Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.
Some of the alternative set theories are:
- the theory of semisets
- the set theory New Foundations
- Positive set theory
- Internal set theory
Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students. It builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF) finite sets and they are called infinite in AST.
Famous quotes containing the words alternative, set and/or theory:
“If you have abandoned one faith, do not abandon all faith. There is always an alternative to the faith we lose. Or is it the same faith under another mask?”
—Graham Greene (1904–1991)
“This is the Scroll of Thoth. Herein are set down the magic words by which Isis raised Osiris from the dead. Oh! Amon-Ra—Oh! God of Gods—Death is but the doorway to new life—We live today-we shall live again—In many forms shall we return-Oh, mighty one.”
—John L. Balderston (1899–1954)
“Psychotherapy—The theory that the patient will probably get well anyway, and is certainly a damned ijjit.”
—H.L. (Henry Lewis)