An exchange between Patrick Grim and Alvin Plantinga regarding Cantorian arguments against the possibility of an omniscient being.

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to (...) the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals. (shrink)

Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.

It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least (...) as articulated here, would not have satisfied Cantor himself. (shrink)

This is a book review of Cantorian set theory and limitations of size by Michael Hallett.

It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1 However, Russell’s reasons for abandoning what he called “denoting concepts”, and (...) his rejection of similar “semantic dualisms” such as Frege’s theory of sense and reference—at least in “On Denoting”—made no explicit mention of any Cantorian paradox. My aim in this paper is to argue that such paradoxes do pose a problem for certain theories such as Frege’s, and early Russell’s, about how definite descriptions are meaningful. My first aim is simply to lay out the problem I have in mind. Next, I shall turn to arguing that the theories of descriptions endorsed by Frege and by Russell prior to “On Denoting” are susceptible to the problem. Finally, I explore what responses a.. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

Tangled Type Theory was introduced by Randall Holmes in [3] as a new way of approaching the consistency problem for$\mathrm {NF}$. Although the task of finding models for this theory is far from trivial (considering it is equiconsistent with$\mathrm {NF}$), ways of constructing models for certain fragments of it have been discovered. In this article, we present a simpler way of constructing models of predicative Tangled Type Theory and consequently of predicative$\mathrm {NF}$. In these new models of predicative$\mathrm {NF}$, the (...) universe is well-orderable and equinumerous to the set of singletons. (shrink)

The usual objections to infinite numbers, and classes, and series, and the notion that the infinite as such is self-contradictory, may... be dismissed as groundless. There remains, however, a very grave difficulty, connected with the contradiction [of the class of all classes not members of themselves]. This difficulty does not concern the infinite as such, but only certain very large infinite classes.

The purpose of this paper is to suggest that we are in the midst of a Cantorian bubble, just as, for example, there was a dot com bubble in the late 1990’s.

Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...) I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the ε-operator. Then, after presenting Cantor's abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo-von Neumann and Frege-Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege's objections to Cantor's proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the ε-operator in the BK definition of cardinal numbers. (shrink)

Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.

Starting from the generalized concept of syntactically and semantically trivial differences between two formal theories introduced by Arsenijević, we show that two systems of the linear continuum, the Cantorian point-based system and the Aristotelian interval-based system that satisfies Cantor's coherence condition, are only trivially different. So, the 'great struggle' (to use Cantor's phrase) between the two contending parties turns out to be 'much ado about nothing'.

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...) the axiomatic setting of BK and the definition of cardinal numbers by means of the \-operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the \-operator in the BK definition of cardinal numbers. (shrink)

In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to systematically integrate highly transfinite (...) concepts of demonstrability and existence. Zermelo incompleteness is then the analogue of the Problem of Proper Classes, and the resolution of these two anomalies is similarly analogous. (shrink)

On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's (...) argument, showing that it fails in two important ways: Its premises are not sufficiently compelling to discredit countervailing intuitions and pragmatic considerations, nor pluralism, and its final inference, from the superiority of Cantor's theory as applied to sets of changeable physical objects to the unique acceptability of that theory for all sets, is irredeemably invalid. (shrink)

This paper explores the connection between Kant’s first antinomy and the set-theoretical paradox of the largest cardinal. The lesson to be drawn is that we should refrain from reifying the collections that generate the antinomies: the collection of all spatio-temporal objects in the case of Kant’s first antinomy, the collection of all objects whatsoever in the case of the Cantorian paradox. This conclusion is not only one that we are entitled to draw but also one that does not generate (...) new philosophical problems of its own. In this respect it is more attractive than other contemporary reactions to the paradoxes of set theory. (shrink)

