There are many contemporary critiques of the use of the actual infinite in mathematics. The most radical come from Chinese logicians led by Wujia Zhu, Department of Computer Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Peoples’ Republic of China. The papers were published in the English language journal “Kybernetes”: (1) Y. Lin (et al.) “Systematic Yoyo Structure in Human Thought and the Fourth Crisis in Mathematics,” “Kybernetes,” vol. 37, no. 3, 2008, pp. 387-425; (2) W. Zhu (et al.) “Cauchy Theater Phenomenon in Diagonal Method and Test Principle of Finite Positional Differences,” “Kybernetes,” vol. 37, no. 3/4, 2008, pp. 469-473; (3) W. Zhu (et al.), “The Inconsistency of the Natural Number System,” “Kybernetes,” vol. 37, no. 3/4, 2008, pp. 482-488; (4) W. Zhu (et al.), “Modern System of Mathematics and a Pair of Hidden Contradictions in its Foundations,” “Kybernetes,” vol. 37, no. 3/4, 2008, pp. 438-445; (5) W. Zhu (et al.), “Modern System of Mathematics and Special Cauchy Theater in its Theoretical Foundation,” “Kybernetes,” vol. 37, no. 3, 2008, pp. 458-464; (6) W. Zhu (et al.), “Wide-Range Co-Existence of Potential and Actual Infinities in Modern Mathematics,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 433-437; (7) W. Zhu (et al.), “Descriptive Definitions of Potential and Actual Infinities,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 424-432; (8) W. Zhu (et al.), “Mathematical System of Potential Infinities (I) – Preparation,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 489-493; (9) W. Zhu (et al.), “Mathematical System of Potential Infinities (II) – Formal Systems of Logical Basis,” “Kybernetes,” vol. 37, no 3/4, 2008, pp. 495-504; (10) W. Zhu (et al.), “Mathematical System of Potential Infinities (III) – Metatheory of Logical Basis,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 505-515; (11) W. Zhu (et al.), “Mathematical System of Potential Infinities (IV) – Set Theoretical Foundation,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 516-525; (12) W. Zhu (et al.), “The Inconsistency of Countable Infinite Sets,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 446-452; (13) W. Zhu (et al.), “Inconsistency of Uncountable Infinite Sets Under ZFC Framework,” “Kybernetes,” vol. 37, no. 3 /4, 2008, pp. 453-457; (14) W. Zhu (et al.) “ Intention and Structure of Actually Infinite, Rigid Sets,” “Kybernetes,” vol. 37, no 3 /4, 2008, pp. 534-542; (15) W. Zhu (et al.), “Problem of Infinity Between Predicates and Infinite Sets,” “Kybernetes,” vol. 37, no. 3, 2008, pp. 526-533.

What is of particular interest is an alleged proof of the inconsistency of the natural numbers, a refutation of Cantor’s diagonal method and a proof of infinite sets. This material is too detailed to review here, but if it proves to be correct, it is “game over” for infinity, and probably for much conventional mathematics. So, all of you pricks on the internet attacking “cranks,” here is something for you to get your claws and beaks into, you geek pieces of shit. Yeah? Well, fuck you too!

As has been said, Cantor’s diagonal argument has been used to “prove” that there are allegedly different levels of infinity; namely that the set of real numbers is “uncountable” or nondenumerable, while the set of natural numbers {1, 2, 3,…}, is countable or denumerable. Thus, the cardinality of the set of real numbers is of a “higher” infinity than the set of natural numbers. Many other results can also be allegedly “proved” by the so-called diagonalization method, such that the set of all subsets of natural numbers is “uncountable.”

The diagonalization argument involves setting up a table with the natural numbers 1,2,3,…, going down the first column. One seeks to put the natural numbers into a 1-1 correspondence with the real numbers. Thus, 1 might be paired with the real number R1=d11d12d13…d1k…, where “d1k” are digits. Proceeding in this way, a table is set up. Cantor then sought to show that a 1-1 correspondence can be counter-exampled by constructing a new real number which does not occur in the table by starting at the left-hand corner of the array and going down the diagonal and changing each decimal by taking 1, if d1k is greater than 1, and if d1k=0, adding 1. This supposedly creates a new real number not on the list, so the 1-1 correspondence doesn’t allegedly exist. If the new number was added to the top of the list and paired off with the number 0, Cantorians claim that a new application of the diagonal method would generate a new real number not on the list, and so on. See G. Hunter, “Metalogic,” (Macmillan, London, 1971), pp. 22-24.

