AXIOM OF CHOICE
CONTENTS ORIGINS HILBERT’S FIRST PROBLEM CHOICE PRINCIPLES SOME EQUIVALENTS TO THE AXIOM OF CHOICE SOME CONCEPTS RELATED TO THE AXIOM OF CHOICE ELEMENTARY OBSERVATIONS HIDDEN CHOICE UNNECESSARY CHOICE CONCEPTS SPLIT UP: COMPACTNESS DISASTERS WITHOUT CHOICE FINITENESS DISASTERS IN CARDINAL ARITHMETIC DISASTERS IN ORDER THEORY DISASTERS IN ALGEBRA I: VECTOR SPACES DISASTERS IN ALGEBRA II: CATEGORIES DISASTERS IN ELEMENTARY ANALYSIS: THE REALS AND CONTINUITY DISASTERS IN TOPOLOGY I: COUNTABLE SUMS DISASTERS IN TOPOLOGY II: PRODUCTS (THE TYCHONO?AND THE ?CECH–STONE THEOREM) DISASTERS IN TOPOLOGY III: FUNCTION SPACES (THE ASCOLI THEOREM) DISASTERS IN TOPOLOGY IV: THE BAIRE CATEGORY THEOREM DISASTERS IN GRAPH THEORY: COLORING PROBLEMS DISASTERS WITH CHOICE DISASTERS IN ELEMENTARY ANALYSIS DISASTERS IN GEOMETRY: PARADOXICAL DECOMPOSITIONS XIV CONTENTS DISASTERS EITHER WAY DISASTERS IN GAME THEORY BEAUTY WITHOUT CHOICE LINDEL¨OF = COMPACT MEASURABILITY (THE AXIOM OF DETERMINATENESS) REFERENCES [FRBALE73, P 84] [BEFR58, P 16] [BEFR58, P 26] — WITHOUT QUESTION THE MOST IN?UENTIAL MATHEMATICAL LECTURE EVER GIVEN — HE FORMULATED THE CONJUNCTION OF CANTOR’S TWO CONJECTURES AS THE ?RST OF HIS FAMOUS PROBLEMS THIS ?ELD IS THE THEORY OF SETS, WHOSE CREATOR WAS GEORG CANTOR, , THIS APPEARS TO ME AS THE MOST MARVELOUS FRUIT OF THE MATHEMATICAL MIND, INDEED AS ONE OF THE HIGHEST ACHIEVEMENTS OF PURELY RATIONAL HUMAN ACTIVITIES NOBODY IS TO BANISH US FROM THE PARADISE CREATED BY CANTOR [HIL26] [COH2002] “DAß ES IMMER M¨OGLICH IST, JEDE WOHLDE?NIERTE MENGE IN DIE FORM EINER WOHLGE- ORDNETEN MENGE ZU BRINGEN, AUF DIESES, WIE MIR SCHEINT, GRUNDLEGENDE UND FOL- GENREICHE, DURCH SEINE ALLGEMEING¨ULTIGKEIT BESONDERS MERKW¨URDIGE DENKGESETZ WERDE ICH IN EINER SP¨ATEREN ABHANDLUNG ZUR¨UCKKOMMEN ” CANTOR (1883)