Re: Well Ordering the Reals

Virgil said:
> In article <MPG.1de5394567f6ae1998a712@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > David R Tribble said:
>
> > > If your N and 2^N-1 were really different (infinite) numbers, then your
> > > ...111(2) and your "unit infinity" also must be different numbers.
> > They are. N=1:000...000. ....1111 might be considered a poorly formed version
> > of N-1, but one really can't tell how that relates to 1:000...000. I am not
> > sure why you say this anyway.
>
>
> TO posits a seqeuence starting with an infinite sequence of 0's and
> ending with an reversed infinite sequence of zeros, which he denotes by
> 000...000.
Yes, it is an "uncountably" infinite string of 0's.
>
> If we call those zeros that are only a finite number of places from the
> left end of his hypothetical infintie string "initial digits" and those
> only a finite number of places from the right end "terminal digits", one
> must ask where the intitial digits and terminal digits get connected to
> each other.
Certainly, no countably infinite subsequence can make it halfway across any
uncountably infinite sequence, so those in finite positions with respect to the
right can never reach those in finite positions with respect to the left.
>
> Clearly there can never be any initial digit immediately followed by a
> terminal digit as that would only make a finite string.
Absolutely!
>
> Are there "middle digits" that are infinitely many places from both ends?
> If so, how do the initial or terminal digits connect up with them.
Yes, there have to be bits between the initial and terminal countably finite
strings. In fact, within the uncountably infinite sequence of bits, there must
be uncountably many disjoint countably infinite subsequences. The question of
the boundaries between these sequences, of where there finite positions from
the left or right end, is one that cannot be clearly answered. On a finite
scale, it certainly seems obvious that no single step can ever go from a finite
to an infinite position, and yet, it is (or should be) obvious that if there
are truly an (uncountably) infinite number of elements in the sequence, that
there must be positions infinitely far apart. This is a paradox, not in that
there is a distinct contradiction logically, but in that we cannot imagine how
this transition could occur.
>
> It is just as clear that no initial digit or terminal digit can be
> adjacent to a middle digit unless the successor to a finite natural can
> be infinite.
On the finite scale it makes no sense. On the infinite scale, if you picture
the number line as the interval [0,1], then the initial sequence appears just
as the points immediately adjacent to 0, and the terminal points are
immediately adjacent to 1, and the vast majority are between them. Ask yourself
whether all points a finite number of points from the endpoints looks any
different from that picture, and ask yourself where the first point is that is
a finite distance from either point. The paradox is there, but it's simply a
matter of perspective. I don't think there is any "semi-infinite" scale that
would resolve this at all. We just have to live with it.
> > >
> > > However, it is provably true that your N, 2^N-1, and ...111(2) are all
> > > exactly the same (infinite) number. I've pointed this out to you
> > > several times, and you have never proven otherwise.
>
> > I certainly can't prove that within your system. Our systems are
> > incompatible.
>
> TO's system is incompatible with every system, including itself.
Oh Virgil, shut up. It's comments like that that make me toss most of your
posts. Only in Virgilogic!!! Ooops!! Looks like Virgil GOOFED again!!! blah
blah blah. Don't you have any humorous Latin quips or something?
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

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