Re: There are uncountably many irrationals

mueckenh_at_rz.fh-augsburg.de
Date: 01/13/05 

Date: 13 Jan 2005 11:06:29 -0800

Emil Vatai wrote:
> mueckenh@rz.fh-augsburg.de wrote:
> > Do you actually teach your
> >
> >>>students that, "To speak of an infinite set of finite (natural)
> >>>numbers (as Cantor did) is a contradictio in adjecto?"
> >
> >
> > Do you think it is not a self-contradiction? As long as n is a
finite
>
> That is not contradictory at all! You should learn English properly.
> infinite set is a set with infinite no of elements = that is there
> exists a bijection between IN and an S real (not trivial) subset (eg
IN
> - {0} if IN ={0,1,2...} ).
>
> If the number of elements (in a set) is infinite, that doesn't
> contradict that all of the elements have the property that they are
finite.

But 1 the set {1,2,3,...,n} has the cardinal number n. And for n
finite, the cardinal number is finite too. Is this so hard to see?
>
> short: set is infinite, elements finite. right?
> is that so hard to understand?

Not, if the elements are rational numbers. But it is impossible for
natural numbers.
>
> > number, we have the ordinal n being the same as the cardianlity of
the
> > sequence 1,2,3,...,n. Do you agree?
> IN != {1,2...n}

No, but there is no element omega within IN.
>
> Do you know the definition of IN (Peano's axioms)
>
> >
> > In short: Ordinal = Cardinal in the finite realm. Agreed?
> >
> > Now you must only get to learn that there is not a single infinite
> > number in IN, to share my conclusion.
> >
> >>I wonder if he tells them that there is no number sqrt(2), and
> >>if so how he tackles Pythagoras's theorem and finding the cosine
> >>of 45 degrees. :-)
> >
> >
> > My students can find the approximation for sqrt(2) as well as
others.
> > (I teach it by using a Cauchy sequence.) And more is impossible.
> > Regards, WM
>
> And what does that method approximate? nothing because there is no
> sqrt(2)? or something different from sqrt(2) ????? isn't THIS
contradictory?

What does the sequence (-1)^n lead to? Wouldn't you like to call it by
a nice name?
>
> sqrt(2), defined as sup{ x in IR | x*x <= 2 }, does exist, because we

> define IR so that such a number does exist in it!
>
> For mathematics something exists if you can describe it! (this might
not
> be 100% true). and sqrt(2) clearly exists as that supremum, and also
as
> a limits of sequences

A number exists, if you can say how often it contains the unit 1.
Regards, WM

(I will a.s.a.p. answer your e-mail.

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