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Difference between revisions of "Reducibility axiom"

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An axiom added by B. Russell and A.N. Whitehead to their ramified theory of types (cf. [[Types, theory of|Types, theory of]]), with the aim of disposing of the stratification of concepts (see [[Non-predicative definition|Non-predicative definition]]). In the ramified theory of types, sets of a given type are divided into orders. Thus, instead of the concept of a set of natural numbers, there appears that of a set of natural numbers of a given order. Here a set of natural numbers defined by formulas without using any sets belongs to the first order. If one uses in a definition a collection of sets of the first order but collections of sets of higher orders are not used, then the defined set belongs to the second order, etc. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080390/r0803901.png" /> is a family of sets consisting of sets of a given order, then the set
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An axiom added by B. Russell and A.N. Whitehead to their ramified theory of types (cf. [[Types, theory of|Types, theory of]]), with the aim of disposing of the stratification of concepts (see [[Non-predicative definition|Non-predicative definition]]). In the ramified theory of types, sets of a given type are divided into orders. Thus, instead of the concept of a set of natural numbers, there appears that of a set of natural numbers of a given order. Here a set of natural numbers defined by formulas without using any sets belongs to the first order. If one uses in a definition a collection of sets of the first order but collections of sets of higher orders are not used, then the defined set belongs to the second order, etc. For example, if $S$ is a family of sets consisting of sets of a given order, then the set
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080390/r0803902.png" /></td> </tr></table>
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M = \{ x : \exists y \in S, x \in y \}
 
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$$
 
must belong to the next order, since its definition contains a quantifier over sets of the given order. The reducibility axioms asserts that for each set there is a set of equal volume (that is, consisting of the same number of elements) of the first order. Thus, the reducibility axiom in fact reduces the ramified theory of types to a simple theory of types.
 
must belong to the next order, since its definition contains a quantifier over sets of the given order. The reducibility axioms asserts that for each set there is a set of equal volume (that is, consisting of the same number of elements) of the first order. Thus, the reducibility axiom in fact reduces the ramified theory of types to a simple theory of types.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  W. Ackerman,  "Grundzüge der theoretischen Logik" , Dover, reprint  (1946)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  W. Ackerman,  "Grundzüge der theoretischen Logik" , Dover, reprint  (1946)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Russell,  A.N. Whitehead,  "Principia mathematica" , '''1–3''' , Cambridge Univ. Press  (1925–1927)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Russell,  A.N. Whitehead,  "Principia mathematica" , '''1–3''' , Cambridge Univ. Press  (1925–1927)</TD></TR>
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</table>
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Latest revision as of 20:37, 22 October 2017

An axiom added by B. Russell and A.N. Whitehead to their ramified theory of types (cf. Types, theory of), with the aim of disposing of the stratification of concepts (see Non-predicative definition). In the ramified theory of types, sets of a given type are divided into orders. Thus, instead of the concept of a set of natural numbers, there appears that of a set of natural numbers of a given order. Here a set of natural numbers defined by formulas without using any sets belongs to the first order. If one uses in a definition a collection of sets of the first order but collections of sets of higher orders are not used, then the defined set belongs to the second order, etc. For example, if $S$ is a family of sets consisting of sets of a given order, then the set $$ M = \{ x : \exists y \in S, x \in y \} $$ must belong to the next order, since its definition contains a quantifier over sets of the given order. The reducibility axioms asserts that for each set there is a set of equal volume (that is, consisting of the same number of elements) of the first order. Thus, the reducibility axiom in fact reduces the ramified theory of types to a simple theory of types.

References

[1] D. Hilbert, W. Ackerman, "Grundzüge der theoretischen Logik" , Dover, reprint (1946)


Comments

References

[a1] B. Russell, A.N. Whitehead, "Principia mathematica" , 1–3 , Cambridge Univ. Press (1925–1927)
How to Cite This Entry:
Reducibility axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducibility_axiom&oldid=42169
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article