Standard Set Theory Research Articles

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

The Many and the One: A Philosophical Study of Plural Logic

<i>The Many and the One: A Philosophical Study of Plural Logic</i>

Sharon Berry.A Logical Foundation for Potentialist Set Theory

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

Copies from "Standard Set Theory"? A Note on the Foundations of Minimalist Syntax in Reaction to Chomsky, Gallego and Ott (2019)

Appeal to standard set theory in (mainstream) minimalist syntax is shown to be in conflict with the goal of analyzing dependency formation, a.k.a. movement, as involving genuine constituent copies. The underlying tension is due to (the axiom of) extensionality, which—other things being equal—favors a perspective on dependencies in terms of multidominance. The above argument is developed against the backdrop of a recent exposition of minimalist syntax (Chomsky et al. in Catal J Linguist (Special Issue):229–261, 2019), which can be seen as exemplary. The resulting critical assessment should be taken as removing obstacles on the way toward proper minimalist foundations for syntactic theory.

A Justification for the Quantificational Hume Principle

In recent work Bruno Whittle has presented a new challenge to the Cantorian idea that there are different infinite cardinalities. Most challenges of this kind have tended to focus on the status of the axioms of standard set theory; Whittle’s is different in that he focuses on the connection between standard set theory and intuitive concepts related to cardinality. Specifically, Whittle argues we are not in a position to know a principle I call the Quantificational Hume Principle (QHP), which connects the application of intuitive, quantificational cardinality concepts (including ‘at least as many’) to claims involving the existence of functions. This paper responds to Whittle’s skeptical arguments by providing an argument justifying the QHP.

On the properties of subsethood measures

This paper explores the properties of subsethood measures, including boundary conditions, monotonicity, duality, and additivity. These properties can be used as candidates of axioms to define specific subsethood measures. The standard set theory introduces the set-inclusion relation with a qualitative nature, that is, a set is either a subset of the other or not. As a quantitative generalization, a subsethood measure considers the degree of inclusion. It should keep essential characteristics of the qualitative set-inclusion and satisfy generalized properties with the qualitative set-inclusion relation as a special case. A systematic study of the relationships between the qualitative and quantitative frameworks results in the four types of properties of subsethood measures presented in this paper. A combination of different types of properties helps in constructing an appropriate subsethood measure in real-world applications.

Gödel's Argument for Cantorian Cardinality

AbstractOn 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: (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.

Letter Games: a metamathematical taster

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

Maximality Principles in Set Theory

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification of maximality principles.

An almost full embedding of the category of graphs into the category of abelian groups

We construct an embedding G:Graphs→Ab of the category of graphs into the category of abelian groups such that for X and Y in Graphs we haveHom(GX,GY)≅Z[HomGraphs(X,Y)], the free abelian group whose basis is the set HomGraphs(X,Y). The isomorphism is functorial in X and Y. The existence of such an embedding implies that, contrary to a common belief, the category of abelian groups is as complex and comprehensive as any other concrete category. We use this embedding to settle an old problem of Isbell as of whether every full subcategory of the category of abelian groups, which is closed under limits, is reflective. A positive answer turns out to be equivalent to weak Vopěnka's principle, a large cardinal axiom which is not provable but believed to be consistent with standard set theory. We obtain some consequences to the Hovey–Palmieri–Strickland problem about existence of arbitrary localizations in a stable homotopy category.Several known constructions in the category of abelian groups are obtained as quick applications of the embedding.

The locations of halo formation and the peaks formalism

We investigate the problem of predicting the halo mass function from the properties of the Lagrangian density field. We focus on a perturbation spectrum with a small-scale cut-off (as in warm dark matter cosmologies). This cut-off results in a strong suppression of low mass objects, providing additional leverage to rigorously test which perturbations collapse and to what mass. We find that all haloes are consistent with forming near peaks of the initial density field, with a strong correlation between proto-halo density and ellipticity. We demonstrate that, while standard excursion set theory with correlated steps completely fails to reproduce the mass function, the inclusion of the peaks constraint leads to the correct number of haloes but significantly underpredicts the masses of low-mass objects (with the predicted halo mass function at low masses behaving like dn/dln m ~ m^{2/3}). This prediction is very robust and cannot be easily altered within the framework of a single collapse barrier. The nature of collapse in the presence of a small-scale cut-off thus reveals that excursion set calculations require a more detailed understanding of the collapse-time of a general ellipsoidal perturbation to predict the ultimate collapsed mass of a peak -- a problem that has been hidden in the large abundance of small-scale structure in CDM. We demonstrate how this problem can be resolved within the excursion set framework.

Sets and Functions in Theoretical Physics

It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.

Objects as Limits of Experience and the Notion of Horizon in Mathematical Theories

Abstract The present work is an attempt to bring attention to the application of several key ideas of Husserl ‘s Krisis in the construction of certain mathematical theories that claim to be altemative nonstandard versions of the standard Zermelo-Fraenkel set theory. In general, these theories refute, at least semantically, the platonistic context of the Cantorian system and to one or the other degree are motivated by the notions of the lifeworld as the pregiven holistic field of experience and that of horizon as the boundary of human perceptions and the de facto constraint in reaching limit-idealizations. Moreover, 1 try to give convincing reasons for the existence of an ultimate constitutional ‘vacuum’ of a subjective origin that is formally refiected in the application of a notion of actual infinity in dealing generally with the mathematical infinite.

