Set Existence Principles Research Articles

Constructive strong regularity and the extension property of a compactification

In contexts in which the principle of dependent choice may not be available, as toposes or Constructive Set Theory, standard locale theoretic results related to complete regularity may fail to hold. To resolve this difficulty, B. Banaschewski and A. Pultr introduced strongly regular locales. Unfortunately, Banaschewski and Pultr's notion relies on non-constructive set existence principles that hinder its use in Constructive Set Theory. In this article, a fully constructive formulation of strong regularity for locales is introduced by replacing non-constructive set existence with coinductive set definitions, and exploiting the Relation Reflection Scheme. As an application, every strongly regular locale L is proved to have a compact regular compactification. The construction of this compactification is then used to derive the main result of this article: a characterization of locale compactifications (and thus, classically, of the compactifications of a space) in terms of their ability of extending continuous functions with compact regular codomains. Finally, an open problem related to the existence of the compact regular reflection of a locale is presented.

Set Existence Principles and Closure Conditions: Unravelling the Standard View of Reverse Mathematics†

AbstractIt is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second-order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the powerset of the natural numbers.

Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

Many mathematical objects arise from equivalence classes, and invite implementation as those classes. Set existence principles that would enable this are incompatible with ZFC’s unrestricted aussonderung but there are set theories (e.g. NF and Church’s CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church’s construction provides equivalence classes for “low” sets, and thus—e.g.—a set of all (low) ordinals. However that set has an ordinal in turn, which is not a member of the set constructed, so no set of all ordinals is obtained thereby. This “recurrence problem” is discussed. This essay can be seen as a non-technical companion to [6] and is, like it, a precursor to—and will be subsumed into—the longer book-length treatment of these matters that I have promised myself I will write. 1

Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories (e.g., NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all (low) ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no set of all ordinals is obtained thereby. This “recurrence problem” is discussed.

Zermelo and Set Theory

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

Characterising polynomial time computable functions using theories with weak set existence principles

Several authors have independently introduced second order theories whose provably total functionals are polynomial time computable functions on strings (e.g. [4], [6] and [7]), including the first author ([3], meant to be the second part of [2]). In this paper we give a detailed proof of the bi-interpretability result between such a second order theory and Buss' first order bounded arithmetic, based on an elegant definition of multiplication due to the second author.

Editorial

This issue of the Journal of Functional Programming is dedicated to work presented at the Workshop on Implicit Computational Complexity in Programming Languages, affiliated with the 1998 meeting of the International Conference on Functional Programming in Baltimore.Several machine-independent approaches to computational complexity have been developed in recent years; they establish a correspondence linking computational complexity to conceptual and structural measures of complexity of declarative programs and of formulas, proofs and models of formal theories. Examples include descriptive complexity of finite models, restrictions on induction in arithmetic and related first order theories, complexity of set-existence principles in higher order logic, and specifications in linear logic. We refer to these approaches collectively as Implicit Computational Complexity. This line of research provides a framework for a streamlined incorporation of computational complexity into areas such as formal methods in software development, programming language theory, and database theory.A fruitful thread in implicit computational complexity is based on exploring the computational complexity consequences of introducing various syntactic control mechanisms in functional programming, including restrictions (akin to static typing) on scoping, data re-use (via linear modalities), and iteration (via ramification of data). These forms of control, separately and in combination, can certify bounds on the time and space resources used by programs. In fact, all results in this area establish that each restriction considered yields precisely a major computational complexity class. The complexity classes thus obtained range from very restricted ones, such as NC and Alternating logarithmic time, through the central classes Poly-Time and Poly-Space, to broad classes such as the Elementary and the Primitive Recursive functions.Considerable effort has been invested in recent years to relax as much as possible the structural restrictions considered, allowing for more exible programming and proof styles, while still guaranteeing the same resource bounds. Notably, more exible control forms have been developed for certifying that functional programs execute in Poly-Time.The 1998 workshop covered both the theoretical foundations of the field and steps toward using its results in various implemented systems, for example in controlling the computational complexity of programs extracted from constructive proofs. The five papers included in this issue nicely represent this dual concern of theory and practice. As they are going to print, we should note that the field of Implicit Computational Complexity continues to thrive: successful workshops dedicated to it were affiliated with both the LICS'99 and LICS'00 conferences. Special issues, of Information and Computation dedicated to the former, and of Theoretical Computer Science to the latter, are in preparation.

Delineating classes of computational complexity via second order theories with weak set existence principles. I

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

A Foundational Delineation of Poly-time

We show that a function over {0, 1}* is poly-time iff it is computed by an equational program which can be proved to be everywhere converging in constructive second-order logic with set-existence (comprehension) restricted to positive quantifier-free formulas, or alternatively with set-existence for positive existential formulas. These set-existence principles convey an ontology of infinite sets as evolving, not completed, totalities. Our characterization results can consequently be viewed as a foundational justification for identifying poly-time with feasibility.

On the Scope and Force of Indispensability Arguments

Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results (of Pour-El and Richards and of the author) are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of physics

George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic

George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic

Continuity and comprehension in intuitionistic formal systems

Within the framework of a formal system, we can ask if one can find any necessary relations between the answers to the two questions posed above. We show that, within the language of second-order arithmetic, one cannot find any such relations; even if one includes Church's thesis, which says that every constructive function is recursive. In earlier work, we have proved independence results related to question (1) in the context of the language of arithmetic. The main tool of the present paper is an extension of our earlier methods to second-order comprehension principles. It is fairly easy to prove the consistency of strong principles of set existence with the continuity of extensional functions, even in the presence of Church's thesis (see discussion in [3]). And, as mentioned, the case where one does not have strong set existence principles has been dealt with in [1]. The main problem, then, is the independence of the continuity of extensional functions from strong set existence principles. Of course, if the formal axioms considered are classically true, this independence is trivial; but we are interested in the independence in an axiomatic framework including nonclassical principles. Foremost among such principles is Church's thesis, which (in a suitable formulation) will reduce members of NN to recursive indices, and functions from NN to N to effective operations, which compute the function value recursively from an index of the argument. Thus, under Church's thesis CT, the statement all functions are continuous reduces to an arithmetical proposition about effective operations. This proposition (for the case of NN) is called KLS, after Kreisel, Lacombe, and Shoenfield, who gave a classical proof of it [5]. It also happens that this sentence KLS lies in a syntactic class for which CT is conservative (over the theories we will consider; see discussion in the text). Thus the difficult part of our problem is to prove the independence of KLS from various principles of set existence.

Set existence principles of Shoenfield, Ackermann, and Powell

Set existence principles of Shoenfield, Ackermann, and Powell