyword(s):  
High Resolution ◽  
Atlantic Ocean ◽  
Large Scale ◽  
South Atlantic ◽  
Surface Relaxation ◽  
The South ◽  
Small Surface ◽  
Basic Model ◽  
Basic Set ◽  
Set Up

Results of the basic model configuration of the REMO project, a Brazilian approach towards operational oceanography, are discussed. This configuration consists basically of a high-resolution eddy-resolving, 1/12 degree model for the Metarea V, nested in a medium-resolution eddy-permitting, 1/4 degree model of the Atlantic Ocean. These simulations performed with HYCOM model, aim for: a) creating a basic set-up for implementation of assimilation techniques leading to ocean prediction; b) the development of hydrodynamics bases for environmental studies; c) providing boundary conditions for regional domains with increased resolution. The 1/4 degree simulation was able to simulate realistic equatorial and south Atlantic large scale circulation, both the wind-driven and the thermohaline components. The high resolution simulation was able to generate mesoscale and represent well the variability pattern within the Metarea V domain. The BC mean transport values were well represented in the southwestern region (between Vitória-Trinidade sea mount and 29S), in contrast to higher latitudes (higher than 30S) where it was slightly underestimated. Important issues for the simulation of the South Atlantic with high resolution are discussed, like the ideal place for boundaries, improvements in the bathymetric representation and the control of bias SST, by the introducing of a small surface relaxation. In order to make a preliminary assessment of the model behavior when submitted to data assimilation, the Cooper & Haines (1996) method was used to extrapolate SSH anomalies fields to deeper layers every 7 days, with encouraging results.

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