Formal Specification and Verification

Published online. — 2006. — 152 p. English. (OCR-слой).[Prof. P.H. Schmitt. Fakultat fur Informatik. Universitat Karlsruhe (TH).
Axiomatic Set Theory. Sommer 2006].Contents.
Zermelo-Fraenkel (ZF) Axiom System (1):
A1 Extensionality.
A2 Foundation.
A3 Subset.
A4 Empty set.
Zermelo-Fraenkel Axiom System (2):
A5 Pair set.
A6 Power set.
A7 Sum.
A8 Infinity.
Zermelo-Fraenkel Axiom System (3):
A9 Replacement.
A10 Axiom of Choice.
Class Terms.
Class Terms as Sets.
Some Abbreviations for Sets.
Existence Claims.
Lemma on Unions and Intersections.
Proof of Existence of Intersections.
Proof of Existence of Unions.
Ordered Pairs.
Relations and Functions.
Existence Proofs.
Natural Numbers N.
Formal Definition of N.
Peano’s Axioms.
Proof of Axiom 4
Set Theoretic Properties of N.
Transitive Sets.
N is transitive.
The order relation on N.
Set Theoretic Properties of N (II).
The Recursion Theorem.
Uniqueness.
Idea of Existence Proof.
Details of Existence Proof.
Addition of natural numbers.
Multiplication of natural numbers.
The Integers.
The construction of Z.
Operations on Z.
Order Relation on Z.