Weak Set Theory Research Articles

Weak set theories in foundational debates.

Perception of the relationship of the discipline of logic to other exact sciences changes with the years. No twentieth-century proposal for a single logical system that would support the whole of mathematics satisfied everyone, so weaker formal systems with applications in many different contexts are now sought, in mathematics, philosophy, computer science or elsewhere. As always, logicians help by asking such questions as 'Is this theory correct for the given context?' and 'Can it be used in other contexts?' The Introduction of this paper surveys the history of logic and its links to other scientific disciplines such as biology, economics, engineering and physics, and some details of current and future research projects are given. Then the very weak set theory PROVI is introduced and its support for the techniques of constructibility (Gödel 1935) and forcing (Cohen PJ 1963 The independence of the continuum hypothesis, I. Proc. Natl Acad. Sci. USA 50, 1143-1148. (doi:10.1073/pnas.50.6.1143)) is given in outline, full details being already in print. Relations with other known techniques are explored and developed. New results coming from the study of illfounded [Formula: see text]-models of PROVI and other systems are given; and new formal systems in the style of Quine (1937 Quine WV. 1936 Set-theoretic foundations for logic. J. Symb. Log. 1, 45-57. (doi:10.2307/2268548)) are described. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Reconciling transparency, low Δ0-complexity and axiomatic weakness in undecidability proofs

Abstract In a first-order theory $\varTheta $, the decision problem for a class of formulae $\varPhi $ is solvable if there is an algorithmic procedure that can assess whether or not the existential closure $\varphi ^{\exists }$ of $\varphi $ belongs to $\varTheta $, for any $\varphi \in \varPhi $. In 1988, Parlamento and Policriti already showed how to tailor arguments à la Gödel to a very weak axiomatic set theory, referring them to the class of $\varSigma _{1}$-formulae with $(\forall \exists \forall )_{0}$-matrix, i.e. existential closures of formulae that contain just restricted quantifiers of the forms $(\forall x \in y)$ and $(\exists x \in y)$ and are writable in prenex form with at most two alternations of restricted quantifiers (the outermost quantifier being a ‘$\forall $’). While revisiting their work, we show slightly less weak theories under which incompleteness for recursively axiomatizable extensions holds with respect to existential closures of $(\forall \exists )_{0}$-matrices, namely formulae with at most one alternation of restricted quantifiers.

Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman–Myhill Result, and Its Immediate Reception†

ABSTRACTThe one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general ($a\!=\!b\vee\neg a\!=\!b$) is derived; then, of sentences in general ($\psi\vee\neg\psi$). Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set theorist with a dispositive reason for not adopting (full) Choice.

Type-two well-ordering principles, admissible sets, and -comprehension

This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalban. The equivalence is interesting because it relates concrete notions from ordinal analysis to abstract notions from reverse mathematics and set theory. A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an almost monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is computed. On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schutte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory. Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic.

Formalizing Scientifically Applicable Mathematics in a Definitional Framework

In [Arnon08, A framework for formalizing set theories based on the use of static set terms.] a new framework for formalizing mathematics was developed. The main new features of this framework are that it is based on the usual first-order set theoretical foundations of mathematics (in particular, it is type-free), but it reflects real mathematical practice in making an extensive use of statically defined abstract set terms of the form { x | p(i) }, in the same way they are used in ordinary mathematical discourse. In this paper we show how large portions of fundamental, scientifically applicable mathematics can be developed in this framework in a straightforward way, using just a rather weak set theory which is predicatively acceptable and essentially first-order. The key property of that theory is that every object which is used in it is defined by some closed term of the theory. This allows for a very concrete, computationally-oriented interpretation of the theory. However, the development is not committed to such interpretation, and can easily be extended for handling stronger set theories (including ZF).

Consequences of Vopěnka’s Principle over weak set theories

It is shown that Vop\v{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $\Delta_0$-Separation and Induction along $\omega$, then ${\rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in \cite{Tz14}, as well as by H. Friedman's \cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+$Foundation$+{\rm VP}$=${\rm ZF+VP}$, and ${\rm EST}+$Foundation$+{\rm AC+VP}={\rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--\{Foundation\}+${\rm VP}$. It is open whether the Axiom of Choice is independent from ${\rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.

