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Michael Dummett famously asked how the serpent of inconsistency entered Frege’s paradise. He himself blamed the impredicative nature of second-order quantification, while many others focused on the inflationary nature of the axiom. Axiom V is, after all, the denial of a higher-order generalization of Cantor’s theorem. Predicativists do not deny this, but they block the derivation of the relevant generalization in predicative fragments of second-order logic. Unfortunately, there is more than one higher-order generalization of Cantor’s theorem, and one of them remains a theorem in predicative fragments of higher-order logic. Our recommendation to predicativists is to respond that only one of them supports the cardinality gloss we associate with Cantor’s theorem and that it is, in fact, false. The other remains a theorem of predicative fragments of higher-order logic but its derivability seemsmore closely related to the Grelling’s paradox than to cardinality considerations.

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Godel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper valueideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms.

On the first page of “What is Cantor’s Continuum Problem?”, Godel 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 Godel’s argument, showing that it fails in two important ways: (i) Its premises are not sufficiently compelling to discredit countervailing intuitions and pragmatic considerations, nor pluralism, and (ii) 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.

Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections (i.e. one-to-one correspondences) between them. These results (which I of course do not dispute) settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity.

We introduce and study finitely supported partially ordered sets.We study the notion of ‘cardinality’ for a finitely supported set, proving several properties related to this concept. Some properties are naturally extended from the non-atomic Zermelo-Fraenkel framework into the world of atomic structures with finite supports. In this sense, we prove that the Cantor theorem and the Cantor-Schroder- Bernstein theorem for cardinalities are still valid in the world of atomic finitely supported sets. Several other cardinality arithmetic properties are preserved from the classical Zermelo-Fraenkel set theory. However, the dual of the Cantor-Schroder- Bernstein theorem (where cardinalities are ordered via surjective mappings) is no longer valid in this framework. Other specific order properties of cardinalities that do not have related Zermelo-Fraenkel correspondents are also proved. Finally, we present a collection of fixed point theorems in the framework of finitely supported ordered structures, preserving the validity of several classical Zermelo-Fraenkel fixed point theorems such as the Bourbaki-Witt theorem, the Scott theorem and the Tarski-Kantorovitch theorem. We also prove several specific fixed point properties in the framework of finitely supported algebraic structures (especially fixed point properties of mappings defined on finitely supported sets that do not contain infinite uniformly supported subsets), results that are not reformulations of some corresponding Zermelo-Fraenkel results.

Every binary relation on an infinite set can be represented by an election in which each voter’s preferences are quasi-transitive and complete (except possibly not reflexive) and in which the electorate has smaller cardinality than or the same cardinality as the set of alternatives, depending on the cardinality of that set.

In this paper, I investigate type theories (TTs) from several perspectives. First, I present and elaborate the philosophical and technical motivations for these theories. I then offer a formal analysis of various TTs, focusing on the cardinality of the set of types contained in each. I argue that these TTs can be divided into four formal categories, which are derived from the cardinality of the set of their basic elementary types and the finiteness of the lengths of their molecular types. The paper provides proofs of the cardinality of the universe of types for each of the specified theories.

A set S of integers is a B 3 -set if all the sums of the form a 1 + a 2 + a 3 , with a 1 , a 2 and a 3 ? S and a 1 ? a 2 ? a 3 , are distinct. We obtain asymptotic bounds for the number of B 3 -sets of a given cardinality contained in the interval n = { 1 , ? , n } . We use these results to estimate the maximum size of a B 3 -set contained in a typical (random) subset of n of a given cardinality. These results confirm conjectures recently put forward by the authors On the number of B h -sets, Combin. Probab. Comput. 25 (2016), no. 1, 108-127.

We study the self-similar subsets of a class of Cantor sets with nice symmetric structure. Our result generalizes the result of Feng, Rao and Wang (2015) [3] , which characterizes all the self-similar subsets of the Middle-Third Cantor set (a question raised by P. Mattila in 1998). We simplify a number-theoretical argument in the paper of Feng, Rao and Wang, and make it applicable to a large class of self-similar sets.

In this paper, we illustrate the paradox concerning maximally consistent sets of propositions, which is contrary to set theory. It has been shown that Cantor paradoxes do not offer particular advantages for any modal theories. The paradox is therefore not a specific difficulty for modal concepts, and it also neither grants advantages nor disadvantages for any modal theory. The underlying problem is quite general, and affects anyone who intends to use the notion of &ldquo, world&rdquo, in its ontology.

Georg Cantor (1845-1918) introduced the notion of the cantor set, which consists of points along a single line segment with a number of remarkable and deep properties. This paper aims to emphasize a proceeding to obtain the Cantor (ternary) set, C by means of the so-called elimination of the open-middle third at each step using a general basic approach in constructing the set.

Given c > 0 a planar Cantor set X with a dim H (X) < 2 is constructed such that each c-monotone subspace of X has a smaller Hausdorff dimension than X.

For every doubling gauge g, we prove that there is a Cantor set of positive finite -measure, -measure, and -premeasure. Also, we show that every compact metric space of infinite -premeasure has a compact countable subset of infinite -premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set E in this class, there exists a countable set F, with , and a doubling gauge g such that has different positive finite -measure and -premeasure.

