Re: The state-of-the-art in mathematics

From: E. E. Escultura (escultur36_at_hotmail.com)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 21:03:42 +0000 (UTC)

On 29 Dec 2004, David C. Ullrich wrote:
>On 29 Dec 2004 05:59:17 -0800, "msherwood1@gmail.com"
><msherwood1@gmail.com> wrote:
>
>>Thanks.
>>
>>In the course I just took, we took it as axiomatic. But this course
>>was a 'warm-up' to Real Analysis, so maybe they thought we just weren't
>>ready for it.
>>
>>Actually, I did just get a book out of the Libary by A.G. Hamilton
>>called "Numbers, Sets, and Axioms"..it looks like it's the book we
>>should have been using for the course I just mentioned. It looks like
>>it does exactly what you just said re: Cauchy sequences.
>>
>>Finally, I just ordered Dedekind's "The Nature and Meaning of
>>Numbers"... that should probably clear up the whole thing... (I'm told
>>that a much better rendering of the title is "What are Numbers, and
>>What are they For?").
>>
>>The argument I made for the difficulty with Trichotomy in the Reals,
>>which my Professor found utterly unconvincing, was as follows:
>>
>>"Assume a Computer Program whose job it is, being fed two real numbers
>>"x" and "y" in decimal representation (note: any base would do), to
>>determine: Case 1) x > y, Case 2) x =y, or Case 3) x<y. The Program
>>knows nothing about "x" and "y" other than the steam of decimals it is
>>being fed.
>>
>>Now assume that the Computer continually receives "identical
>>information" about "x" and "y". At no point can said Computer make the
>>above "Case 1/Case 2/Case 3". determination.
>>
>>Conclude that Trichotomy breaks down in the Reals."
>>Is this sophistry, or is there merit to this Argument?
>
>The argument shows that it can be hard to decide certain things
>given certain sorts of information, and that can be very
>important in some situations. But it certainly does not
>show that trichotomy fails for the reals.
>
>Trichotomy says that for any x, y in R exactly one of x < y,
>x = y and x > y is true. That doesn't say anything about
>the possibility of deciding which one holds in a finite
>amount of time given a stream of arbitrarily many decimals
>in the expansion of x and y; whether or not that program
>can ever determine that two sequences are the same (of
>course it can't in the situation above) they _are_ the
>same, or not.
>
>(Which is exactly what the professor said when he explained
>why he found the argument utterly unconvincing. Not blindly
>believing everything you're told is a good thing. When a
>student points out an error of mine I'm happy about that
>for several different reasons. But when you point out
>that something's wrong, the professor listens to what
>you have to say and is not convinced, then it would be
>a good thing to consider the possibility that _you_
>may be wrong, and listen carefully to his explanation
>of why.)
>
>>Thanks again!
>
>
>************************
>
>David C. Ullrich

I can see David sneaking by the side door because he cannot well-define the concept “real number.

E. E. Escultura

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