In this paper, I develop a view on set-theoretic ontology I call Universe-Indeterminism, according to which there is a unique but indeterminate universe of sets. I argue that Solomon Feferman’s work on semi-constructive set theories can be adapted to this project, and develop a philosophical motivation for a semi-constructive set theory closely based on Feferman’s but tailored to the Universe-Indeterminist’s viewpoint. I also compare the emergent Universe-Indeterminist view to some more familiar views on set-theoretic ontology.

Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. (...) Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them. (shrink)

The distinction between the syntactic and the semantic approach to scientific theories emerged in formal philosophy of science. The semantic approach is commonly considered more advanced and more successful than the syntactic one, but the transition from the one approach to the other was not brought about without any loss. In essence, it is the formal analysis of atomic propositions and the analysis of deductive reasoning that dropped out of consideration in at least some of the elaborated versions of the (...) semantic approach. In structuralist theory of science, as founded by Sneed and Stegmüller, the focus is on global propositions concerning the question of whether or not certain empirical systems satisfy a set-theoretic predicate that encodes the axioms of a scientific theory. Hence, an analysis of deductive reasoning from atomic premisses with the help of a given theory is missing. The objective of the present paper is to develop a deductive system on the basis of the structuralist framework. This system comes with a novel formulation of empirical propositions in structuralism. (shrink)

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

We prove a theorem stating that any semantics can be encoded as a compositional semanties, which means that, essentially, the standard definition of compositionality is formally vacuous. We then show that when compositional semantics is required to be systematic (that is, the meaning function cannot be arbitrary, but must belong to some class), it is possible to distinguish between compositional and noncompositional semantics. As a result, we believe that the paper clarifies the concept of compositionality and opens the possibility of (...) making systematic formal comparisons of different systems of grammar. (shrink)

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in (...) contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...) complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

One objective of this paper is the determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with Δ 2 1 comprehension and bar induction. As Martin-Löf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a (...) strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts.Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. (shrink)

What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the (...) nature of the equivalence of PA and ZF−inf and corrects some errors in the literature. We also survey the restrictions of the Ackermann interpretation and its inverse to subsystems of PA and ZF−inf, where full induction, replacement, or separation is not assumed. The paper concludes with a discussion on the problems one faces when the totality of exponentiation fails, or when the existence of unordered pairs or power sets is not guaranteed. (shrink)

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or (...) \\)th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. (shrink)

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II11 reflection . We show that admissible sets such as H and Lω2 which fail to have strict-II11 reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms.

We prove a completeness theorem for Kmath image, the infinitary extension of the graded version K0 of the minimal normal logic K, allowing conjunctions and disjunctions of countable sets of formulas. This goal is achieved using both the usual tools of the normal logics with graded modalities and the machinery of the predicate infinitary logics in a version adapted to modal logic.

KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $\Sigma (L_{\omega _1^c } )$ sentences, whereω 1 c denotes the least nonrecursive ordinal. This paper is self-contained, at least from a technical (...) point of view. (shrink)

We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.

We investigate the extent to which structures consisting of sequences of forests on the same underlying set are well-quasi-ordered under embeddings.

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91].

A general result is proved about the existence of maximally consistent theories satisfying prescribed closure conditions. The principle is then used to give streamlined proofs of completeness and omitting-types theorems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory respects a certain class of inference rules.

Appeal to standard set theory in 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 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 :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. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...) of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. The main result shows that these models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main step towards this characterization shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We then construct models (...) with this property as generic extensions of iterates of mice under the assumption that the corresponding projective ordinal is below ω2ω2. On the way, we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For instance, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably View the MathML source Δ1/3 equivalence relations, and generalize this to all projective levels. (shrink)

In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By (...) Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φ ω00 is also the ordinal of KPi0+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundation. (shrink)

The paper contains an axiomatic treatment of the intuitionistic theory of hereditarily finite sets, based on an induction axiom-schema and a finite set of single axioms. The main feature of the principle of induction used (due to Givant and Tarski) is that it incorporates Foundation. On the analogy of what is done in Arithmetic, in the axiomatic system selected the transitive closure of the membership relation is taken as a primitive notion, so as to permit an immediate adaptation of the (...) theory to cases of restricted induction (in particular primitive recursive induction). At the end of the paper several different forms of induction, which play an important role in the development of the theory, are compared. An alternative axiomatization of the theory, which is of intrinsic interest, is also discussed. (shrink)

We extend a theorem of Barwise and Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad (...) hoc nor unnatural. (shrink)

We prove an extension of the Lemma of Rasiowa and Sikorski and give some applications. Moreover, we analyze the relationship to corresponding results on the omission of types.

