Authors: Philip Molyneux
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.
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