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Paper on representation of the Reals (and others from Darmstadt)


The following paper is now available by anonymous ftp from the site at 
Imperial College.

        A Faithful Computational Model of the Real Numbers
             
                    Philipp S{\"u}nderhauf


ABSTRACT. We investigate the representation of real numbers by
  sequences of digits, thought of as radix expansions. ``Faithful''
  refers to the fact that we overcome the classical problem of
  multiple representations for certain numbers. This is established by
  employing a suitable quasi-uniform structure on the set of finite
  sequences.  (The paper contains a motivating introduction to
  quasi-uniformities.)  The completion of this space adds exactly one
  representative for each real number. Moreover, the quasi-uniformity
  induced on the set of total elements is exactly the usual uniformity
  on the reals.  Hence we do also give a faithful representation of
  the topological structure of the real numbers.

  The quasi-uniformity on our model may be described in a finitary
  fashion: There is a base consisting of relations U_n such that in
  order to determine whether x U_n y holds, one needs to know only 
  the first n digits of the sequences x and y.

  Among the continuous endofunctions on our model, the uniformly
  continuous ones turn out to play a prominent role: They
  correspond to continuous endofunctions on the reals.

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ftp instructions

$ftp theory.doc.ic.ac.uk

Name: anonymous

Password: <your e-mail address>

ftp> cd papers/Jung
ftp> binary
ftp> get reals.ps.Z
ftp> bye

$uncompress reals.ps.Z
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This is the place to draw your attention to the directory papers/Jung
at theory.doc.ic.ac.uk which contains several papers from members of the 
Darmstadt "AG Domains". 

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Philipp S"underhauf          
Fachbereich Mathematik (AG1)
Technische Hochschule Darmstadt
Schlo{\ss}gartenstr. 7          fax: x49-6151-164011
D-64289 Darmstadt               email: sunderhauf@mathematik.th-darmstadt.de