Re: Well Ordering the Reals


Tony Orlow wrote:
> William Hughes said:
> >
> > Tony Orlow wrote:
> > > William Hughes said:
> > > >
> > > > Tony Orlow wrote:
> > > > > William Hughes said:
> > > > > >
> > > > > > Tony Orlow wrote:
> > > > > > > Matt Gutting said:
> > > > > > > > Tony Orlow wrote:
> > > > > > > > > Virgil said:
> > > > > > > > >> In article <MPG.1df53bd49cfd891098a779@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > > > > > >> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > > > > > > > >>
> > > > > > > > >>> David R Tribble said:
> > > > > > > > >>>> I admit that I can't because I can't create a workable definition for
> > > > > > > > >>>> infinite numbers, so that addition, comparison, etc. work properly.
> > > > > > > > >>>> But your definitions so far are no better.
> > > > > > > > >>>>
> > > > > > > > >>>>
> > > > > > > > >>> define x>y if the most significant digit where they differ is a 1 in x and a
> > > > > > > > >>> 0
> > > > > > > > >>> in y.
> > > > > > > > >>> prove: n+1>n
> > > > > > > > >>
> > > > > > > > >> But in TO's "infinite sequences of digits with two ends" how does one
> > > > > > > > >> tell which digit is most significant? TO keeps glossing over this
> > > > > > > > >> critical issue.
> > > > > > > > > As in all regular digital systems, the leftmost nonzero digit is most
> > > > > > > > > significant. You knew that.
> > > > > > > > >> Given two such TO-strings, each with a single 1 and the rest zeros, how
> > > > > > > > >> does on test them to see which , if either, is larger? What objects is
> > > > > > > > >> TO using to index the digit positions?
> > > > > > > > > Actually, one need not index the entire string of bits. Knowing that all bits
> > > > > > > > > are the same except some set of corresponding bits is sufficient to order any
> > > > > > > > > pair of T-riffic numbers. The number with a 1 in the most significant differing
> > > > > > > > > bit is larger.
> > > > > > > >
> > > > > > > > But how does one decide *which* is the most significant differing bit, without
> > > > > > > > some sort of index system?
> > > > > > > Of course you have "some sort" of indexing system. If we consider two numbers
> > > > > > > to have the same number of bits, then that is sufficient. Say I add 1:000...000
> > > > > > > and 0:010...101. Do I need to know the exact number of bits? If they are the
> > > > > > > same, the sum is 1:010...101. If I subtract I get 0:101...011. I don't need to
> > > > > > > know how many bits the ellipses cover. It could be finite, countably infinite,
> > > > > > > uncountably infinite, or worse. it doesn't make a difference. You have what you
> > > > > > > call "relative indexes".
> > > > > > >
> > > > > > > Consider a real world example. We have a straight two-lane road. At one point,
> > > > > > > the lanes split up, one continues straight, while the other takes a detour
> > > > > > > around the other side of the mountain. Otherwise, they are together. Do we need
> > > > > > > to know how long this road is, to know that the second lane is longer than the
> > > > > > > first? No, as long as we know that everywhere else they are equal, we can say
> > > > > > > that the lane that takes the detour is longer. Hope that helps.
> > > > > > > >
> > > > > > > > >>> Does that do it for you? If not, why not?
> > > > > > > > >> It does not explain how one tells which of two digit different positions
> > > > > > > > >> is the more significant.
> > > > > > > > > Whichever is to the left of the other, as usual.
> > > > > > > > >
> > > > > > > >
> > > > > > > > Without an indexing system, how can you tell which is to the left? Remember, our
> > > > > > > > typical left-to-right representation is simply one form of an indexing by the
> > > > > > > > natural numbers.
> > > > > > > You can have relative finite indexes in the middle of the infinite bit string.
> > > > > > > All else being equal, ...101... is larger than ...010... and smaller than
> > > > > > > ...110...
> > > > > >
> > > > > > And presumably, ...010000.... is greater that both ...000101....
> > > > > > and ...000110...
> > > > > > So how can I tell the difference between ...010... and ...010000... ?
> > > > >
> > > > > You really can't without additional specification,
> > > >
> > > >
> > > > BINGO! The whole question is "what is this additional specification"
> > > The places of the bits, either relative or absolute.
> > > >
> > > > >because you don't know which
> > > > > three bits the left ones correspond to in the right ones. If you say the
> > > > > leftmost 0 is in the same place,
> > > >
> > > > What does it mean to say "the leftmost 0 is in the same place" ?
> > > >
> > > > - William Hughes
> > > >
> > > >
> > > Oh come on. It means the bits are the in the same position, the exact same
> > > number of places from the digital point
> >
> > Ok, where is the digital point and what exactly do you mean
> > by "the exact same number of places". Please illustrate using
> > an example where exactly one digit is not 0.
> >
> > -William Hughes
> >
> >
> Say you have ...001000.... and ...000100..., where the ellipses denote infinite
> strings of 0's in corresponding places, or any infinite identical strings, for
> that matter. The first is obviously more than the second. If the ellipses ARE
> all zeroes, the first is twice the value of the second. We don't even need to
> know where the digital point is. It could be infinitely far to the left or
> right, or right in the middle of the known digits, as long as it's in the same
> place in both strings.


If we don't know where the digital point is how can we tell that is
is the same place in both string?. What does "the same place" mean
exactly?

-William Hughes

.

Relevant Pages

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