A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is [formula omitted]-equivalent to KP

We introduce a version of operational set theory, OST−, without a choice operation, which has a machinery for Δ0 separation based on truth functions and the separation operator, and a new kind of applicative set theory, so-called weak explicit set theory WEST, based on Gödel operations. We show that both the theories and Kripke–Platek set theory KP with infinity are pairwise Π1 equivalent. We also show analogous assertions for subtheories with ∈-induction restricted in various ways and for supertheories extended by powerset, beta, limit and Mahlo operations. Whereas the upper bound is given by a refinement of inductive definition in KP, the lower bound is by a combination, in a specific way, of realisability, (intuitionistic) forcing and negative interpretations. Thus, despite interpretability between classical theories, we make “a detour via intuitionistic theories”. The combined interpretation, seen as a model construction in the sense of Visser's miniature model theory, is a new way of construction for classical theories and could be said the third kind of model construction ever used which is non-trivial on the logical connective level, after generic extension à la Cohen and Krivine's classical realisability model.

Set Theory, Topology, and the Possibility of Junky Worlds

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds; Watson (2010) suggests that Bohn’s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo–Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: topology (and possibly other “algebraic” mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds.

THE CLASSICAL CONTINUUM WITHOUT POINTS

AbstractWe develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary “actual infinity”. Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishop’s (1967) constructivism, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a nonpunctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind–Cantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

The strength of extensionality II — Weak weak set theories without infinity

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 (2009) 234–268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse mathematics–Friedman–Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory–can be reformulated within one language so that we can compare them more directly.

The strength of extensionality I — weak weak set theories with infinity

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg (and the axiom of choice AC). We first introduce a weak weak set theory Basic (which has the axioms of infinity and of collapsing) as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals: 1. | Basic | = ω ω and | Basic + Ext | = ε 0 , 2. | Basic + Δ 0 - Sep | = ε 0 and | Basic + Δ 0 - Sep + Ext | = Γ 0 . We also show that neither Reg nor AC affects the proof-theoretic strength, i.e., | T | = | T + Reg | = | T + AC | = | T + Reg + AC | where T is Basic plus any combination of Ext and Δ 0 - Sep .

Pairs, sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.

The axiom of choice and the law of excluded middle in weak set theories

AbstractA weak form of intuitionistic set theoryWSTlacking the axiom of extensionality is introduced. WhileWSTis too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng upWSTwith moderate extensionality principles or quotient sets enables the derivation to go through. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The natural numbers in constructive set theory

AbstractConstructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

LÉVY HIERARCHY IN WEAK SET THEORIES

We investigate the interactions of formula complexity in weak set theories with the axioms available there. In particular, we show that swapping bounded and unbounded quantification preserves formula complexity in presence of the axiom of foundation weakened to an arbitrary set base, while it does not if the axiom of foundation is further weakened to a proper class base. More attention is being paid to the necessary axioms employed in the positive results, than to the combinatorial strength of the positive results themselves.

Alternative Translation Techniques for Propositional and First-Order Modal Logics

We describe and analyze techniques, other than the standard relational/functional methods, for translating validity problems of modal logics into first-order languages. For propositional modal logics we summarize the s-as-Pow method, a complete and automatic translation into a weak set theory, and then describe an alternative method, which we call i>algebraic, that achieves the same full generality of s-as-Pow but is simpler and computationally more attractive. We also discuss the relationships between the two methods, showing that s-as-Pow generalizes to the first-order case. For first-order modal logics, we describe two extensions, of different degrees of generality, of s-as-Pow to logics of rigid designators and constant domains.

Type inference for set theory

Weak set theories are employed for set-theoretic specification. We develop and explore type inference systems for such set theories.

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.

Algebraic proofs of consistency and completeness

An embedding of the relations in the predicate transformers, analogous to that of the integers in the rationals, is exploited to provide simple algebraic proofs for the consistency and completeness of a calculus of program refinement. The calculus of refinement is derived by almost direct translation of the Hoare logic inference rules, and so alternatively the proofs may be viewed as demonstrating the soundness and completeness of Hoare logic. The main attributes of the embedding used in the proofs are that it supports a weak form of inversion (i.e. Galois connection) of relations, and that it supports an operator on predicate transformers that behaves like the floor operator on rationals: the operator maps arbitrary predicate transformer down in the natural ordering to the nearest embedded relation. A more general use for the floor-like operator in extending the relational calculus is suggested by its providing decomposition of the weakest prespecification operator. A weak algebraic set theory is used as a foundation for proving all required properties of the floor-like operator.