Standard Set Theory Research Articles

Why quasi-sets?

Quasi-set theory was developed to deal with collections of indistinguishable objects. In standard mathematics, there are no such kind of entities, for indistinguishability (agreement with respect to all properties) entails numerical identity. The main motivation underlying such a theory is of course quantum physics, for collections of indistinguishable (’identical’ in the physicists’ jargon) particles cannot be regarded as ’sets’ of standard set theories, which are collections of distinguishable objects. In this paper, a rationale for the development of such a theory is presented, motivated by Heinz Post’s claim that indistinguishability of quantum entities should be attributed ’right at the start’.

Computational type theory

Computational type theory provides answers to questions such as: What is a type? What is a natural number? How do we compute with types? How are types related to sets? Can types be elements of types? How are data types for numbers, lists, trees, graphs, etc. related to the corresponding notions in mathematics? What is a real number? Are the integers a subtype of the reals? Can we form the type of all possible data types? Do paradoxes arise in formulating a theory of types as they do in formulating a theory of sets, such as the circular idea of the set of all sets or the idea of all sets that do not contain themselves as members? Is there a type of all types? What is the underlying logic of type theory? Why isn’t it the same logic in which standard set theories are axiomatized? What is the origin of the notion of a type? What distinguishes computational type theory from other type theories? In computational type theory, is there a type of all computable functions from the integers to the integers? If so, is it the same as the set of Turing computable functions from integers to integers? Is there a type of computable functions from any type to any type? Is there a type of the partial computable functions from a type to a type? Are there computable functions whose values are types? Do the notations of an implemented computational type theory include programs in the usual sense? What does it mean that type theory is a foundational theory for both mathematics and computer science? There have been controversies about the foundations of mathematics, does computational type theory resolve any of them, do these controversies impact a foundation for computing theory? This article answers some of these questions and points to literature answering all of them.

Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic Semantics

Church's Higher Order Logic is a basis for influential proof assistants -- HOL and PVS. Church's logic has a simple set-theoretic semantics, making it trustworthy and extensible. We factor HOL into a constructive core plus axioms of excluded middle and choice. We similarly factor standard set theory, ZFC, into a constructive core, IZF, and axioms of excluded middle and choice. Then we provide the standard set-theoretic semantics in such a way that the constructive core of HOL is mapped into IZF. We use the disjunction, numerical existence and term existence properties of IZF to provide a program extraction capability from proofs in the constructive core. We can implement the disjunction and numerical existence properties in two different ways: one using Rathjen's realizability for IZF and the other using a new direct weak normalization result for IZF by Moczydlowski. The latter can also be used for the term existence property.

Discerning Fermions

We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrödinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. 1. Introduction: The Currently Dominant View1.1. Weyl on Leibniz's principle 1.2. Intermezzo: Terminology and Leibnizian principles 1.3. The rise of the currently dominant view 1.4. Overview 2. Elements of Quantum Mechanics2.1. Physical states and physical magnitudes 2.2. Composite physical systems of similar particles 2.3. Fermions and bosons 2.4. Physical properties 2.5. Varieties of quantum mechanics 3. Analysis of Arguments3.1. Analysis of the Standard Argument 3.2. Van Fraassen's analysis 3.3. Massimi's analysis 4. The Logic of Identity and Discernibility4.1. The language of quantum mechanics 4.2. Identity of physical systems 4.3. Indiscernibility of physical systems 4.4. Some kinds of discernibility 5. Discerning Elementary Particles5.1. Preamble 5.2. Fermions 5.3. Bosons 6. Concluding Discussion

FREGE MEETS ZERMELO: A PERSPECTIVE ON INEFFABILITY AND REFLECTION

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

ON ADOPTING KRIPKE SEMANTICS IN SET THEORY

Several philosophers have argued that the logic of set theory should be intuitionistic on the grounds that the open-endedness of the set concept demands the adoption of a nonclassical semantics. This paper examines to what extent adopting such a semantics has revisionary consequences for the logic of our set-theoretic reasoning. It is shown that in the context of the axioms of standard set theory, an intuitionistic semantics sanctions a classical logic. A Kripke semantics in the context of a weaker axiomatization is then considered. It is argued that this semantics vindicates an intuitionistic logic only insofar as certain constraints are put on its interpretation. Wider morals are drawn about the restrictions that this places on the shape of arguments for an intuitionistic revision of the logic of set theory.

Hume’s Big Brother: counting concepts and the bad company objection

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising.

The natural logic of language and cognition

This paper aims at an explanation of the discrepancies between natural intuitions and standard logic in terms of a distinction between NATURAL and CONSTRUCTED levels of cognition, applied to the way human cognition deals with sets. NATURAL SET THEORY (NST) restricts standard set theory cutting it down to naturalness. The restrictions are then translated into a theory of natural logic. The predicate logic resulting from these restrictions turns out to be that proposed in Hamilton (1860) and Jespersen (1917). Since, in this logic, NO is a quantifier in its own right, different from NOT-SOME, and given the assumption that natural lexicalization processes occur at the level of basic naturalness, single-morpheme lexicalizations for NOT-ALL should not occur, just as there is no single-morpheme lexicalization for NOT-SOME at that level. An analogous argument is developed for the systematic absence of lexicalizations for NOT-AND in propositional logic.

