Epsilon-induction: 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: |
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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Revision as of 04:17, 22 July 2009
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. -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 .