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| | In [[mathematics]], '''<math>\in</math>-induction''' ('''epsilon-induction''') is a variant of [[transfinite induction]], which can be used in [[axiomatic set theory|set theory]] to prove that all [[Set (mathematics)|sets]] satisfy a given property ''P''[''x'']. If the truth of the property for ''x'' follows from its truth for all elements of ''x'', for every set ''x'', then the property is true of all sets. In symbols: |
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| | : ''<math>\forall x \Big(\forall y (y \in x \rightarrow P[y]) \rightarrow P[x]\Big) \rightarrow \forall x \, P[x]</math>'' |
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| | This principle, sometimes called the '''axiom of induction''' (in set theory), is equivalent to the [[axiom of regularity]] given the other [[Zermelo–Fraenkel set theory|ZF]] axioms. <math>\in</math>-induction is a special case of [[well-founded relation#Induction and recursion|well-founded induction]]. |
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| | The name is most often pronounced "epsilon-induction", because the set membership symbol <math>\in</math> historically developed from the Greek letter <math>\epsilon </math>. |
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| | [[Category:Mathematical induction]] |
| | [[Category:Wellfoundedness]] |
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| | {{settheory-stub}} |
Revision as of 16:27, 2 December 2013
In mathematics, -induction (epsilon-induction) is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a given property P[x]. If the truth of the property for x follows from its truth for all elements of x, for every set x, then the property is true of all sets. In symbols:
This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity given the other ZF axioms. -induction is a special case of well-founded induction.
The name is most often pronounced "epsilon-induction", because the set membership symbol historically developed from the Greek letter .
Template:Settheory-stub