> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/list-of-axioms-of-set-theory.md).

# list-of-axioms-of-set-theory

filename : axioms-of-set-theory.md section : math subsection : set title : Axioms of Set Theory date created : 2018-09-23 date modified : 2020-04-19

## Axioms of Set Theory

* Axiom of Extension: 

  two sets are equal iff they contain the same elements.
* Axiom of Regularity: 
* Axiom Schema of Specification: 
* Axiom of Pairing: 

  for any two sets, there exists a set to which only those two sets belong.
* Axiom of Union:  

  for every collection of sets, there exists a set that contains all the elements that belong to at least one of the sets in the collection.
* Axiom of Replacement: 

  for any set $$S$$, there exists a set $$x$$ such that, for any element $$y$$ of $$S$$, if there exists an element $$z$$ satisfying the condition $$P(y,z)$$, where $$P(y,z)$$ is a propositional function, then such $$z$$ appear in $$x$$.
* Axiom of the Empty Set: 

  there exists a set that has no elements.
* Axiom of Subsets: 

  for every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.
* Axiom of Powerset: 

  for each set there exists a collection of sets that contains amongst its elements all the subsets of the given set.
* Axiom of Infinity: 

  there exists a set containing a set with no elements and the successor of each of its elements.
* Axiom of Foundation: 

  for all non-null sets, there is an element of the set that shares no member with the set.
* Axiom of Choice: 

  for every set, we can provide a mechanism for choosing one element of any non-empty subset of the set.

[Choice](https://www.wikiwand.com/en/Axiom_of_choice) [countable](https://www.wikiwand.com/en/Axiom_of_countable_choice) [dependent](https://www.wikiwand.com/en/Axiom_of_dependent_choice) [Constructibility (V=L)](https://www.wikiwand.com/en/Axiom_of_constructibility) [Determinacy](https://www.wikiwand.com/en/Axiom_of_determinacy) [Extensionality](https://www.wikiwand.com/en/Axiom_of_extensionality) [Infinity](https://www.wikiwand.com/en/Axiom_of_infinity) [Limitation of size](https://www.wikiwand.com/en/Axiom_of_limitation_of_size) [Pairing](https://www.wikiwand.com/en/Axiom_of_pairing) [Power set](https://www.wikiwand.com/en/Axiom_of_power_set) [Regularity](https://www.wikiwand.com/en/Axiom_of_regularity) [Union](https://www.wikiwand.com/en/Axiom_of_union) [Martin's axiom](https://www.wikiwand.com/en/Martin%27s_axiom) [Axiom schema](https://www.wikiwand.com/en/Axiom_schema) [replacement](https://www.wikiwand.com/en/Axiom_schema_of_replacement) [specification](https://www.wikiwand.com/en/Axiom_schema_of_specification)

<https://proofwiki.org/wiki/Axiom:Axiom_of_Extension> <https://proofwiki.org/wiki/Axiom:Zermelo-Fraenkel_Axioms>