Re: Is continuum completely filled up?

>, MoeBlee writes

Andy Smith wrote:
OK, another ignorant question.

A real number can be defined by a Cauchy sequence

A real number is not a Cauchy sequence, but rather a real number is an
equivalence class of Cauchy sequences.

But if a real number can be represented by a sequence of rationals, why
do we need any other numbers than rationals?

To be the least upper bounds and greatest lower bounds of sets of
rationals, which allows that there exists things such as the square
root of 2.

Or is it that the actually
infinite set of a sequence of rationals is logically something other?

A convergent sequence of rationals is a Cauchy sequence. But a real
number is not just a Cauchy sequence but rather is an equivalence class
of Cauchy sequences. The members of the equivalence class all converge
to the same limit; so a real number is that set of Cauchy sequences
that converge to that real number (the actual definition, though, is
not circular as it would seem from what I just wrote, as you can see
from such treatments as Suppes's 'Axiomatic Set Theory').

Thanks. I had a similar clarification a few posts back from Dave Marcus who observed that defining all the reals is different from defining one instance of the reals once the reals have been defined, and I need to read up on "defining the reals".

Following a rather chaotic line of thought re the title of this thread, what about space filling curves - covering the plane with a curve is the same issue as covering the line with points.

Space filling curves e.g. Hilbert, Peano are fractals; there is a generator function - a suitably line defined in e.g. a square. At each iteration the generator is e.g. reduced in linear scale by a factor 2, replicated by 4 (with some defined rotations for Hilbert) and the scheme is then iterated. As I understand it, the gist of the argument that it is space filling is that, as a function of iteration number n, one can show that one can choose an n such that the distance from any point is less than any eta>0 and that the distance reduces for increasing n.

What about foreground and background - like Escher's array of white birds flying one way with black gaps, but on inspection it is equally black birds flying the other on a white background (Dave Marcus mentioned Godel, Escher, Bach - a book I haven't looked at for 25 years, but it has nice examples of all of that as illustrations). Anyway, what I mean is, suppose that instead of the generator function being a geometric line, make it a thick line, such that the area of the line is the same as the area of its enclosing space - if you wish, you can consider it as two geometric curves of zero width, delineating the boundaries between white and black.

Then, when we iterate using the same argument as for a single line, we can conclude that any point on the plane is covered by both curves. However, by construction, there are as many white points (or holes) as there are black points. And I think that points alternate between black and white on subsequent iterations, so that their black/white status is undefined in the limit? I am now very aware that I shouldn't apply finite reasoning to infinite sets, but in this case there is a complete symmetry between the status of black/white, so anything applicable to black is equally so to white, finite or infinite.

One could maybe do something similar for defining the reals (I have still to read up on that). But one could, following an analogy with the 2D case above do something like - define the reals as a subset of the rationals in a recursive fashion, proceeding 1 bit at a time, so starting with {.0,.1}, then {.00,.01,.10,.11} etc. However, at each iteration n divide the set of rationals into two sets - black (exist) or white(holes), according to : if n even, let black = all even bit strings (last bit = 0), white = all odd (last bit = 1); for odd n, the reverse. At the limit the black/white status of any real number is undefined (the last bit, there isn't one, = either 0 or 1), but any real can still be defined with the rationals in either set - from the point of view of the Cauchy sequence defining a real, it doesn't matter if the last bit (there isn't one) is a 0 or a 1? So maybe it is a matter of definition as to whether the continuum is filled up?


--
Andy Smith
.

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