Difference between revisions of "Set theory axioms"
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! Axiom | ! Axiom | ||
! Description | ! Description | ||
| + | ! Formal statement | ||
|- | |- | ||
| 1 | | 1 | ||
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| 2 | | 2 | ||
| Extensionality (Equality) | | 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}} | + | | 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 | | R | ||
| Line 24: | Line 26: | ||
| 3 | | 3 | ||
| Schema of Comprehension | | 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)}} | + | | 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 | | R | ||
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| Pair | | 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> | | 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 | | 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) | | 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 | | 5 | ||
| Union | | 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> | | 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 | | R | ||
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| | | | ||
| {{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> | | {{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 | | 6 | ||
| Power set | | 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]]) | | 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]] | [[Category:Set Theory]] | ||
Latest revision as of 10:25, 11 March 2015
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 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
| Number | Axiom | Description | Formal statement |
|---|---|---|---|
| 1 | Existence | There exists a set with no elements | |
| 2 | Extensionality (Equality) | If every element of [ilmath]X[/ilmath] is also an element of [ilmath]Y[/ilmath] and every element of [ilmath]Y[/ilmath] is also an element of [ilmath]X[/ilmath] then |
[math]\forall X\forall Y(\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y)[/math] |
| R | The empty set is unique can now be proved, and thus denoted [math]\emptyset[/math] | ||
| 3 | Schema of Comprehension | For a property [ilmath]P(x)[/ilmath] of x, given a set [ilmath]A[/ilmath] there is a set [ilmath]B[/ilmath] such that [ilmath]x\in B\iff x\in A\text{ and }p(x)[/ilmath]. 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 [ilmath]A[/ilmath] and a property [ilmath]P[/ilmath] 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 [ilmath]A[/ilmath] and [ilmath]B[/ilmath] there is a set [ilmath]C[/ilmath] 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 [ilmath]S[/ilmath] there exists a set [ilmath]U[/ilmath] 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 | [ilmath]A[/ilmath] is a subset of [ilmath]B[/ilmath] if and only if every element of [ilmath]A[/ilmath] belongs to [ilmath]B[/ilmath], 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 [ilmath]S[/ilmath] there eixsts a set [ilmath]\mathcal{P} [/ilmath] such that [ilmath]X\in\mathcal{P}\iff X\subset S[/ilmath] (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] |
