‘Cantor on Infinity in Nature, Number, and the Divine Mind’

Cybernetics and Systems, 2015

I heap up monstrous numbers, Pile millions upon millions, I put aeon upon aeon and world upon world, And when from that awful height Reeling, again I seek thee, All the might of number increased a thousand fold Is still not a fragment of thee. I remove them and thou liest wholly before me.

Studies in History and Philosophy of Science Part A, 2009

The actual infinity Aristotle-Cantor , potential infinity . The origins of Cantor’s infinity, aleph null, the diagonal argument The natural infinity , continuum The mathematical infinity A first classification of sets Three notable examples of countable sets The 1-1 correspondence, equivalent sets, cardinality . The theory of transfinite numbers Existence and construction, existence proofs

Philosophia Scientiae, 2005

ExistencePS Press: https://epspress.com/NTF/CantorAndTheAttemptToRefuteAristotle.pdf, 2020

Aristotle famously only accepts potential infinities regarding realities that exist fully at any moment (temporal infinities, with one part going away as a new part comes into existence, is a different story). In opposition to the Northern Greek, Cantor attempts to justify actual mathematical infinite sets by holding that God holds them in mind as a possibility and because of the Principle of Plenitude, the possibility must be actualized at some moment in time. This article reveals how Aristotle would reject Cantor's reasoning.

2019

It might seem surprising to talk about the relationship between a theologian and a mathematician. One deals with matters of faith while the other deals with hard, logical arguments — or not? The relationship might not seem so surprising if I could, in as non-technical terms as possible, explain Cantor’s theory of infinite sets, the objections raised against it, and what an eminent defender of his theory said. I’ll try to do this in these few minutes, without risking going out of point, because this is basically what Florenskij does in his 1904 paper The symbols of the infinite (An essay on the ideas of G. Cantor) (Italian translation), and on which I was asked to comment for this session.peer-reviewe

The paper shows the histroy of the concept of in infinity starting with the Aristotelian understanding and the way it was perceived by the church fathers. The watershed in the writings of Nicolaus of Cusa (Cusanus) is elucidated and show how continually it changes from theology to metaphysics and mathematics culminating in Georg Cantors revolutionary mathematics

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.

Studies in History and Philosophy of Science Part A, 2009

Georg Cantor, the founder of set theory, cared much about a philosophical foundation for his theory of infinite numbers. To that end, he studied intensively the works of Baruch de Spinoza. In the paper, we survey the influence of Spinozean thoughts onto Cantor's; we discuss Spinoza's philosophy of infinity, as it is contained in his Ethics; and we attempt to draw a parallel between Spinoza's and Cantor's ontologies. Our conclusion is that the study of Spinoza provides deepening insights into Cantor's philosophical theory, whilst Cantor can not be called a 'Spinozist' in any stricter sense of that word.