Those rejecting Cantor’s diagonal argument, on the internet, are regarded as “cranks” by the mathematical establishment. However, one Cantorian has asked; “What if Cantor’s Proof is Wrong?” at http://rjlipton.wordpress.com/2011/10/21/what-if-cantos-proof-is-wrong/

It is pointed out there that there is in fact a well-grounded mathematical reason to doubt the diagonal argument in this context, because of the Löwenheim-Skolem Theorem: if a first order theory has a model, then it has a denumerable model, if the theory is consistent. This model is one with no uncountable/nondenumerable sets. As the above mathematician puts it: “In such a model, everything we think we are saying about the real numbers is translated into an equally meaningful assertion about a set that is actually countable. This is ultimately because in a logical language we can say only countably many things before breakfast, and in a first-order language we can talk about only one thing at a time. We cannot actually say – or believe – uncountably many things before breakfast.” Not a good start to the Cantorian day. This consequence of the Löwenheim-Skolem theorem, known as Skolem’s paradox, is not a paradox in the sense of a contradiction, but a result which flies in the face of Cantorianism. It does not show that there are no nondenumerable sets, only that under the conditions of the theorem, such sets can be effectively eliminated. See B. Slater, “Logical Paradoxes,” “Internet Encyclopedia of Philosophy,” at http://www.iep.utm.edu/par-log/. It is worth noting, as Slater does in this piece, that it is arguable that Cantor’s diagonal argument is actually a paradox, because a direct application of it yields Richard’s paradox, which is generally accepted as a real paradox. If one cannot distinguish “good” against “bad” applications of Cantor’s argument, then the whole thing will need to be rejected.

P. O. Johnson, “Wholes, Parts and Infinite Collections,” “Philosophy,” vol. 67, 1992, pp. 367-379, argued against the idea that infinite sets can be placed in a 1-1 correspondence: “We cannot match terms unless we know in advance that each series or set contains the same number, and there is no other justification for saying that they ‘correspond.’ This is, of course, exactly the opposite argument to Cantor’s. Where Cantor argues that, where there is a one-to-one correspondence between two infinite classes, both must contain the same number of objects, I say that unless some classes can be shown to contain the same number of objects, their terms cannot be said to correspond,” (p. 372) It is an assumption that the notion of a 1-1 correspondence between infinite sets is meaningful, and can be established. Cantor’s proof is non-constructive, and the skeptic should ask, to see the “supertask” of actually producing the diagonal number completed. Who knows what could happen in reality as one attempts to construct such a number; maybe an evil demon will slay one, or the sky will fall? Perhaps there is some unknown physical law that would prevent one “completing” the supertask? Who the fuck knows?

Those mathematicians and logicians critical of Cantor’s diagonal argument, are usually critical of received set theory. For example, N. J. Wildberger, “Set Theory: Should You Believe?” At http://web.maths.unsw.edu.au/~norman/views2.htm, has said: “If you have an elaborate theory of ‘hierarchies upon hierarchies of infinite sets,’ in which you cannot even in principle decide whether there is anything between the first and second ‘infinity’ on your list, then it’s time to admit that you are no longer doing mathematics.” The same skepticism has been expressed by Wildberger in his paper, “Numbers, Infinities and Infinitesimals,” at http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf. If forcing axioms are added to ZFC set theory, then the continuum hypothesis is false, but if the “inner-model” axiom “V=ultimate L” is added, then the continuum hypothesis is true. So, at least from the perspective of mathematical Platonism, which holds that these entities have some sort of existence, what does “God’ see: is there an infinity between the smallest infinity (the set of counting numbers) and the continuum, or not? If everything depends upon what arbitrary assumptions one begins with, then at least mathematical Platonism will bite the dust.

Ludwig Wittgenstein (1889-1951), was highly critical of set theory, especially Cantor’s transfinite set theory, seeing it as “utter nonsense.” See V. Rodych, “Wittgenstein’s Critique of Set theory,” “Southern Journal of Philosophy,” vol. 38, 2000, pp. 281-319. He rejected the concept of actual infinity, and said that there was no actual set of all natural numbers. One of his arguments against the notion of an actual infinity was this: “Let’s imagine a man whose life goes back for an infinite time and who says to us: “I’m just writing down the last digit of π, and it’s a 2.” Every day of his life he has written down a digit, without ever having begun; he has just finished. This seems utter nonsense, and a reductio ad absurdum of the concept of an infinite totality.” See L. Wittgenstein, “Philosophical Remarks,” (Basil Blackwell, Oxford, 1975), § 145.

