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Über dieses Buch

H. Hermes: Basic notions and applications of the theory of decidability.- D. Kurepa: On several continuum hypotheses.- A. Mostowski: Models of set theory.- A. Robinson: Problems and methods of model theory.- S. Sochor, B. Balcar: The general theory of semisets. Syntactic models of the set theory.



Basic Notions and Applications of the Theory of Decidability

Preliminary Remarks. The first three lectures contain an exposition of the fundamental concepts of some main theorems of the theory of recursive functions. One of the more difficult theorems of the theory of recursive functions is Friedberg-Mučniks theorem which asserts the existence of non-trivial enumerable degrees. In Lectures 4 and 5 we prove this theorem, following the treatment given by Sacks, but stressing somewhat more the combinatorial part of the proof (Lecture 4). Lecture 6 deals with problems in the theory of primitive recursive functions. As a typical example of the application of the theory of recursitivy we give in Lecture 7 in detail a proof for the unsolva-bility of the domino problem in the simplest case of the origin-restricted problem and show in Lecture 8 how the domino problem is connected with the ∧∨∧- case of the Entscheidungsproblem.
Lecture 6 has been given before Lectures 4 and 5. The interchange is due to systematical reasons.
H. Hermes

On Several Continuum Hypotheses

The classical Cantor's continuum hypothesis states that for every infinite set S the cardinality of the set PS of all the subsets of S is the immediate follower of the cardinality kS of S.i.e.
$$ {\text{kPS}} = \left( {{\text{k}}\,{\text{S}}} \right) $$
Djurio Kurepa

Models of Set Theory

Aim of the lectures: to outline various methods used recently in construction of models for axioms of set theory. No completeness in pursuing this aim is attempted.
In the introductory lecture I we describe three systems of axioms for abstract set theory. In all these systems there are two primitive notions: “class” and “membership”. We define sets as classes which are capable of being members of other classes: x is a set if and only if there is a class y such that x ∈ y. We also define atoms as objects which have no elements.
A. Mostowski

Problems and Methods of Model Theory

1. Introduction, Over the last century, the axiomatic approach has pervaded Mathematics. According to this approach, a mathematical discipline starts from a specified list of conditions or axioms, which are concerned with a set of basic notions, otherwise undefined. The discipline then consists of a detailed investigation of the structures which are models of, i.e. which satisfy, the system of axioms in question. In order that such structures may be assumed to exist, it is necessary that the given set of axioms be devoid of contradictions and this is proved either absolutely, or relative to another system, which is itself supposed to be devoid of contradiction, or else it is simply assumed.
Abraham Robinson

The General Theory of Semisets

The General Theory of Semisets
It is the purpose of this article to explain briefly some concepts and methods, especially so called the theory of semites, which are studied in Prague seminar. The authors of the theory of semisets are P. Vopénka and P. Hájek. We present here some results (not in the most general form) that are contained in their book “Sets, Semisets, Models” (to be published) with their kind permission. This article was written as material to our lecture that was held in the Summer Institute in Varenna (Italy) and contains no our new results.
At first we give the following illustration in order to acquire some idea about semisets. The reader is already acquainted with the Gö el-Bernays' set theory (GB) from the lecture of prof. Mostowski (in what follows we shall denote this lecture by [M]) where also the universal class V and the class L of all constructible sets were defined.
Bohuslav Balcar
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