## ZF: Zermelo-Fraenkel Set Theory

This directory contains the ML sources of the Isabelle system for ZF Set Theory, based on FOL.

There are several subdirectories of examples:

AC
  subdirectory containing proofs from the book "Equivalents of the Axiom of Choice, II" by H. Rubin and J.E. Rubin, 1985. Thanks to Krzysztof Gr`abczewski.

Coind
  subdirectory containing a large example of proof by co-induction. It is by Jacob Frost following a paper by Robin Milner and Mads Tofte.

IMP
  subdirectory containing a semantics equivalence proof between operational and denotational definitions of a simple programming language. Thanks to Heiko Loetzbeyer & Robert Sandner.

Resid
  subdirectory containing a proof of the Church-Rosser Theorem. It is by Ole Rasmussen, following the Coq proof by G�ard Huet.

ex
  subdirectory containing various examples.

Isabelle/ZF formalizes the greater part of elementary set theory, including relations, functions, injections, surjections, ordinals and cardinals. Results proved include Cantor's Theorem, the Recursion Theorem, the Schroeder-Bernstein Theorem, and (assuming AC) the Wellordering Theorem.

Isabelle/ZF also provides theories of lists, trees, etc., for formalizing computational notions. It supports inductive definitions of infinite-branching trees for any cardinality of branching.

Useful references for Isabelle/ZF:

- Lawrence C. Paulson,
  Set theory for verification: I. From foundations to functions.
  J. Automated Reasoning 11 (1993), 353-389.
- Lawrence C. Paulson,
  Set theory for verification: II. Induction and recursion.
  Report 312, Computer Lab (1993).
- Lawrence C. Paulson,
  A fixedpoint approach to implementing (co)inductive definitions.
  In: A. Bundy (editor),
  CADE-12: 12th International Conference on Automated Deduction,
  (Springer LNAI 814, 1994), 148-161.

Useful references on ZF set theory:

- Paul R. Halmos, Naive Set Theory (Van Nostrand, 1960)
- Patrick Suppes, Axiomatic Set Theory (Dover, 1972)
- Keith J. Devlin,
  Fundamentals of Contemporary Set Theory (Springer, 1979)
- Kenneth Kunen
  Set Theory: An Introduction to Independence Proofs
  (North-Holland, 1980)