Re: Cardinality of Set of Computable Numbers?

From: Arthur J. O'Dwyer (ajo_at_nospam.andrew.cmu.edu)
Date: 12/27/03 

Date: Fri, 26 Dec 2003 19:30:08 -0500 (EST)

[Strangely, only the r.o.mensa crosspost was added by Herc.
Why is this thread crossposted to all these groups?]

On Sat, 27 Dec 2003, |-|erc wrote:
>
> This is my point,
>
> We label rationals as countable, and irrationals as not countable!

  We label *the set of* rational numbers "countable," and we label
*the set of* irrational numbers "uncountable." Correct. It's also
true that the set of irrational numbers is uncountable *whether we
label it as such or not.* This is an important point which we
should not lose sight of. :)

> Rational is just counting functions over_the_division_operation!

  [I'm assuming the _s indicate emphasis. I'd prefer to see _this_
style of emphasis, with the bars on the "ends" of the emphasized
section, too -- but that's purely for my own convenience.]
  True, as long as you define what you mean by "over the division
operation." It sounds like you can have a pretty good definition
worked out with a bit of thought, though: the set of rationals is
the set you get by "attempting to close" the set of nonzero integers
over the division operation, if you see what I mean.

> SQRT(2) and pi are equally countable as rationals, in the same list
> as rationals.

  Yes, the set {rationals, and SQRT(2), and pi} is also countable.

> A lot of irrationals are countable!

  A *whole* lot of irrationals are countable -- in fact, if you
give me a list of irrational numbers, no matter how long it is,
even an infinitely long list, I *guarantee* you the number of
elements in that list will be countable! :)
  However, it's impossible to count *all* the irrational numbers.
You can't map R onto N. The *set of* irrational numbers is
uncountable, unlike the *set of* rational numbers.

> All the computable ones.
> Why give division special precedence when we are concerned with countability
> and forget the majority of useful numbers.

  These two phrases don't make sense on their own.

> begin 666 countables.gif

  Rude. I've taken the time to extract and post your image at
http://www.contrib.andrew.cmu.edu/~ajo/herc_is_rude.gif

  Note that your diagram is perfectly accurate: the set of rationals
is countable; the set of irrationals is not; and the set of
computable reals is. I don't understand what you mean by the
notation "NC numbers" in the bottom left corner, nor what you
mean by labeling the two halves of the chart "Todays Understanding"
and "Tomorrows Understanding" -- are you hoping the OP will
understand "tomorrow"? :)

-Arthur

Relevant Pages

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