Constructive toposes with countable sums as models of constructive set theory

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive set theory. We show that any constructive topos with countable coproducts provides a model of a standard constructive set theory, CZFExp (that is, the variant of Aczel’s Constructive Zermelo–Fraenkel set theory CZF obtained by weakening Subset Collection to the Exponentiation axiom). The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk’s programme of algebraic set theory. A curiosity is that our model always validates the axiom V=Vω1 (in an appropriate formulation). It follows that the full Separation schema is always refuted.

Indestructibility of Vopěnka’s Principle

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders.

The bias and mass function of dark matter haloes in non-Markovian extension of the excursion set theory

The excursion set theory based on spherical or ellipsoidal gravitational collapse provides an elegant analytic framework for calculating the mass function and the large-scale bias of dark matter haloes. This theory assumes that the perturbed density field evolves stochastically with the smoothing scale and exhibits Markovian random walks in the presence of a density barrier. Here, we derive an analytic expression for the halo bias in a new theoretical model that incorporates non-Markovian extension of the excursion set theory with a stochastic barrier. This model allows us to handle non-Markovian random walks and to calculate perturbatively these corrections to the standard Markovian predictions for the halo mass function and halo bias. Our model contains only two parameters: κ, which parametrizes the degree of non-Markovianity and whose exact value depends on the shape of the filter function used to smooth the density field, and a, which parametrizes the degree of stochasticity of the barrier. Appropriate choices of κ and a in our new model can lead to a closer match to both the halo mass function and the halo bias in the latest N-body simulations than the standard excursion set theory.

The bargaining set in strategic market games

This paper presents a hybrid equilibrium notion that blends together the ‘cooperative’ and the ‘noncooperative’ theories of competition. In particular, the Mas-Colell bargaining set has been modified in order to accommodate the features of strategic market games. In other words, allocations, objections and counter objections of the standard bargaining set theory are described for an economy, where trades among groups of individuals are conducted via the Shapley–Shubik mechanism. In the main part of the paper, it is proved that in atomless economies the allocations resulting from this equilibrium notion are competitive.

Regular variation without limits

Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 1]) explores functions f for which the limit function g ( λ ) : = f ( λ x ) / f ( x ) exists (as x → ∞ ) and for which g ( λ ) = λ ρ subject to mild regularity assumptions on f. Further Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 2]) explores functions f for which the upper limit f * ( λ ) : = lim sup f ( λ x ) / f ( x ) , as x → ∞ , remains bounded. Here the usual regularity assumptions invoke boundedness of f * on a Baire non-meagre/measurable non-null set, with f Baire/measurable, and the conclusions assert uniformity over compact λ-sets (implying upper bounds of the form f ( λ x ) / f ( x ) ⩽ K λ ρ for all large λ, x). We give unifying combinatorial conditions which include the two classical cases, deriving them from a combinatorial semigroup theorem. We examine character degradation in the passage from f to f * (using some standard descriptive set theory) and thus identify natural classes in which the theory may be established.

Hierarchical Agent Monitored Parallel On-Chip System

In this paper, the authors present a formal specification of a novel design paradigm, hierarchical agent monitored SoCs (HAMSOC). The paradigm motivates dynamic monitoring in a hierarchical and distributed manner, with adaptive agents embedded for local and global operations. Formal methods are of essential importance to the development of such a novel and complex platform. As the initial effort, functional specification is indispensable to the non-ambiguous system modeling before potential property verification. The formal specification defines the manner by which the system can be constructed with hierarchical components and the representation of run-time information in modeling entities and every type of the monitoring operations. The syntax follows the standard set theory with additional glossary and notations introduced to facilitate practical SoC design process. A case study of hierarchical monitoring for power management in NoC (Network-on-chip), written with the formal specification, is demonstrated.

Hartry Field, Saving Truth from Paradox

This is a difficult, challenging, thick book (approximately four hundred pages) consisting of five parts and 27 chapters. A substantial amount of it consists of the author's original contributions, but it also offers a detailed comparison with different competing approaches. The influential work in the area of formal semantics, at least the one closer to Field's own views (Tarski, Kripke, Friedman and Sheard, Priest) is thoroughly considered, but not in the spirit of a systematic neutral attitude: in the author's own words, "the book is a survey, but an opinionated one, of philosophical work on paradoxes, with occasional side steps into issues like vagueness, validity, incompleteness theorems". According to Field, there is a coherent concept of full truth after all, but in dealing with it one must allow for a generalization of classical logic. Indeed, the central result of the book is a new model for self-referential truth. In particular, the unrestricted truth schema T( 'A]) A (for A arbitrary) is maintained, and the object language includes its own metalanguage (Here [A] is a fixed effective Godei numbering of A; henceforth we simply write TA instead of the proper T( [A] )). More generally, the intersubstitutivity principle holds: if D results from Cby replacing (in transparent contexts) some occurrence of A in C with TA, then D implies C and conversely. Due to these features, the author's solution can be summarized as follows: (1) the application of classical logic in particular tertium non datur TND is restricted to conditions that do not involve the truth predicate or, more generally, predication, or instantiation; (2) a non-standard implication, which reduces to full classical reasoning in "ordinary contexts", including standard set theory, is defined by means of a non-trivial construction. Feature (1) corresponds to the so called paracomplete solutions; these are in contrast with (and in a sense dual to) the so-called