Re: Cantor K.O.'d -- again! - Further explication

From: Matt Gutting (tchrmatt_at_yahoo.com)
Date: 01/07/05 

Date: Thu, 06 Jan 2005 22:14:35 -0500

Mark Adkins wrote:

<snip>

>
> By starting with a list consisting of one member, then adding
> additional members one by one, as the list grows in length,
> the length of the list members also grows, and the length of
> the diagonal grows with them; and it can be seen that the
> diagonal must grow *exactly* as the list members do, otherwise
> the diagonal cannot grow. The diagonal is a derivative entity,
> mirroring the growth of the list members.
>
> Is it possible for such a list to cover the range of natural
> numbers, N? This is equivalent to asking whether it is
> possible to have an L such that: (a) The quantity of list
> members is "infinite"; (b) The length of the diagonal is
> "infinite"; (c) The length of the members remains finite.
>
> Call this hypothetical list L*. If it exists, it may be regarded
> as a kind of limiting case for the process of list growth just
> described. In L* the diagonal is still made of the terminals
> of list members; however, the diagonal must now contain all
> possible terminal symbols.
>
> It may not always be explicitly recognized that a list is
> intrinsically a geometric entity: but this is the case, when
> each member is a string of symbols; each symbol occupies
> an ordinal location in an abstract space, and possesses
> spatial relations of "right" and "left" relative to its neighbors;
> and different members occupy similar ordinal locations,
> only their spatial relations, relative to one another, are those
> of "above" and "below"; and all of this implies that individual
> symbols and list members are separated by spaces so as
> to preserve their status as distinct entities. Thus, a "list" is
> a geometric structure. Metrics, however, being arbitrary,
> are rarely considered.
>
> It is clear that L*, if it exists, can be mapped into a unit
> square in the plane. One such scheme, using standard
> orthogonal coordinates, maps each symbol according to
> the following formulae:
>
> x = 1 - 1/2^(m-1)
> y = -1 + 1/2^(n-1)
>
> It is understood that the mapping specifies these points
> as the centers of the circular symbols.
>
> Thus, the symbols of the diagonal, running from upper-left
> to lower-right, take the coordinates:
> (0,0) ; (1/2,-1/2) ; (3/4,-3/4) ; (7/8,-7/8) ; etc.
>

---Emphasis Added:Begin

> The coordinates of the unit square's corners are:
> UL (0,0) ; UR (0,1) ; LL (0,-1) ; LR (1,-1). It can be seen that
> except for the Origin at the upper-left, these are limit points
> and do not have list symbols mapped to them.

--- Emphasis Added:End

<more snip>

> we will be
> taking a mind's-eye tour, travelling along the diagonal line
> connecting (0,0) with (1,-1), stopping at each of the diagonal's
> mapping points at each stage of its construction, to make
> sure the job is done right. Our average travel speed is a
> leisurely square-root-of-two units per hour: thus we shall
> traverse the entire diagonal in one hour. Those who maintain
> that it is impossible to traverse an infinite number of points in
> a finite time should consult Zeno; and those who remain skeptical
> should take this opportunity to announce their rejection of
> point-set geometry and the calculus.
>
> Our little day-hike of infinity may thus be considered illustrative
> of the adage solvitur ambulando. (Despite Randy Poe's hysterical
> invective to the contrary, this does not properly translate as "proof
> by foot-stomping", even if it does entail a kind of forced march.)
>
> Our tour originates at the Origin. Here, the first symbol of the
> diagonal is laid. It is also, of course, the first (and only) symbol
> of the first list member. At this point, the diagonal and the first
> list member are identical.
>
> On we go: The second symbol of the diagonal is laid. The
> diagonal, now consisting of two symbols, is identical to the two
> symbols of the second list member. That is, in both length
> and content, the diagonal and the second list member are
> identical.
>
> Here we are at the third stage. The diagonal and the third list
> member are now identical. The pattern is clear. It is, I fear,
> rather monotonous. We shall skip ahead some considerable
> way...
>
> Here we are at the 1,013_th stage. The diagonal is identical,
> in both content and length, to the 1,013_th list member.
> Goodness, this is boring! When *will* the diagonal become
> infinite, thus distinguishing itself from the list members, all
> of which must remain finite? Hmmm...that's a toughie. The
> Cantorists would probably answer: "When the list is completed".
> We shall see.
>
> Meanwhile, *how* can the diagonal ever become infinite, since at
> each stage n the diagonal is identical to the n_th list member,
> and every possible n is a finite natural number?
>
> The diagonal is a derivative entity, and merely mirrors the list
> members from which it is derived. It *should* therefore be
> clear that in order for the diagonal to become infinite and
> contain all possible symbols, some eventual list member must
> itself become infinite and contain all possible symbols. After
> all, the diagonal we are constructing can grow no faster -- and
> no longer -- than the list members from which it is derived. Well,
> let's address the matter again when we are done...
>
> ...And here we are, after one hour of travel, at coordinates (1,-1),
> the bottom-right corner of the unit square. Have we not constructed
> every member of L* using our straightforward mapping formulae?

No. As you yourself stated, in the "Emphasis Added" section, one doesn't in fact
reach the coordinates (1,-1), since no point (i.e. no natural number) is mapped
to it. One can get arbitrarily close, but your one hour of travel won't get you
there; there will always be another member of L* which you haven't mapped yet.

The idea which you state in that section, that the point (1,-1) is a "limit
point" of your mapping, implicitly assumes the idea of an infinite collection of
points in a finite amount of space. It looks to me as if you've just constructed
the list whose impossibility you've been asserting. Congratulations.

Matt

Relevant Pages

  • Re: infinity
    ... > containing an infinite number of elements. ... > It's obvious that f maps every member of S to every member of T, ... > not included in the one-to-one mapping defined by f and g. ... > between any FINITE subset of it and another finite subset of it. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... conclude that there is a set that has no largest member. ... And to distinguish two infinite decimals, ... two reals is always finite. ...
    (sci.math)
  • Re: Cantor Confusion
    ... because it doesn't satisfy the trichotomy law. ... set of crossings is not well ordered because the ... subset does not have a greatest member? ... a countably infinite ordered set with finite boundaries. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... and is greater than the sum of the string to the right of it. ... How does not being a member of a collection lead you ... infinite value in the set. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... > to be finite if the set of naturals up to and including it is a finite ... > set by the Cantor definition, is, by the Cantor definition, an infinite ... > member of the set. ... Even worse, I posit that the first may come after the last, and the last before ...
    (sci.math)