Re: Dial 999 for the real number line

On May 6, 12:59 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Sun, 06 May 2007 16:49:07 +0100, Six Letters wrote:
On Tue, 1 May 2007 13:44:03 +0000 (UTC), Dave Seaman <dsea...@xxxxxxxxxxxx>
wrote:
But there are only countably many real numbers that can be specified by a
formula or an algorithm, leaving uncountably many that cannot be so
specified.
There aren't uncountably many anything. One of the important
consequences of this way of understanding infinity is that the distinction
between countable and uncountable is undermined. There is only one
infinity.

Do you claim that there is a bijection between the naturals and the
reals?

....

You will never understand real numbers until you get over your hangup on
decimal representations. A real number is an equivalence class of Cauchy
sequences of rationals. Or, a real number is Dedekind cut. Either
definition will do, and neither makes any reference to decimal digits.
You are only confusing yourself by dragging in irrelevant considerations.



....

Every real number has an infinite decimal expansion. It exists, whether
we can write down all the digits or not.



....

Do you accept the axiom of the power set?


....

Yawn. Another long-winded way of pointing out that your table of numbers
is dense in [0,1].

....
There is no axiom that says we can count to infinity. There is, however,
an axiom that implies the existence of an infinite set. There is a
difference.

....
Wrong. The computable reals are an ordered field, but they are not
complete. That is, there is a set of computable reals that is bounded
above, but for which the least upper bound is not computable.
That would be the conventional point of view. You introduced the
notion of

...

read more »

Hi Dave,

You can represent each Cauchy sequence or Dedekind cut, of a number
between zero and one, as a string of digits, no, ie a sequence no
different in value? Their definitions are equivalent (and some don't
find Dedekind/Cauchy sufficient to represent all the real numbers,
only some of them).

Yeah, 1 = 1.0000..., and 0 = (0,0,0,0,0,...).

I don't accept the axioms of regularity or infinity.

Do you accept that there are everywhere and only real numbers between
zero and one? There are. (Well-order the reals, divide by zero.)

If an algorithm describes constructing an infinite sequence, then have
it do each of them. Then, where as above you claim no algorithm can
ever construct all of them, constructing Cauchy sequences or Dedekind
cuts won't, and there are thus some that aren't so built. Perhaps
then above you meant computable.

Ross

--
Finlayson Consulting
C/C++/Java

.

Relevant Pages

  • Re: Different Kinds of Infinity
    ... hypothesis that it is in fact the correct algorithm and that it has ... successful prediction. ... other substrings within the normal number that match that finite sequence. ... the infinity of "correct" sequence ...
    (talk.origins)
  • Re: Different Kinds of Infinity
    ... hypothesis that it is in fact the correct algorithm and that it has ... successful prediction. ... other substrings within the normal number that match that finite sequence. ... the infinity of "correct" sequence ...
    (talk.origins)
  • Re: aleph_0 1 2 but not 3
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    (sci.math)
  • Re: Different Kinds of Infinity
    ... hypothesis that it is in fact the correct algorithm and that it has ... successful prediction. ... other substrings within the normal number that match that finite sequence. ... the infinity of "correct" sequence ...
    (talk.origins)
  • Re: Different Kinds of Infinity
    ... hypothesis that it is in fact the correct algorithm and that it has ... successful prediction. ... other substrings within the normal number that match that finite sequence. ... the infinity of "correct" sequence ...
    (talk.origins)
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