We prove two conservation results involving a generalization of the principle of strict ${\Pi^1_1}$ -reflection, in the context of bounded arithmetic. In this context a separation between the concepts of bounded set and binary sequence seems to emerge as fundamental.

Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π 1 l −TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast growing) extended Grzegorczyk (...) hierarchy $(F_a )_{a \in T(\Omega )_0 }$ and the slow growing hierarchy $(G_a )_{a \in T(\Omega )_0 }$ are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way: $$\begin{array}{*{20}c} {G_0 (n): = 0,} & {F_0 (n): = (n + 1)^2 ,} \\ {\begin{array}{*{20}c} {G_{a + 1} (n): = G_a (n) + 1,} \\ {G_l (n): = G_{l[n]} (n),} \\ \end{array} } & {\begin{array}{*{20}c} {F_{a + 1} (n): = \underbrace {F_a (...F_a }_{n + 1 - times}(n)...),} \\ {F_l (n): = F_{l[n]} (n),} \\ \end{array} } \\ \end{array}$$ wherel is a countable limit ordinal (term) and (l[n]) n ∈N is the distinguished fundamental sequence assigned tol. For each ordinal (term)a in T(Ω) and each natural numbern letC n (a) be the formal term which results from the ordinal terma by successively replacing every occurence ofψ a by $\psi _{ - 1 + C_n (a)}$ whereψ −1 is considered as a defined function symbol, namely $\psi _{ - 1} b: = F_{\psi _0 b + 1} (n + 1)$ . (Note thatψ a 0=Ω a ) In this article it is shown that for each ordinal termψ 0 a in T(Ω) there exists a natural numbern 0 such thatψ 0 C n (a) ∈ T(Ω) and $G_{\psi _0 a} (n) \leqslant F_{\psi _0 C_n (a) + 1} (n + 1)$ holds for alln≥n 0. This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals — i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question — of various theories of iterated inductive definitions (and subsystems ofKPi) and on the number and size of the subrecursively inaccessible ordinals — i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up — below the proof-theoretic ordinal ofACA 0+(Π 1 l −TR). In particular these subrecursively inaccessibles ordinals are necessarily of the form $\psi _0 \Omega ..._{\Omega _\omega }$. (shrink)

In ordinal analysis of impredicative theories so-called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly on admissible (...) sets also renders a new and perspicuous approach to well-ordering proofs possible. MSC: 03F15, 03F35. (shrink)

Hoover [2] proved a completeness theorem for the logic L[MATHEMATICAL SCRIPT CAPITAL A]. The aim of this paper is to prove a similar completeness theorem with respect to product measurable biprobability models for a logic Lmath image with two integral operators. We prove: If T is a ∑1 definable theory on [MATHEMATICAL SCRIPT CAPITAL A] and consistent with the axioms of Lmath image, then there is an analytic absolutely continuous biprobability model in which every sentence in T is satified.

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is animpassable barrierin the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted inbut also what cannot be so (...) interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

We apply the method of asymmetric interpretation to the basic fragment of bounded arithmetic, endowed with a weak collection schema, and to a system of “feasible analysis”, introduced by Ferreira and based on weak König's lemma, recursive comprehension and NP-notation induction. As a byproduct, we obtain two conservation results.

The paper characterizes the second order arithmetic theorems of a set theory that features a recursively Mahlo universe; thereby complementing prior proof-theoretic investigations on this notion. It is shown that the property of being recursively Mahlo corresponds to a certain kind of β-model reflection in second order arithmetic. Further, this leads to a characterization of the reals recursively computable in the superjump functional.