From: Stephan Kepser
Subject: Re: arbitrary-precision real arithmetic
Message-ID: <424459$>
In article <············> ··· (Bert  
THOMPSON) writes:

> There are no `real' real numbers. Anything that models the real number
> axioms is real. The Loewenheim-Skolem-Tarski theorem states that any
> axiomatization that has an infinite model has infinite models of all
> possible infinite cardinalities, including countable.

The Loewenheim-Skolem theorem holds only for first order languages. Part  
of the axiomatisation of the real numbers is the axiom that every Cauchy  
sequence converges (or an aequivalent statement for example using dedekind  
intersections). These statements involve quantification over functions or  
sets. And it takes at least second order languages to express that. Thus  
the Loewenheim-Skolem Theorem cannot be applied.

The axiomatisation of the reals contains an axiomatisation of the  
rationals and the Cauchy axiom (or something equivalent). No model of  
cardinality omega of the rationals fullfils the Cauchy axiom. Thus a model  
of the reals must have (at least) cardinality Aleph1, i.e. it must be  

Thus Steve Pipkin is right. If you want to use real numbers in  
computation, you must restrict yourself. There you have two main choices.  
Either you work with floats. Floats are - theoretically speaking -  
representatives of equivalence classes of real numbers. You must fix the  
calculation precision before you start the calculation. And no matter how  
precise you calculate, most floats will always represent countably  
infinite sets of reals. If you want arbitrary precision, you must choose a  
countable subset of the reals. And again this choice must be made, before  
you start the computation.

That doesn't mean you can't do anything with "the reals" in practise, if  
you do the right approximations. But in theory, the set of reals as a  
whole is beyond reach.

Stephan Kepser           ······
CIS   Centrum fuer Informations- und Sprachverarbeitung
LMU Muenchen, Wagmuellerstr. 23/III, D-80538 Muenchen,  Germany
Tel: +49 +89/2110666     Fax: +49 +89/2110674