Re: Cantor was Right!



Tony Orlow (aeo6) wrote:
> anzaurres1@xxxxxxxxxxx said:
> > Tony Orlow (aeo6) wrote:
> > > anzaurres1@xxxxxxxxxxx said:
> > > > Look. If I came to sci.physics, claiming that I have a refutation of
> > > > the idea that atoms contain nucleii and then admitted that I don't know
> > > > what an atom is, I would be laughed out of sci.physics, wouldn't I?
> > > >
> > > > That's exactly what you are doing at sci.math. The concept of a Cauchy
> > > > sequence is the basis for understanding real numbers. If Cauchy
> > > > sequences didn't have limits, real numbers could very well be countble.
> > > > But because they do - the real numbers are uncountable.
> > > >
> > > > Explain something to me. Judging by your interest in the countability
> > > > of real numbers, you are a fan of mathematics and real numbers. Then
> > > > why have you denied yourself the real pleasure of learning what other
> > > > people, interested in math, have discovered about real numbers? And I
> > > > don't mean anything advanced. Just your basic sophomore calculus.
> > > >
> > > > Why do you have time to post zillions post to sci.math but don't have
> > > > time to learn calculus? Wouldn't learning be more fun?
> > > >
> > > >
> > > I have leanred a lot here, actually. It's a long time since i have been able to
> > > afford to go to school, and now have a good sized family to take care of. The
> > > countability of the reals is something that perhaps I am not understanding. It
> > > seems to depend on there being only finite numbers of digits in each, to be
> > > countable?
> >
> > If you define reals as sequences of decimal digits, yes. Sort of. That
> > is, if you consider only the numbers that have finite numbers of digits
> > in them, then this set is countable.
> That seems rather arbitrary, especially in light of the fact that an infinite
> set of naturals necessarily contains infinite naturals, and therefore, by this
> thinking, is uncountable. Is this why you cling to your concept of finite
> naturals, while claiming the set is infinite, so you can pretend to be talking
> about infinity?
> >
> > But the converse is not true. The set of rational numbers contains
> > members whose expression in decimals is infinite, such as 1/3 or 1/7.
> > Yet this set is still countable.

> Even if the numerators and denominators can assume infinite values?

What do you mean by that?

Well, you can certainly add 2 extra terms to your rational field R:
"-oo" and "+oo" to denote minus and plus infinity. Then we can define
devisions invloving them as:

a/(+oo) = 0 for any a in R
a/(-oo) = 0 for any a in R
(+oo)/a = +oo for any non-zero a in R
(-oo)/a = -oo for any non-zero a in R
(-oo)/(+oo) is undefined
(-oo)/(-oo) is undefined
(+oo)/(+oo) is undefined
(+oo)/(-oo) is undefined
(-oo)/0 is undefined
(-oo)/0 is undefined

But adding 2 extra members to an infinite set won't change its
cardinality by much, will it? :-)

.

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