Re: Cantor was Right!

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.

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.

In any case, defining reals as infinite sequences of digits may not be
the best way to go. I don't remeber why but I remember that it wasn't.

I forgot most other, more productive ways to define reals in terms of
rationals. Russian textbooks are usually very good at that. It's been
many decades since I went through these different definitions and
derivations.

But what I find important is that any Cauchy sequence of rational (or
real) numbers has a real-valued limit.

A sequence A1, A2, A3, ... of numbers is Cauchy if for any e > 0,
there exists an index i such that |Aj - Ak| < e for any k > i and j >
i.

An interpretation is that a Cauchy sequence is "asking for a limit".

> I am not sure what the criterion is, but I have come up with a nice
> enumeration of them that I hope will turn into something useful.

My advice to you is to give it up.

Why? For example, wasn't it you who told me that if the set of reals
were enumeratable, then the area of the 1 x 1 square is equal to 0?

That can't be so, can it?

Do you REALLY think that a square 1 foot x 1 foot has area equal to 0
square feet?!

Or do you think that basic human notion of area is false or
self-contradictory?

.

Relevant Pages

  • Re: A consideration concerning the diagonal argument of G. Cantor
    ... belong to a sequence of sequences, i.e., belong to a countable set. ... all reals" and "There is a sequence containing all reals" mean exactly ... So give us a definition countability that cover ALL countable sets. ...
    (sci.math)
  • Re: Multiple infinities - one more look
    ... continued for lager length of digit sequences without limit. ... infinite digit sequences... ... so the resulting reals have an order. ... (i.e. having a finite program to output their digits in sequence). ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... The way you get reals by a neverending process of generating digits ... and which thereby constructs a sequence of nested intervals whose ... It isn't *already* in the rationals you started with. ...
    (sci.math)
  • Re: limitation to induction on finite bounds
    ... >> if we don't know and never can know all of its digits. ... >> as whether it is normal (AIUI, an expansion is normal if, ... >> sequence, therefore making the real numbers countable. ... The reals are countable if we use similar logic against ...
    (sci.math)
  • Re: limitation to induction on finite bounds
    ... >> if we don't know and never can know all of its digits. ... >> as whether it is normal (AIUI, an expansion is normal if, ... >> sequence, therefore making the real numbers countable. ... The reals are countable if we use similar logic against ...
    (sci.logic)