The List Problem: a pons asinorus for Cantorists

msadkins04_at_yahoo.com
Date: 12/27/04 

Date: 27 Dec 2004 12:18:23 -0800

Let L be a finite list indexed to natural numbers, such that:
(I) Each list member, n, consists of n symbols; (II) For m from
1 to n, the m_th symbol of the n_th list member is a circle with
m dots inside.

Thus, the first list member has one symbol: a circle with one dot
inside. The second member has two symbols: a circle with one dot,
followed by a circle with two dots. The third member has three
symbols: a circle with one dot, then a circle with two dots, then
a circle with three dots. Subsequent list members follow the same
plan.

This plan ensures that each subsequent list member contains a
concatenation of all previous members' terminal symbols.

The list contains an imbedded diagonal string; each of its symbols
is the terminal symbol of a list member.

The first symbol of the diagonal is identical to the first list
member; the first two symbols of the diagonal are identical to
the second list member; the first three symbols of the diagonal
are identical to the third list member; and in general the first
n symbols of the diagonal are identical to the n_th list member.

In considering the growth of L from one member up through
arbitrarily large sizes, 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, in the abstract, to instantaneously grow L "to
infinity", i.e., so that it covers the range of natural numbers,
N? This is equivalent to asking whether it is possible to grow L
such that: (a) The quantity of list members goes from finite to
"infinite"; (b) The length of the diagonal goes from finite to
"infinite"; (c) The length of the members remains finite.

It remains true that:

(1) Each subsequent list member contains a concatenation of all
previous members' terminal symbols.
(2) The diagonal is made of the terminals of list members.
(3) The diagonal must grow *exactly* as the list members do;
otherwise the diagonal cannot grow.

Now, however, the diagonal must grow to contain all possible
terminal symbols.

>>From (2) it is clear that the diagonal grows by concatenating the
terminals of list members. From (3) it is clear that this
concatenation must mimic that of the members themselves (the
diagonal is a derivative entity). Given (2) and (3), it is clear
that (1) requires a list member to concatenate all possible
terminal symbols if the diagonal is to do so. This, however, is
impossible if all members remain of finite length, as they must
since they are indexed to the natural numbers.

By design, the diagonal cannot be longer than all of the list
members: though there could be no "largest" member under such a
scheme, all members remain finite, and an "infinite" diagonal
would be larger than any and all of them.

There is no point at which the length of the list members can
become infinite: therefore there is no point at which the diagonal
may transcend them, since it is constructed from their terminals.
Nor is there a completion point or "point at infinity" for the
list; thus there can be no "completed infinite" diagonal.

The solution to the problem is the realization that "infinite"
lists, diagonals, etc., are impossible because their very
postulation entails logical inconsistency. One cannot have an
entity which is simultaneously limitless and completed, because
that is a contradiction in terms.

The phrase "all the naturals" is internally inconsistent: one
cannot have "all" of something that is limitless. In fact,
however, the term "natural numbers" does not even actually specify
a set of individual members, but merely references a particular
concept of number; a set of rules or template through which
individual, specific entities can be created.

Mark Adkins
msadkins04@yahoo.com

Relevant Pages

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