Re: Calculus XOR Probability

David R Tribble said:
Tony Orlow wrote:
So, for infinite sets, you want to claim that size
is something OTHER than a number???


Brian Chandler said:
Well, the "size" of an unending sequence can't really be the last
number you shout (oh, or was it 'sing') from the ditty, can it, since
there isn't a last number.


Tony Orlow wrote:
Is that true of all infinite sets? Isn't "1" the last number chanted in the
ditty of reals in [0,1]?


Tony Orlow wrote:
Do remind me how this ditty starts? I mean, I assume "0, ..." but what
follows 0?


Tony Orlow wrote:
(sigh) 0.000...000, 0.000...001, 0.000...010, ... , 0.111...110, 0.111...111,
1.000...000. That's the series (x=0->Big'un: Lil'un*x).

If these "numbers" of yours are infinitesimals, they do not exist in
the standard real number line. If these are standard reals, they do
not exist as consecutive well-ordered points. Either way, they do
not comprise a sequential denumeration (the "ditty") of the reals.

They do not exist in the standard mathemtical definitions. They're obviously
non-standard. Are you just getting that now? These are discrete infinitesimal
real intervals ala Ross' degenerate intervals and well ordered reals,
represented as extended digital strings.


These "numbers" of yours also contradict your other posting:
With x<z -> x<y<z we have an intermediate value between any two distinct real
values, and by recursive logic we then have an infinite number of nested
intervals between x and z. That's internal infinity. Peano defines external
infinity such that y -> y<z, which can be restated in two directions as
y -> x<y<z. While standard arguments claim the Peano set only generates finite
values, any infinite number of recursive iterations will produce an infinite
number of elements in the external set.

This is a contradiction because your 0.000...000 looks like some x,
and your 0.000...001 looks like some z, so where is y=(x+z)/2, x<y<z,
in your list?


Uh, did you deliberately answer this question with the one before it? Those are
not distinct standard reals, because they're only infinitesimally different. At
any point where you have subdivided the interval an infinite number of times,
no more subdivisions are necessary to ensure one has a number arbitrarily close
to any standard real, at least in terms of being less than any finite distance
from that real.

On the other hand, one *can* define internal infinity within one of these
intervals, and maintain continuity at the subinfinitesimal level. It's just not
necessary for distinguishing standard reals.

--
Smiles,

Tony
.

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