The age dependence of halo clustering

Abstract We use a very large simulation of the concordance Λ cold dark matter cosmogony to study the clustering of dark matter haloes. For haloes less massive than about 1013h−1 M⊙ the amplitude of the two-point correlation function on large scales depends strongly on halo formation time. Haloes that assembled at high redshift are substantially more clustered than those that assembled more recently. The effect is a smooth function of halo formation time and its amplitude increases with decreasing halo mass. At 1011h−1 M⊙ the ‘oldest’ 10 per cent of haloes are more than five times more strongly correlated than the ‘youngest’ 10 per cent. This unexpected result is incompatible with the standard excursion set theory for structure growth, and it contradicts a fundamental assumption of the halo occupation distribution models often used to study galaxy clustering, namely that the galaxy content of a halo of given mass is statistically independent of its larger scale environment.

The Reference Principle

Certain features of Horwich's theory make the situation even worse. Namely, as Horwich explicitly notes (Horwich 1998: 20, n. 4), the minimal theory cannot be regarded as the set of propositions of the form ((p) is true <-4 p). What this actually means is that it is a proper class larger than any, however highly infinite, set. Assuming standard set theory, the totality of the axioms of the minimal theory has thus the same extreme level of infinity as the whole set theoretic universe (at least the power of the first inaccessible cardinal). Certainly this is nobody's theory of truth. This may be worrying enough, but in the present context the main issue is that neither the o-rule nor anything analogous to it makes sense if the intended universe is uncountable. For such rules require that there is a canonical name for every element of the universe, and this is totally incredible if the universe is uncountable. But this is the case with Horwich's totality of propositions. Under closer scrutiny, Horwich's rescue strategy emerges as desperately implausible. Minimalism as it stands is still in deep trouble with the problem of generalizations.

Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies

This paper demonstrates two meta-mathematical propositions concerning the increasingly popular “intuitionistic” (= vague) approaches to fuzzy sets and fuzzy topology, as well as the closely related interval-valued (= grey) sets and interval-valued “intuitionistic” sets: (1) the term “intuitionistic” in these contexts is historically inappropriate given the standard mathematical usage of “intuitionistic”; and (2), at every level of existence—powerset level, topological fibre level, categorical level—interval-valued sets, interval-valued “intuitionistic” sets, and “intuitionistic” fuzzy sets and fuzzy topologies are redundant and represent unnecessarily complicated, strictly special subcases of standard fixed-basis set theory and topology. It therefore follows that theoretical workers should stop working in these restrictive and complicated programs and instead turn their efforts to substantial problems in the simpler and more general fixed-basis and variable-basis set theory and topology, while applied workers should carefully document the need or appropriateness of interval-valued or “intuitionistic” notions in applications.

Consistency of a counterexample to Naimark's problem.

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming that these axioms are consistent). We prove that the statement "there exists a counterexample to Naimark's problem which is generated by aleph (1) elements" is undecidable in standard set theory.

The cumulative hierarchy and the constructible universe of ZFA

AbstractWe present two results which shed some more light on the deep connection between ZFA and the standard ZF set theory: First of all we refine a result of Forti and Honsell (see [5]) in order to prove that the universe of ZFA can also be obtained (without appealing to choice) as the least fixed point of a continuous operator and not only as the greatest fixed point of the powerset operator. Next we show that it is possible to define a new absolute Gödel operation in addition to the standard ones in order to obtain the “constructible” model of ZFA as the least fixed point of the continuous operator of Gödel closure with respect to the standard and the new Gödel operations. (© 2003 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A Strong Model of Paraconsistent Logic

The purpose of this paper is mainly to give a model of paraconsistent logic satisfying the Frege comprehension scheme in which we can develop standard set theory (and even much more as we shall see). This is the continuation of the work of Hinnion and Libert. © 2004 University of Notre Dame.

A Revenge-Immune Solution to the Semantic Paradoxes

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A within the language.

Realism, nonstandard set theory, and large cardinals

Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST + (Basic Nonstandard Set Theory plus additional Idealization axioms). BNST + has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST +, and compare it with other existing nonstandard set theories.

Information granulation and rough set approximation

Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarse-grained views of the universe. Approximation of concepts, represented by subsets of the universe, deals with the descriptions of concepts using granules. In the context of rough set theory, this paper examines the two related issues. The granulation structures used by standard rough set theory and the corresponding approximation structures are reviewed. Hierarchical granulation and approximation structures are studied, which results in stratified rough set approximations. A nested sequence of granulations induced by a set of nested equivalence relations leads to a nested sequence of rough set approximations. A multi-level granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored. © 2001 John Wiley & Sons, Inc.

Modal deduction in second-order logic and set theory - I

We investigate modal deduction through translation into standard logic and set theory. In a previous paper, using a set-theoretic translation method, we proved that derivability in the minimal modal logic K, corresponds precisely to derivability in a weak, computationally attractive set theory ω In this paper, this approach is shown equivalent to working with standard first-order translations of modal formulae in a theory of general frames. The employed techniques are mainly model-theoretic and set-theoretic, and they admit extensions to richer languages and modal deductive systems than that of basic modal logic. Some of these extensions are discussed in the last part of the paper.

Mathematical constraints on a theory of human memory

Colonius suggests that, in using standard set theory as the language in which to express our computational-level theory of human memory, we would need to violate the axiom of foundation in order to express meaningful memory bindings in which a context is identical to an item in the list. We circumvent Colonius's objection by allowing that a list item may serve as a label for a context without being identical to that context. This debate serves to highlight the value of specifying memory operations in set theoretic notation, as it would have been difficult if not impossible to formulate such an objection at the algorithmic level.

Set theoretic foundations for a theory of human memory

A formal language to implement a computational-level theory of memory cannot be based on standard (Zermelo-Fraenkel) set theory. For correct models of a memory database to exist, non-well-founded sets must be permitted. This is illustrated by modifying an example from Barwise & Moss (1991).