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| ``Re: more refined argument against ``is a triple'' -type definitions''
by rspuzio on 2006-03-19 21:57:31 |
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| > The fact that when you ``look inside'' a real number you see Cauchy
> sequences of rational numbers is irrelevant to what the real numbers
> *are*.
Alright then, what is a real number, really?
The only way I know to answer this question is to first pick some
axiomatic framework and work within it. In one framework, real
numbers are equivalence classes of Cauchy sequences. In another
framework, they are Dedekind cuts. In a third, they are objects which
satisfy certain properties.
Of course, one should complememnt this with a description of how to
relate these different axiomatic descriptions. Which sequence
corresponds to which set of rational numbers? How does one assign a
Cauchy sequence or a Dedekind cut to an object satisfying the axioms?
In a way, this is much like motion in Physics. To describe the motion
of an object, we need to first pick a frame of reference, then say
what is going on in that reference frame and be careful to not
indiscrimainately mix up things from different reference frames
without first transforming them into a common reference frame.
Just as it is meaningless to ask whether an object is at rest except
witin the context of a particular reference frame, so I would say that
this business of looking inside real numbers is similarly meaningless
except in the context of a specific axiomatic framework. In the first
system, you see sequences; in the second you sets of rational numbers;
in the third, you don't see anything becasue looking inside is
undefined.
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