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1. Basic Concepts of Set Theory. 1.1. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe
Set theory was created by George Cantor (1845-1918) in the years 1874-1897. He was led to his ideas by certain problems in real analysis (Fourier series). In 1874 Cantor discovered that one cannot enumerate the real numbers by the natural numbers, showing that in nite sets come in di erent \sizes".
what a set is, we will describe what can be done with sets. Intuitivelly, a set is a collection of objects of any kind, which we call the elements of a set. The second primitive notion of set theory is the notion of belonging. We write x ∈ X meaning ‘x belongs to the set X’, or ‘x is an element of X’ (Tipically
Set theory is a rich and beautiful subject whose fundamental concepts perme-ate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In
losophy of set theory. By the end of this book, students reading it might have a sense of: 1.why set theory came about; 2.how to embed large swathes of mathematics within set the-ory + arithmetic; 3.how to embed arithmetic itself within set theory; 4.what the cumulative iterative conception of set amounts to;
Chapter 0 Introduction. Set Theory is the true study of in nity. This alone assures the subject of a place prominent in human culture. But even more, Set Theory is the milieu in which mathematics takes place today. As such, it is expected to provide a rm foundation for all the rest of mathematics.
itive concepts of set theory the words “class”, “set” and “belong to”. These will be the only primitive concepts in our system. We then present and briefly dis-cuss the fundamental Zermelo-Fraenkel axioms of set theory. 1.1 Contradictory statements. When expressed in a mathematical context, the word “statement” is viewed in a
