Re: The Dedekind Snap

On Mon, 08 Dec 2008 23:03:43 +0000, John Jones <jonescardiff@xxxxxxx>
said:
Chris Menzel wrote:
On Mon, 08 Dec 2008 19:44:07 +0000, John Jones <jonescardiff@xxxxxxx>
said:
MoeBlee wrote:
On Dec 6, 7:31 am, John Jones <jonescard...@xxxxxxx> wrote:
David C. Ullrich wrote:

Similarly for the question of whether

Dedekind cuts "fall on a line" - whether you like it or not, the
real line is _defined_ to be the set of all Dedekind cuts.
That doesn't square with the idea that a Dedekind cut falls on two
of the same points on the number line, one point from each half.
You're getting yourself confused with whatever your notion of
"falling on a line" is.

And remember that the Dedekind cut construction of the reals
involves Dedekind cuts on the RATIONAL line.
It's made on the reals in the wiki article.

Do have another go at the section "The cut construction of the real
numbers". There's a good lad.

Yes, they said that if the cut doesn't fall on a rational then an
irrational is created by the mathematician.

This nonsense about creation by the mathematician is your own fevered
imagining.

But that's strange, if the line is continuous the irrational is
already there.

The idea of the real line is informal and intuitive. In mathematics,
informal ideas need to be made precise, and pushing too hard on an
intuition without giving it formal expression often leads to confusion,
your case being especially instructive...or comical...or sad...or
something. The Dedekind construction, by contrast, answers the call for
precision and provides a rigorous, mathematical characterization of the
reals in terms of sets of rationals. For more detail (not for you of
course, as you do not believe in self-education), see the complementary
article http://en.wikipedia.org/wiki/Construction_of_the_real_numbers .

This is what I wrote elsewhere:

"But then it looks as if the rationals can express the irrationals.

Because if a loaf of bread (line) of rationals can be cut to reveal an
irrational (a bug in the loaf), then it's not because of the knife
(dedekind) that the bug is found there, but because of the loaf (the
rationals). "

And this is what I wrote in reply:

"A lovely example of how ignorance and imagination together conspire to
foster misleading analogies. You'd make a great flat-earther."

.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... sequences or Dedekind cuts are useful in defining completeness. ... sequences nor Dedekind cuts add anything essential, ... view of what sort of reasoning about reals, functions of reals, and so on, ... The motivation for completing the rationals lies mainly in analysis, ...
    (sci.math)
  • Re: The Dedekind Snap
    ... It's made on the reals in the wiki article. ... I recognized that there is a notion of Dedekind cuts of reals, ... RATIONALS. ... But then it looks as if the rationals can express the irrationals. ...
    (sci.logic)
  • Re: Are the Dedekind cuts uncountable?
    ... >> Of course the Dedekind cuts are uncountable as the reals are defined by ... >> of the subsets of the rationals and linearly moving along the rationals. ... sandwich theorem states that in between any two real numbers there is ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... sequences nor Dedekind cuts add anything essential, ... view of what sort of reasoning about reals, functions of reals, and so on, ... and compare it with our expectations of completeness, ...
    (sci.math)
  • Re: Are the Dedekind cuts uncountable?
    ... > Of course the Dedekind cuts are uncountable as the reals are defined by ... this sequence does not terminate, then the a_i's, being a nonempty ... while every countable subset has measure 0. ...
    (sci.math)