Lebedev–Milin inequality: Difference between revisions

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In [[non-standard analysis]], a '''hyperinteger''' ''N'' is a [[hyperreal number]] equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer.  An example of an infinite hyperinteger is given by the class of the sequence (1,2,3,...) in the ultrapower construction of the hyperreals.
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==Discussion==
The standard [[integer part]] function:
:<math>[x]</math>
is defined for all real ''x'' and equals the greatest integer not exceeding ''x''.  By the [[transfer principle]] of non-standard analysis, there exists a natural extension:
:<math>^*[\,\cdot\,]</math>
defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if:
:<math> x = {}^*\![x]</math>.
Thus the hyperintegers are the image of the integer part function on the hyperreals.
 
==Internal sets==
The set <math>^*\mathbb{Z}</math> of all hyperintegers is an [[internal set|internal subset]] of the hyperreal line <math>^*\mathbb{R}</math>. The set of all finite hyperintegers (i.e. <math>\mathbb{Z}</math> itself) is not an internal subset. Elements of the complement
 
:<math>^*\mathbb{Z}\setminus\mathbb{Z}</math>
 
are called, depending on the author, ''non-standard'', ''unlimited'', or ''infinite'' hyperintegers. The reciprocal of an infinite hyperinteger is an [[infinitesimal]].
 
Positive hyperintegers are sometimes called ''hypernatural'' numbers.  Similar remarks apply to the sets <math>\mathbb{N}</math> and <math>^*\mathbb{N}</math>.  Note that the latter gives a [[non-standard model of arithmetic]] in the sense of [[Skolem]].
 
==References==
* [[Howard Jerome Keisler]]: ''[[Elementary Calculus: An Infinitesimal Approach]]''. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
 
 
{{Number systems}}
{{Infinitesimals}}
 
[[Category:Non-standard analysis]]
[[Category:Infinity]]
[[Category:Calculus]]

Latest revision as of 21:16, 25 November 2014

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