In article <············@mulga.cs.mu.OZ.AU> ···@munta.cs.mu.OZ.AU (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
uncountable.
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.
--
Stephan Kepser ······@cis.uni-muenchen.de
CIS Centrum fuer Informations- und Sprachverarbeitung
LMU Muenchen, Wagmuellerstr. 23/III, D-80538 Muenchen, Germany
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