Dickey–Fuller test: Difference between revisions

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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>''
 
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]].
 
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>.
 
[[Category:Mathematical induction]]
[[Category:Wellfoundedness]]
 
{{settheory-stub}}

Latest revision as of 19:50, 28 December 2014

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