## Axiomatic Set Theory, Part 1 |

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### Contents

1 | |

9 | |

17 | |

A MORE EXPLICIT SET THEORY¹ | 49 |

SETS SEMISETS MODELS | 67 |

THE BOOLEAN PRIME IDEAL THEOREM DOES NOT IMPLY THE AXIOM OF CHOICE | 83 |

ON MODELS FOR SET THEORY WITHOUT AC | 135 |

PRIMITIVE RECURSIVE SET FUNCTIONS | 143 |

PREDICATIVE CLASSES | 247 |

ON SOME CONSEQUENCES OF THE AXIOM OF DETERMINATENESS | 265 |

EMBEDDING CLASSICAL TYPE THEORY IN INTUITIOMSTIC TYPE THEORY | 267 |

ORDINAL DEFINABILITY | 271 |

AN AXIOM OF STRONG INFINITY AND ANALYTIC HIERARCHY OF ORDINAL NUMBERS | 279 |

LIBERAL INTUITIONISM AS A BASIS FOR SET THEORY | 321 |

FORCING WITH PERFECT CLOSED SETS | 331 |

UNRAMIFIED FORCING¹ | 357 |

END EXTENSIONS OF MODELS OF SET THEORY | 177 |

OBSERVATIONS ON POPULAR DISCUSSIONS OF FOUNDATIONS | 189 |

INDESCRIBABILITY AND THE CONTINUUM | 199 |

THE SIZES OF THE INDESCRIBABLE CARDINALS | 205 |

ON THE LOGICAL COMPLEXITY OF SEVERAL AXIOMS OF SET THEORY | 219 |

CATEGORICAL ALGEBRA AND SETTHEORETIC FOUNDATIONS | 231 |

THE SOLUTION OF ONE OF ULAMS PROBLEMS CONCERNING ANALYTIC RECTANGLES | 241 |

THE CONSISTENCY OF THE GCH WITH THE EXISTENCE OF A MEASURABLE CARDINAL | 391 |

REALVALUED MEASURABLE CARDINALS¹ | 397 |

TRANSFINITE SEQUENCES OF AXIOM SYSTEMS FOR SET THEORY | 429 |

HYPOTHESES ON POWER SET | 439 |

MULTIPLE CHOICE AXIOMS | 447 |

467 | |

### Common terms and phrases

absolute arithmetic Assume G.C.H. axiom of choice axiom of constructibility Boolean algebra complete boolean algebra consistent constructible sets continuum hypothesis contradiction COROLLARY countable definition den(u denote disjoint element elementary extension equivalent exists first-order formal formula free variables functor Godel graph Hajnal hence holds hyperarithmetic implies inaccessible cardinal induction hypothesis infinite language Lemma logic mapping Math mathematics measurable cardinal measure zero model of ZF morphism nonempty nontrivial notion obtain one-one ordinal number positive measure power set predicate Prim Prim0 primitive recursive Primo Problem proof of Theorem prove quantifiers recursive ordinal regular cardinal relation replacing satisfies schema semisets sentence set functions set theory Skolem Solovay statement subset Suppose symbols term transitive set Turing degree ultrafilter ultraproduct universe

### Popular passages

Page 15 - ... into the unsafe ground of set theory. This is our fate, to live with doubts, to pursue a subject whose absoluteness we are not certain of, in short to realize that the only "true" science is itself of the same mortal, perhaps empirical, nature as all other human undertakings.

Page 11 - Platonist] position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

Page 9 - The authors would like to acknowledge the support of the Air Force Office of Scientific Research through the program, "Theoretical and Experimental Studies of Vibroacoustic Systems", under the management of Major Brian Sanders. 1. Clark, RL and Fuller, CR, "Optimal placement of piezoelectric actuators and polyvinylidene fluoride error sensors in active structural acoustic control approaches,

Page 11 - To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hubert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency. . . . The Realist ] Platonist] position is probably the one which most mathematicians would prefer to take.