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Difference between revisions of "Transfinite sequence"

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''of elements of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937101.png" />''
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$#C+1 = 8 : ~/encyclopedia/old_files/data/T093/T.0903710 Transfinite sequence
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A mapping of an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937102.png" /> of (transfinite) ordinal numbers into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937103.png" /> (cf. also [[Ordinal number|Ordinal number]]). By an element, or term, of the transfinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937104.png" /> is meant an ordered pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937107.png" />; this term is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093710/t0937108.png" />.
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''of elements of a given set  $  X $''
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A mapping of an interval $  [ 0, \beta ) $
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of (transfinite) ordinal numbers into $  X $(
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cf. also [[Ordinal number|Ordinal number]]). By an element, or term, of the transfinite sequence $  f:  [ 0, \beta ) \rightarrow X $
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is meant an ordered pair $  ( \alpha , x) $,  
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where $  \alpha \in [ 0, \beta ) $
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and $  x= f( \alpha ) $;  
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this term is often denoted by $  x _  \alpha  $.

Latest revision as of 08:26, 6 June 2020


of elements of a given set $ X $

A mapping of an interval $ [ 0, \beta ) $ of (transfinite) ordinal numbers into $ X $( cf. also Ordinal number). By an element, or term, of the transfinite sequence $ f: [ 0, \beta ) \rightarrow X $ is meant an ordered pair $ ( \alpha , x) $, where $ \alpha \in [ 0, \beta ) $ and $ x= f( \alpha ) $; this term is often denoted by $ x _ \alpha $.

How to Cite This Entry:
Transfinite sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transfinite_sequence&oldid=49010
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article