Authors: Philip Molyneux
Comments: 10 Pages.
This paper critically examines the Cantor Diagonal Argument (CDA) that is used in
set theory to draw a distinction between the cardinality of the natural numbers and
that of the real numbers. In the absence of a verified English translation of the original
1891 Cantor paper from which it is said to be derived, the CDA is discussed here
using a consensus from the forms found in a range of published sources (from
"popular" to "professional"). Some general comments are made on these sources. The
discussion then focusses on the CDA as applied to the correspondence between the set
of the natural numbers, and the set of real numbers in the open range (0,1) in their
expansion from decimal digits (0.123… etc.).
Four points critical of the CDA are raised: (1) The conventional presentation of the
CDA forms a putative new real number (X) from the "diagonal" of the chosen list of
real numbers and which is therefore not on this initial list; however, it omits to
consider that it may indeed be on the later part of the list, which is never exhausted
however far the "diagonal" list is extended. (2) This aspect, combined with the fact
that X is still composed of decimal digits, that is, it is a real number as defined,
indicates that it must be on the later part of the list, that is, it is not a "new" number at
all. (3) The conventional application of the CDA leads to one putative "new" real
number (X); however, the logical extension of this in its "exhaustive" application, that
is, by using all possible different methods of alteration of the decimal digits on the
"diagonal", and by reordering the list in all possible ways, leads to a list of putative
"new" real numbers that become orders of magnitude longer than the chosen
"diagonal" list. (4) The CDA is apparently considered to be a method that is
applicable generally; however, testing this applicability with the natural numbers
themselves leads to a contradiction.
Following on from this, it is found that it indeed is possible to set up a one-to-one
correspondence between the natural numbers and the real numbers in (0,1), that is, ℕ
⇔ ℝ; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension
of the natural number is intended to be a mirror image of the left hand extension of
the real number. It is also shown how this may be extended to real numbers outside
the range (0,1).
Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998
critical review of "hopeless papers" dealing with the CDA; this form is also examined
from the same viewpoints, and to the same conclusions.
Finally, some comments are made on the concept of "infinity", pointing out that to
consider this as an entity is a category error, since it simply represents an absence, that
is, the absence of a termination to a process.
Category: Set Theory and Logic