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 countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification

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