Plato’s philosophy is important to Badiou for a number of reasons, chief among which is that Badiou considered Plato to have recognised that mathematics provides the only sound or adequate basis for ontology. The mathematical basis of ontology is central to Badiou’s philosophy, and his engagement with Plato is instrumental in determining how he positions his philosophy in relation to those approaches to the philosophy of mathematics that endorse an orthodox Platonic realism, i.e. the independent existence of a realm of (...) mathematical objects. The Platonism that Badiou makes claim to bears little resemblance to this orthodoxy. Like Plato, Badiou insists on the primacy of the eternal and immu- table abstraction of the mathematico-ontological Idea; however, Badiou’s reconstructed Platonism champions the mathematics of post-Cantorian set theory, which itself af rms the irreducible multiplicity of being. Badiou in this way recon gures the Platonic notion of the relation between the one and the multiple in terms of the multiple-without-one as represented in the axiom of the void or empty set. Rather than engage with the Plato that is gured in the ontological realism of the orthodox Platonic approach to the philosophy of mathematics, Badiou is intent on characterising the Plato that responds to the demands of a post-Cantorian set theory, and he considers Plato’s philosophy to provide a response to such a challenge. In effect, Badiou reorients mathematical Platonism from an epistemological to an ontological problematic, a move that relies on the plausibility of rejecting the empiricist ontology underlying orthodox mathematical Platonism. To draw a connec- tion between these two approaches to Platonism and to determine what sets them radically apart, this paper focuses on the use that they each make of model theory to further their respective arguments. (shrink)

We will explicate Cantor’s principle of set existence using the Gödelian intensional notion of absolute provability and John Burgess’ plural logical concept of set formation. From this Cantorian Comprehension principle we will derive a conditional result about the question whether there are any absolutely unprovable mathematical truths. Finally, we will discuss the philosophical significance of the conditional result.

(1997). Dimensional symmetry breaking, information and the arrow of time in cantorian space. World Futures: Vol. 49, The Quest for a Unified Theory of Information, pp. 391-400.

In a recent paper, Kit Fine offers a reconstruction of Cantor's theory of ordinals. It avoids certain mentalistic overtones in it through both a non-standard ontology and a non-standard notion of abstraction. I argue that this reconstruction misses an essential constructive and computational content of Cantor's theory, which I in turn reconstruct using Martin-Löf's theory of types. Throughout, I emphasize Kantian themes in Cantor's epistemology, and I also argue, as against Michael Hallett's interpretation, for the need for a constructive understanding (...) of Cantorian ?existence principles? (shrink)

I bring to light a set-theoretic reason to think that there are more mental properties than shapes, sizes, masses, and other characteristically “physical” properties. I make use of a couple counting principles. One principle, backed by a Cantorian-style argument, is that pluralities outnumber particulars: that is, there is a distinct plurality of particulars for each particular, but not vice versa. The other is a principle by which we may coherently identify distinct mental properties in terms of arbitrary pluralities of (...) physical properties. I motivate these principles and explain how they together imply that there are more mental properties than physical properties. I then argue that certain parody arguments fail for various instructive reasons. The purpose of my argument is to identify an unforeseen “counting” cost of a certain reductive materialist view of the mind. (shrink)

How-many numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed how-many number in terms of one-to-one correspondence between two sets. That is to say, one arrived at numbers by either abstracting from the concept of correspondence, in the case of Cantor, or by using it to provide an out-and-out definition, in the case of Frege.

This paper continues a thread in Analysis begun by Adam Rieger and Nicholas Denyer. Rieger argued that Frege’s theory of thoughts violates Cantor’s theorem by postulating as many thoughts as concepts. Denyer countered that Rieger’s construction could not show that the thoughts generated are always distinct for distinct concepts. By focusing on universally quantified thoughts, rather than thoughts that attribute a concept to an individual, I give a different construction that avoids Denyer’s problem. I also note that this problem for (...) Frege’s philosophy was discovered by Bertrand Russell as early as 1902 and has been discussed intermittently since. (shrink)

A Cantorian argument that there is no set of all truths. There is, for the same reason, no possible world as a maximal set of propositions. And omniscience is logically impossible.

Does Cantorian set theory alter our intuitive conception of number? Yes. In particular, Cantorian set theory revises our intuitive conception of when two sets have the same size (cardinal number). Consider a variant of Galileo’s Paradox, which notes that the members of the set of natural numbers, N, can be put in one-to-one correspondence with the members of the set of even numbers, E.

Axioms in set theory typically have the form , where is a relation which links with in some way. In this paper we introduce a particular linkage relation and a single axiom based on from which all the axioms of (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of pertinent features of the (...) system of set theory to be erected on the single axiom. is shown to be somewhat stronger than , but much weaker than (Zermelo-Fraenkel set theory). (shrink)