Vopěnka's "Alternative Set Theory" 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. The following axioms hold for sets. •
Extensionality - Sets with the same elements are the same. •
Empty set: ∅ exists. •
Successor: For any sets x and y, x \cup \{y\} exists. •
Induction: Every formula \phi expressed in the language of sets only (all parameters are sets and all quantifiers are restricted to sets) and true of ∅ and true of x \cup \{y\} if it is true of x is true of all sets. •
Regularity: Every set has an element disjoint from it. The following axioms hold for all classes. •
Existence of classes: If \phi (x) is any formula, then the class \phi (x) of all sets x such that \phi (x) exists. (The set x is identified with the class of elements of x.) Note that Kuratowski pairs of sets are sets, and so we can define (class) relations and functions on the universe of sets much as usual. •
Extensionality for classes: Classes with the same elements are equal. •
Axiom of proper semisets: There is a proper semiset. •
Prolongation axiom: Each countable function F can be extended to a set function. •
Axiom of extensional coding: Every collection of classes which is codable is extensionally codable. Vopenka considers representations of superclasses of classes using relations on sets. A class relation R on a class A is said to code the superclass of inverse images of elements of A under R. A class relation R on a class A is said to extensionally code this superclass if distinct elements of A have distinct preimages. •
Axiom of cardinalities: If two classes are uncountable, they are the same size. ==References==