View source for Set theory axioms

This page is supposed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page)

==List of axioms==
The number column describes the order of introduction in the [[Motivation for set theory axioms|motivation for set theory axioms page]], note that "R" denotes a result. Only "major" results are shown, they are covered in the motivation for set theory page, and "D" denotes "definition" - which is something that is natural to define at that point

{| class="wikitable" border="1"
|-
! Number
! Axiom
! Description
! Formal statement
|-
| 1
| Existence
| There exists a set with no elements
|-
| 2
| Extensionality (Equality)
| If every element of {{M|X}} is also an element of {{M|Y}} and every element of {{M|Y}} is also an element of {{M|X}} then {{M|X=Y}}<br/>
|<math>\forall X\forall Y(\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y)</math>
|-
| R
| 
| [[Motivation for set theory axioms#The empty set is unique|The empty set is unique]] can now be proved, and thus denoted <math>\emptyset</math>
|-
| 3
| Schema of Comprehension
| For a property {{M|P(x)}} of x, given a set {{M|A}} there is a set {{M|B}} such that {{M|x\in B\iff x\in A\text{ and }p(x)}}.<br/>A property may be <math>P(x):=x\in A=\phi(x,A)</math> where <math>\phi</math> is a ''formula''
| <math>\forall X\forall p\exists Y\forall u(u\in Y\leftrightarrow[u\in X\wedge\phi(u,p)])</math>
|-
| R
|
| For a set {{M|A}} and a property {{M|P}} the set known to exist by axiom 3 is unique, thus we may write <math>\{x\in A|P(x)\}</math> to denote it unambiguously
|-
| 4
| Pair
| For any {{M|A}} and {{M|B}} there is a set {{M|C}} such that <math>x\in C\iff x=A\text{ or }x=B</math>
| <math>\forall A\forall B\exists C\forall x(x\in C\leftrightarrow x=A\vee x=B)</math>
|-
| R
|
| The set known to exist from axiom 4 is unique, we denote it by <math>\{A,B\}</math> or <math>\{A\}</math> if <math>A=B</math> (at this point we "just write" this, we have no concept of cardinality yet)
|-
| R
|
| [[Ordered pair]]
| Kuratowski: <math>(a,b)=\{\{a\},\{a,b\}\}</math>
|-
| 5
| Union
| For any set {{M|S}} there exists a set {{M|U}} such that <math>x\in U\iff[x\in A\text{ for some }A\in S]</math>
| <math>\forall X\exists U\forall s(s\in U\leftrightarrow \exists A(A\in X\wedge S\in A))</math>
|-
| R
|
| The union of a set S is unique, and thus denoted by <math>\cup S</math>
|-
| D
| 
| {{M|A}} is a subset of {{M|B}} if and only if every element of {{M|A}} belongs to {{M|B}}, that is <math>\forall x:x\in A\implies x\in B</math> - we denote this <math>A\subset B</math>
| <math>\forall u(u\in A\rightarrow u\in B)</math><math>\iff \subset(A,B)\iff A\subset B</math>
|- 
| 6
| Power set
| For any set {{M|S}} there eixsts a set {{M|\mathcal{P} }} such that {{M|X\in\mathcal{P}\iff X\subset S}} (see [[Power set]])
| <math>\forall X\exists\mathcal{P}\forall U(\forall s(s\in U\rightarrow s\in X)\leftrightarrow U\in\mathcal{P})</math>
|-
| 7
| Infinite set
| There exists an [[Inductive property]]
| <math>\exists S(\emptyset \in S\wedge \forall x(x\in S\rightarrow x\cup\{x\}\in S))</math>
|}


[[Category:Set Theory]]