Re: Well Ordering the Reals

imaginatorium@xxxxxxxxxxxxx said:
> Tony Orlow wrote:
> > imaginatorium@xxxxxxxxxxxxx said:
> > >
> > > Tony Orlow wrote:
> > > > imaginatorium@xxxxxxxxxxxxx said:
> > > > >
> > > > > Tony Orlow wrote:
> > > > > > Daryl McCullough said:
> > > > > <snip>
> > > > > > > On the other hand, if you include the *infinite*
> > > > > > > bit strings, then there is no (known) well-ordering of
> > > > > > > the infinite bitstrings. Your ordering of the reals
> > > > > > > relies on the lexicographic ordering of the
> > > > > > > set of bitstrings, and that is not a well-ordering.
> > > > > > Why is that? The dictionary certainly nevers seems to be confused about what
> > > > > > comes after what, or what comes first. What condition of well ordering does
> > > > > > that ordering violate?
> > > > >
> > > > > You know, the one truly astonishing thing about all this is that as you
> > > > > spew out your endless nonsense, people like me are actually learning
> > > > > something from the process. OK, a slim volume I have here is the
> > > > > Penguin Dict. of Math., and it says "A set is well-ordered if it is
> > > > > ordered [I suppose that means a total order] and every subset has a
> > > > > first element."
> > > > >
> > > > > Well, in the lexicographic ordering of infinite bit strings, the set
> > > > > {1000..., 0100..., 0010..., ...} has no first element. Ergo it isn't
> > > > > well-ordered.
> > > > >
> > > > > Go on - now babble a bit. <g>
> > > > If you are working with the infinite bit strings, then it is only your
> > > > restriction of finiteness on the number of intermediate zeroes which makes that
> > > > sequence not end. If you allow the intermediate zeroes to be infinite in
> > > > number, then the bottom of that chain is ...00001, wouldn't you say?
> > >
> > > Yes, I suppose if I were at the point at which infinity ends, I woud
> > > say that, but I would also say everything else, and I would be the King
> > > of Peru, because there is no such place.
> > >
> > > Yawn oh yawn, you can now do your "Bongo" trick, where you say "No
> > > largest natural yahooo!", "No end to the unending.... Spliftjhgy!" Or
> > > whatever. You simply cannot grasp the concept of an infinite set. You
> > > can understand a big set, or a verrrry big set, or a very very really
> > > hugely enormous set, but you cannot understand the idea of considering
> > > an unending sequence in its totality. Of course not, because you have
> > > this simplistic idea that "totality" must mean you got to the last one.
> > > Well, it doesn't. Never mind.
> > Uh, what was that leftmost bit up there? Is that in an infinite position or
> > not? Did I put that there? When I talk about the sequence ending, I am talking
> > about YOUR sequence, which it would appear to continue down to ...00001.
>
> MY sequence appears to YOU to "continue down to" where it ends, because
> you cannot conceive of anything that doesn't end; fairly clearly you
> believe that by letting go of some inhibitions you can "get to"
> infinity, where unending sequences end. But this is just your
> misunderstanding.
No, sorry Brian, you are confused. My strings have one end, where the root x is
in the exponentiation. The first bit represents the overall sign of the
expression, and subsequent bits represent the signs of the exponents. The sign
all the way to the left is the root bit, but in the bit strings it is on the
right end, as the least sgnificant bit, like the binary naturals. So, there is
a right end to the bit string, not the left. If you have turned the string
around to make it seem otherwise, that does not change the fact that you have
an ascending sequence. Since I have a countably infinite number of bits, any
bit you choose will only be a finite distance from the root bit, and therefore
will have a finite number of bits and a finite value.
>
> OK, if it helps, consider the sequence (using - just to separate into
> pieces):
>
> 0-000...
> 01-00-000...
> 001-000-000...
>
> In other words, element 0 is 000...., element n for n>0 consists of n-1
> zeros followed by a 1, then n zeros, then indefinitely many zeros. In
> this sequence, the first marker '-' can never be closer to the second
> marker than to the beginning, so however powerful your infinity machine
> is, anything ending in ...000001 will never appear.
You need to take a closer look at how the bit strings are defined. You're not
making any sense in the context of my well ordering. Thise strings should be
starting at the right, at the root bit. You are trying to pretend we have an
infinite string of bits to the right of the root bit, but this is NOT a digital
representation in the normal fashion. There are NO bits, implied or otherwise,
to the right of the root bit.
>
> You still have not grasped the notion of an endless sequence.
You don't seem to grasp that these sequences of bits have a beginning which is
the least significant bit at the right end, and that adding bits leftward from
there is an ascending sequence in this well ordering.
>
> Brian Chandler
> http://imaginatorium.org
>
>

--
Smiles,

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

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