Introduction to Set Theory
A Solution Manual for Hrbacek and Jech (1999)

Jianfei Shen
School of Economics, The University of New South Wales

Sydney, Australia

The Lord by wisdom founded the earth, by understanding he established the heavens. — Proverbs 3:19

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sets 1.1 1.2 1.3 1.4 ............................................................... Introduction to Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …ver más…

1.2 Properties
No exercises.

1.3 The Axioms
I Exercise 1 (1.3.1). Show that the set of all x such that x 2 A and x … B exists.
Proof. Notice that

˚

« ˚ « x W x 2 A and x … B D x 2 A W x … B :

Then by the Axiom Schema of Comprehension, we know that such a set does exist. t u

I Exercise 2 (1.3.2). Replace The Axiom of Existence by the following weaker postulate: Weak Axiom of Existence: Some set exists. Prove the Axiom of Existence using the Weak Axiom of Existence and the Comprehension Schema.
Proof. Let A be a set known to exist. By the Axiom Schema of Comprehension, there is a set X such that

˚ « X D x2AWx¤x :

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CHAPTER 1

SETS

There is no subjects x satisfying x ¤ x , so there is no elements in X , which proves the Axiom of Existence. t u

I Exercise 3 (1.3.3). a. Prove that a “set of all sets” does not exist.
b. Prove that for any set A there is some x … A. Proof. (a) Suppose that there exists a universe set (a set of all sets) V . Then by the Axiom Schema of Comprehension, there is a set B D fx 2 V W x … xg; that is x 2 B () x 2 V and x … x: (1.1) Now we show that B … V , that is, B is not a set. Indeed, if B 2 V , then either B 2 B , or B … B . If B 2 B , then, by the “H)” direction of (1.1), B 2 V and B … B . A contradiction; if B … B , then, by the “(H” direction of (1.1), the assumption B 2 V and B … B yield B 2 B .