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set C? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset F of C containing more than one point, every linear generating IFS of F must consist of similitudes with contraction ratios ± 3 − n , n ∈ N . In particular, a simple criterion is formulated to characterize self-similar subsets of C with equal contraction ratio in modulus.

In order to solve a precise problem on trigonometric series, “Can a function have more than one representation by a trigonometric series” the great German mathematician Georg Cantor created set theory and laid the foundations of the theory of real numbers. This had a profound impact on mathematics. In this article, we narrate this fascinating story.

I bring to light a set-theoretic reason to think that there are more (identifiable) mental properties than (identifiable) 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.

In this work we prove that a C1+α-hyperbolic Cantor set contained in S1 that is close to an affine Cantor set is not C1-minimal.

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness” in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a random Cantor set translate with a given constructive dimension.

In the “intended” models of (first-order) ZFC — Zermelo-Fraenkel set theory with the axiom of Choice — there are only sets. ZFCU is ZFC modified to allow for the existence of urelements, or atoms, i.e., things that can be members of sets but are not themselves sets and do not themselves have members. Consider, then, the following consequence of David Lewis’s (1986) unqualified principle of Recombination

The space of Cantor subsets of R 3 , denoted C ( R 3 ) , is a Polish space. We prove this space is path connected and locally path connected. The group of autohomeomorphisms of R 3 , denoted Aut ( R 3 ) , acts on C ( R 3 ) naturally. This action gives us natural invariant classes of Cantor sets and we show that these classes are in the lower levels of the Borel hierarchy, in fact they are open, closed, F σ or G δ in C ( R 3 ) . Moreover, we prove that the classification problem of Cantor sets arising from this action is at least as complicated as the classification of countable linear orders.

We prove a generalization of the hyper-game theorem by using an abstract version of inductively generated formal topology. As applications we show proofs for Cantor theorem, uncountability of the set of functions from N to N and Godel theorem which use no diagonal argument.

In this paper, I argue that the thesis of Composition as Identity blocks the plural version of Cantor’s Theorem, and that this in turn has implications for our use of Cantor’s theorem in metaphysics. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano, and might be turned around to become an abductive argument for Composition as Identity

Although recent evidence is somewhat ambiguous, if not confusing, Patrick Grim still seems to believe that his Cantorian argument against omniscienceis sound. According to this argument, it follows by Cantor’s power set theorem that there can be no set of all truths. Hence, assuming that omniscience presupposes precisely such a set, there can be no omniscient being. Reconsidering this argument, however, guided in particular by Alvin Plantinga’s critique thereof, I find it far from convincing. Not only does it have an enormously untoward side effect, but it is self-referentially incoherent as well.

Some decision making models are discussed from the point of view of neurophysiology and quantum mechanics. The main feature of these models is that a straight line segment is replaced by the Cantor set. In this direction, many interesting results have been obtained by methods of number theory, p-adic analysis, and the theory of dynamical systems. Some generalizations of existing models are also discussed, which are formulated in terms of the so-called Cantor cubes, that is, Cartesian products of infinitely many standard two-point spaces D (as is known, the Cantor cube $$D^{\aleph _0 }$$ is homeomorphic to the Cantor set). This approach involves difficulties caused by the nonmetrizability and nonseparability of the Cantor cubes D m for m > ℵ0 and nonseparable for m > c, respectively.

We consider a question raised by John Cobb: given positive integers n > l > k is there a Cantor set in R n such that all whose projections onto l-dimensional planes are exactly k-dimensional? We construct in R n a Cantor set such that all its shadows (projections onto hyperplanes) are k-dimensional for every 0 ⩽ k ⩽ n − 1 . We also consider the extension of Cobbʼs question to Hilbert space.

Through a series of six guided classroom discoveries, students create, via targeted questions, a definition for deciding when two sets have the same cardinality. The program begins by developing basic facts about cardinalities of finite sets. Extending two of these facts to infinite sets yields two statements on comparing infinite cardinalities that contradict each other. The experiment “More circles or more squares?” resolves this dilemma in favor of the definition of “same cardinality” that Georg Cantor adopted over a century ago.

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and,kappa-analyticity, thereby adding to the 'Baire space recognition literature' (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak alpha-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by kappa-analytic (in particular analytic) sets that are 'heavy' (everywhere large in the sense of some sigma-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy-Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach-Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in kappa-analytic form.

International audience; Our aim is to quantify how complex is a Cantor set as we approximate it better and better. We formalize this by asking what is the shortest program, running on a universal Turing machine, which produces this set at the precision ε in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, that we call the "ε-distortion complexity". How does this quantity behave as ε tends to 0? And, moreover, how does this behaviour relates to other characteristics of the Cantor set? This is the subject of the present work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems (IFS's) and exhibit very different behaviours. For instance, the ε-distortion complexity of a C^k Cantor set is proven to behave like ε^{−D/k}, where D is its box counting dimension.

A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em relative infinitesimals} and an {\em inversion} rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor set $\tilde C$ where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set $C$ is identified with the closure of the set of gaps of $\tilde C$. The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpretd as (every where) locally constant on these extended ultrametric spaces. An interesting phenomenon, called {\em growth of measure}, is studied on such an ultrametric space. Using the reparametrisation invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get {\em deformed} leading to a {\em deformed} Cantor set with a positive measure. The definition of a new {\em valuated exponent} is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension $\log_3 2$ and thickness 1 are constructed explicitly., Comment: Final Published Version