The Australian logician H. Slater, is another critic of set theory. In “Numbers are Not Sets,” “The Reasoner,” vol. 4, no. 12, December 2010, pp. 175-176, he argues that there is a grammatical confusion in taking the number zero to be the empty set, as it is the number of elements in the empty set which is zero, not the empty set itself. Along the same lines, it can be argued that the definition of the empty set, as a set containing no (i.e. no number) of elements, is circular, because the concept of a number is presupposed. Consequently, defining 0 as { }, will be flawed, as defining { }, will presuppose the concept of zero in specifying that this set has no number of elements. See H. Slater, “The De-Mathematisation of Logic,” (Polimetrica, Milano, 2007), p. 19, and H. Slater, “Grammar and Sets,” “Australasian Journal of Philosophy,” vol. 84, 2006, pp. 59-73. Edward Nelson, “Predicative Arithmetic,” (Princeton University Press, 1986), also viewed the standard definition of a natural number as circular. On the metaphysics of sets see: Max Black, “The Elusiveness of Sets,” “Review of Metaphysics,” vol. 24, 1971, pp. 614-636.

Further, if physical quantities are not treated using set theory, so that continua are not viewed as an infinite collection of points, then, “since if there is no number of points in some stuff then there is no question of whether that number is, or is not greater than some other.” See H. Slater, “Aggregate Theory versus Set Theory,” “Erkenntnis,” vol. 59, 2003, pp. 189-202, p. 189; H. Slater, “Against the Realisms of the Age,” (Ashgate, Aldershot, 1998), chapter 7, “Set Theory,” pp. 144-156. Set theory is not an accurate account of the way we use collectives, and of the mathematics of collectives, such as groups, flocks, and so on.

In H. Slater, “Logic Reformed,” (Peter Lang, Bern, 2002), Slater rejects the idea that there is a determinate number of natural numbers, and even if two sets were allegedly put into 1-1 correspondence, and had the same “power,” they may not have the same number, as they, if “infinite,” may have no determinate number at all. (p. 34) He follows Aristotle in rejecting the notion of completed infinities, so that there is no number of the natural numbers or of the continuum. (pp. 35-39) By the same line of reasoning, there are no “irrational” numbers either: “if we define them not in terms of impossible Platonic limits but merely convergent sequences of rational numbers, then we are identifying ‘irrational numbers’ with certain functions, since sequences are functions from the natural numbers. But the description ‘number’ is then strictly a misnomer, since a function is not a number, even if each of its values is one.” (p. 38)

In a paper often cited by the anti-crank mathematical thought police, Wilfred Hodges, “An Editor Recalls Some Hopeless Papers,” “Bulletin of Symbolic Logic,” vol. 4, 1998, pp. 1-16, he says that the main attacks against Cantor’s diagonal argument attack the elementary version of the argument using the matrix representation of the sequence of decimal real numbers, but “none of the authors showed any knowledge of Cantor’s theorem about the cardinalities of power sets.” (p. 2) Cantor allegedly generated a hierarchy of cardinal numbers, an infinite sequence of such infinities, each one generated by taking the power set, or set of all subsets of the preceding infinite set. However, inaccessible cardinals, the so-called “higher infinite,” cannot be obtained from smaller cardinals in this way: A. Kanamori, “The Higher Infinite,” (Springer, 2003)

Although there are paradoxes associated with Cantor’s theorem, these are taken to be ruled out by axiomatic set theory: K. C. Klement, “Russell, His Paradoxes, and Cantor’s Theorem: Part 1,” “Philosophy Compass,” vol. 5, no. 1, 2010, pp. 16-28. Nevertheless, paradoxes still escape these strictures, such as Grim’s paradox of the set of all truths: P. Grim, “There is No Set of All Truths,” “Analysis,” vol. 44, 1984, pp. 206-208. Suppose T is the set of all truths. P(T) is the power set of T. Then for each element si of P(t), there is a truth tn. So, there will be as many elements of T as P(t), contrary to Cantor’s power set theorem.

There was a lengthy debate for many years about the significance of this argument for omniscience. It was noted by some logicians e.g. R. Sylvan, “Grim Tales Retold: How to Maintain Ordinary Discourse about –and Despite –Logically Embarrassing Notions and Totalities,” “Logique et Analyse,” vol. 139-140, 1992, pp. 349-374; G. Priest, “Beyond the Limits of Thought,” (2002), pp. 230-232, that the set of all truths, is unobjectionable, and certainly set theory should not rule out what objects it can encompass.

N. Rescher and P. Grim, “Beyond Sets: A Venture in Collection-Theoretic Revisionism,” (Transaction Books, 2011), have moved to be highly critical of set theory because of its unrealistic treatment of collectivities. In particular; “Set theory was born in paradox, was shaped by paradox, and continues to carry the threat of paradox into its current adolescence.” (p. 6) An outstanding philosophical difficulty is: “coherently conceptualizing a set of all things, the realm of unrestricted quantification (or even the sense of restricted quantification), the totality of all events, all facts, all propositions, or all that is true.” (p. 6) And they conclude: “Sets are structurally incapable of handling any of these.” (p. 6)

With the rejection of the concept of a set, we can reject as well the Cantorian worldview that goes with it.

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