Why Cantor's Diagonalization Proof Comes Up Short

This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered table T could be a rational number from which different rational antidiago-nals (elements of (0, 1) that cannot be in T) could be defined. If that were the case, and for the same reason as in Cantor's diagonal argument, the open rational interval (0, 1) would be non-denumerable, and we would have a contradiction in set theory , because Cantor also proved the set of rational numbers is denumerable.

2022

The first theorem of Cantor's 1891 paper introduced the diagonal method by example; namely, as an argument for showing that any list of the reals is necessarily incomplete and concluding the reals are not denumerable. The present paper, one of three on the diagonal method, argues that the diagonal method is not a valid form of argument (i.e., not a valid proof schema). Based on any of three independent refutations presented here, the diagonal argument cannot logically lead to the conclusion(s) alleged for it. We generalize these arguments to apply to the diagonal method and its generalizations and variations, in a second, companion paper [13]. We review the profound and pervasive impact of the diagonal method on meta-mathematics, mathematics, logic and computer science, noting the central position arguments based on the diagonal method hold in the foundations of these subjects, in a third companion paper [14], thereby providing a context for the results presented in both this and the second papers. "And as of 'no-go theorems'… One always has to take the assumptions into consideration, just as the small print in a contract."-Gerard t'Hooft (1999 Nobelist in Physics) [18] I.

In "Cantor's 1891 Diagonal Argument: Three Refutations" (www.academia.edu, February 4, 2020), three refutations of Cantor's diagonal argument (DA) were presented. Numerous reviewers claimed that the first refutation failed to identify any flaw in the DA. Some claimed that the discussion had no relevance to Cantor's proof. The present note is to clarify these points, explicitly overcoming those reviewer's objections.

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 apparently 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 this 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, ! ⇔ "; this takes the form: … a 3 a 2 a 1 ⇔ 0. a 1 a 2 a 3 …, 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. This may be extended to the general case of real numbers-that is, not limited to the range (0,1)-by intercalation of the digit sequence of its decimal fraction part into the sequence of the natural number part, giving the one-to-one-correspondence: … A 3 a 3 A 2 a 2 A 1 a 1 ⇔ ... A 3 A 2 A 1. a 1 a 2 a 3 … 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.

Cardinality of $\Sigma^*$ and Cantor's Diagnolization discussed herein. , 2023

Let Σ = {0, 1}, herein in the discussion below: Theorem 0.1 Here we show that Cantor's Diagonalization is wrong in his proof of the Uncountable Cardinality of the Reals or that the Cardinality of the reals is Countable Infinity. Proof: 1 Let us partition the integers into the N∪{0} and the the negative integers. Let T = (0, 1) be a subset of the reals. Let the set T be listable or have a bijection with the countably infinite set of natural numbers. Map the number 0 to the first real r 0 and the second number 1 to r 1 the second real, the number 2 to r 2 and so on in binary till countable infinity. Let all r i be different. Now I assume the reader here is familiar with Cantor's Argument in his Diagonalization Method. We simply choose a diagonal starting from the 0 th line such that we flip the bits of the diagonal. Now this real number of the flipped diagonal is different from the ones listed so far. However instead of claiming we have a contradiction, we claim that we just have not listed the reals from set T completely. So we put this diagnol say t −1 on the top of the list at the −1th position above the number 0 that is on top of the r 0 th real.Then we rediagonalize from the −1th and the t −1 th real, position and flip bits of our diagonal and again place it on the −2th position above the −1th number for the real now denoted as t −2 noting that this number t −2 is different from all those below that is t −1 , r 0 , r 1 , r 2 , r 3 • • •. Or we could form another different seperate list and so on to place these new reals of diagonals in. We could carry this method of Cantor's Diagonalization forever and come to the conclusion that we have not listed the reals completely. This could imply that there is no contradiction as Cantor claimed in listing the reals from the set T .

This chapter explains in detail the first Cantor's proof of the uncoun-table nature of the real numbers, and discusses the conditions under which it could also be applied to the set of the rational numbers.

Arxiv preprint math/0304310, 2003

Whatever other beliefs there may remain for considering Cantor's diagonal argument as mathematically legitimate, there are three that, prima facie, lend it an illusory legitimacy; they need to be explicitly discounted appropriately. The first - Cantor's diagonal argument defines a non-countable Dedekind real number; the second - Goedel uses the argument to define a formally undecidable, but interpretively true, proposition; and the third - Turing uses the argument to define an uncomputable Dedekind real number.

ABOUT CANTOR'S UNCOUNTABILITY PROOFS ROBERTO MUSMECI, 2021

In this paper I analyse the demonstrations of not-denumerability of Real Numbers to point out what are the fallaciousness' of these kind of proofs.

An illustration is provided for how the real numbers can be counted. This establishes a one-to-one correspondence between the real numbers and the natural numbers. Cantor's diagonal argument is examined to resolve the apparent inconsistency with this finding. Watch the paper explained at https://youtu.be/NQsSLpZ_NjE https://youtu.be/aAHTHBRU2Io

Bertrand D. THEBAULT, 2024

RCDA2: this article is the second installment of a series titled 'Refuting Cantor's Diagonal Argument' In RCDA1, I proposed to apply formal acceptance procedures to Cantor's Diagonal Argument. The concept involves employing CDA in scenarios beyond its original intent to rigorously assess its validity. In this RCDA2 article, I assume the presumably uncountable real numbers interval [0, 1) can be put in one-to-one correspondance with uncountable transfinite ordinals relying on their unique Cantor Normal Form (CNF) [2]. Afterward, I employ Cantor's Diagonal Argument twice on this same oneto-one mapping: (1) between the transfinite ordinals (CNF) and the interval of real numbers [0, 1), and (2) conversely, between the interval of real numbers [0, 1) and the transfinite ordinals (CNF). " I obtain contradictory results, implying the assumption is false which refutes both the validity of transfinite ordinals and the consistency of Cantor's theory on transfinite numbers.