Axiomatic set theory
In mathematics, an axiomatic set theory is a form of set theory devised to avoid paradoxes.
Description
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox.
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:
- Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
- Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
- General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
- Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
- Sets and proper classes. These include:
- Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone
- Morse-Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC
The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members.
The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything, " relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.
Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.
An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.
See also
- Axiom
- Axiom of choice
- Axiom of power set
- Class (set theory)
- Finite sets
- General set theory
- Kripke–Platek set theory
- Morse-Kelley set theory and Tarski–Grothendieck set theory
- Peano axioms
- Set theory
- Urelement
- Von Neumann–Bernays–Gödel set theory
- Zermelo set theory
- Zermelo–Fraenkel set theory
External links
- Set theory # Axiomatic set theory @ Wikipedia.org
