Re: Well Ordering the Reals

Virgil said:
> In article <MPG.1df793669defa5c198a7a6@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1df62e22f9ea958f98a781@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > >
> > > > Virgil said:
>
> > > > > TO is assuming that whenever two strings are to be compared this is
> > > > > possible, but without some mechanism to compare the realtive
> > > > > positions of arbitrary digits, this is not possible.
> > >
> > > > If you do not specify the number of bits, then you don't know what number
> > > > you
> > > > are talking about in the first place
> > >
> > > Then precisely how does one specify the "number of bits"? That has been
> > > my question all along!
>
> > I answered it. Maybe you haven't gotten to that post yet. One defines the
> > number of bits as a formula using N, such as log2(N) bits in 1:000...000
> > denotes N, log2(N)-1 bits denotes N/2, N/2 bits denotes sqrt(N), N bits
> > denotes
> > 2^N, etc. etc.
>
> Since TO's N is variable, that means that his "numbers" are variable too
> so that which of two numbers is larger could depend on what the value of
> N is at that moment, but cold change momentarily.

Sure, you have to assume equal N, like equal ellipses, for comparison. Infinity
is pretty relative.

> > >
> > >
> > >
> > > > but it is not necessary to know the
> > > > exact
> > > > number in order to compare it with another number. It is only necessary
> > > > to
> > > > determine which has the most significant 1 bit where the other has a 0.
> > > > Depending on how you define your numbers, this may not always be
> > > > possible, but such numbers are not well-formed.
> > >
> > > One issue is how TO defines HIS numbers, as they are not numbers by
> > > anyone else's standards. And the second is, how does TO know that his
> > > "numbers" are orderable if there are such ill formed numbers that defy
> > > comparisons?
>
> > Those numbers are not part of the set.
>
> A TO-number starting with a zero and having infinitely many zeros
> following it then a 1 then infinitely many more zeros then a final zero
> must always be a part of his set, at least as he has described it.

It's also part of your set, if that 1 is in a finite position, realtive to the
digital point. See? You have one anchor, the digital point, and everything must
be within a finite number of positions of it. But, you can have multiple anchor
points defined whch are infinitely distant from each other, with finite
specifications around them, and infinite bit patterns between, or the
assumption of equal infinite strings, and then comparison is possible.
>
> But two such TO-numbers cannot be compared for size unless one is told
> in advance which is larger or that they are equal.

Just like in finite digital systems, you have to know where your anchor point
is.

>
> > Can I say that the normal binary
> > system
> > doesn't work because you can't tell me what 10100.10100.00100 is? You would
> > say
> > that's not a number in that system. Well, ...010101 is not a T-riffic number.
> > It could be 1:010....0101 or 0:10101....0101, and we have no idea of the
> > number
> > of bits, so it is not specified correctly. That doesn't make 1:0101...0101
> > ill-
> > defined. It's 4N/3 (really (4N-1)/3, since there is another 1/3 not included,
> > to the right of the binary point).
>
> Then 0:0000....000010000....0000 and 0:000...010...000 are both
> properly defined "numbers" in TO-numerics, so there must be some rule
> for determining from their representations which is larger.
Not unless you specify where the central parts correspond.
>
> So how does one tell, strictly from their representations, which is
> larger?
>
> > > The issue is whether there is a mechanism to compare sizes of any two
> > > given "TO-numbers" or not. Apparently not.
>
> > If you give two actual T-riffic numbers, thay can always be compared.
>
> 0:0000....000010000....0000 and 0:000...010...000?
Okay my statement wasn't quite right. You need two T-riffic numbers with
corresponding bit positions. You haven't specified that for the middle part.
> > >
> > > Since there is such a method for natural numbers, TO-numbers are
> > > unnatural.
> >
> > Oh they are not only natural, but truly organic.
>
> They are as organic as compost and no more mathematically relevant.
Around here they have a composter's guild. Pretty neat, huh?
>

--
Smiles,

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

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