In the
foundations of mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
, Aczel's anti-foundation axiom is an
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
set forth by , as an alternative to the
axiom of foundation in
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
. It states that every
accessible pointed directed graph corresponds to exactly one
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a
Quine atom
Quine may refer to:
* Quine (computing), a program that produces its source code as output
* Quine's paradox, in logic
* Quine (surname), people with the surname
** Willard Van Orman Quine (1908–2000), American philosopher and logician
See ...
. A set theory obeying this axiom is necessarily a
non-well-founded set theory
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axio ...
.
Accessible pointed graphs
An
accessible pointed graph is a
directed graph with a distinguished
vertex (the "root") such that for any node in the graph there is at least one
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desir ...
in the directed graph from the root to that node.
The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of exactly one set. For example, the directed graph with only one node and an edge from that node to itself corresponds to a set of the form ''x'' = .
See also
*
von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory ( ...
References
*
*
*
{{Set theory
Axioms of set theory
Directed graphs
de:Fundierungsaxiom#Mengenlehren ohne Fundierungsaxiom