A model of idealized scientific inquiry is presented in which scientists are required to infer the nature of the structure that makes true the data they examine. A necessary and sufficient condition is presented for scientific success within this paradigm.

First we prove that the set of countable linear orders of the form I + I form a complete analytic set. As a consequence of this we improve a result of Humke and Laczkovich, who showed in [HL] that the set of functions of the form f ⚬ f form a true analytic set in C[0, 1]. We show that these functions form a complete analytic set, solving a problem mentioned on p. 215 of [K1] and on p. 4 (...) of [B]. (shrink)

In this note we show that proof-theoretic uniform boundedness or bounded collection principles which allow one to formalize certain instances of countable Heine–Borel compactness in proofs using abstract metric structures must be carefully distinguished from an unrestricted use of countable Heine–Borel compactness.

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere (...) club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $, $R(\ell,D)$ implies $R(\ell,C)$, there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$, $R(\ell,\Lambda (\ell ))$. In particular, under these assumptions, for all $n \in \omega $, $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization. (shrink)

In this paper, we discuss what we argue is a newly observed use of nouns like woman, man, and lawyer, in the sort of morphosyntax characteristic of count nouns. We argue that the relevant data constitutes normative uses of the relevant nouns, and we build an analysis on the assumption that such nouns are polysemous between descriptive and normative senses (Leslie 2015), using the formal account of polysemy in Pustejovsky (1998), and the analysis of count- ability in Rothstein (2010). In (...) doing so, we provide evidence in support of the aforementioned kinds of analyses of social terms and countability, as opposed to others which do not seem to be able to account for the data in as straight forward a way. (shrink)

Two reverse supertasks—one new and one invented by Pérez Laraudogoitia —are discussed. Contra Kerkvliet and Pérez Laraudogoitia, it is argued that these supertasks cannot be used to conduct fair infinite lotteries, i.e., lotteries on the set of natural numbers with a uniform probability distribution. The new supertask involves an infinity of gods who collectively select a natural number by each removing one ball from a collection of initially infinitely many balls in a reverse omega-sequence of actions.

We examine the impact of Suzumura’s (Economica 43:381–390, 1976) consistency property when applied in the context of collective choice rules that are independent of irrelevant alternatives, neutral, and monotonic. An earlier contribution by Blau and Deb (Econometrica 45:871–879, 1977) establishes the existence of a vetoer if the collective relation is required to be complete and acyclical. The purpose of this paper is to explore the possibilities that result if completeness and acyclicity are dropped and Suzumura consistency is imposed instead. A (...) conceptually similar but logically independent version of the combined axiom that requires the collective decision mechanism to be independent, neutral, and monotonic is employed. In the case of a finite population, we obtain an alternative impossibility theorem if a collective choice rule is assumed to be non-degenerate and a modified no veto requirement is imposed instead of Blau and Deb’s (1977) condition. If the population is countably infinite, the impossibility can be avoided but it resurfaces if our new no veto property is extended to a coalitional variant. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than (...) the ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

First published in 1998. Routledge is an imprint of Taylor & Francis, an informa company.

We show that a $$ -simplified morass can be added by a forcing with working parts of size smaller than $\kappa $. This answers affirmatively the question, asked independently by Shelah and Velleman in the early 1990s, of whether it is possible to do so.Our argument use a modification of a technique of Mitchell’s for adding objects of size $\omega _2$ in which collections of models – all of equal, countable size – are used as side conditions. In our (...) modification, whilst the individual models are, as in Mitchell’s technique, taken ad hoc from quite general classes, the collections of models are very highly structured, in a way that is somewhat different from, perhaps more stringent than, Mitchell’s original, arguably making the method more wieldy and giving the prospect of further uses with more delicate working parts. (shrink)

This paper argues that professionals who make boundary-related decisions should be guided by relationship-based practice. In our roles as service users and professionals, drawing from our lived experiences of professional relationships, we argue we need to move away from distance-based practice. This includes understanding the boundary stories and narratives that exist for all of us – including the people we support, other professionals, as well as the organisations and systems within which we work. When we are dealing with professional boundary (...) issues, we should centre relationship-building skills that are central to many other aspects of our work. Skills that foster relationships at all levels – between professionals, service users, and services – need to be revalued. Our final recommendation is to create, develop, and foster safer spaces within and outside of organisations, as well as inter-professionally, for the discussion and exploration of boundary-related issues and practice. We are interested in hearing from those with experiences of being marginalised by boundaries so that they can inform a reshaping of our collective ideas around boundary related practices. To foster relationship-based practices in organisations, we have outlined several recommendations here; however, we recognize that these do not go far enough, and that collective action is needed to inform systemic change. (shrink)

We introduce a suitable notion of eight-shaped curve in the product S × ℝ of a Suslin line S for the real line ℝ, and we prove that if S is dense in itself, then every collection of pairwise disjoint eight-shaped curves in S × ℝ is countable. This parallels a folklore result which holds for the real plane.

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ (...) classes and for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)

Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov sum (...) rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers. (shrink)

We introduce a hierarchy of sets which can be derived from the integers using countable collections. Such families can be coded into countable algebraic structures preserving their algorithmic properties. We prove that for different finite levels of the hierarchy the corresponding algebraic structures have different classes of possible degree spectra.

We show in ZFC, assuming all reals have sharps, that a countable collection of ▵ 1 3 -degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.

We show in ZFC, assuming all reals have sharps, that a countable collection of $\triangle^1_3$-degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.

This article addresses nominal quantification in English in relation to discrete and continuous quantity, the two semantic categories of discrete and continuous / mass being analysed as interpretations of syntax. It re-examines the hypothesis of non-quantifiable continuous nouns as well as some theoretical questions such as overloaded definitions, unexploited oppositions or notions found without an explicit definition, as is sometimes the case with the concept of collective. The study then proceeds to examine semantic multiplicity in connection with grammatical number. Plural (...) is examined in association with numerals and fuzzy quantifiers. The singular category also also yields interpretations of multiplicity, whether with mass-nouns or with discrete collective nouns. This, together with the possible conversion of the latter into mass-nouns, draws one’s attention to a mode of reference shared by mass nouns and collective discrete nouns. (shrink)

Let L be a countable language, let I be an isomorphism-type of countable L-structures, and let a∈2ω. We say that I is a-strange if it contains a computable-from-a structure and its Scott rank is exactly ω1a. For all a, a-strange structures exist. Theorem : If C is a collection of ℵ1 isomorphism-types of countable structures, then for a Turing cone of a’s, no member of C is a-strange.

Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize (...) these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs. (shrink)

Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim Skolem theorem says that if a first order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first (...) order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers? (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a (...) way that vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

We analyze the decisiveness structures associated with acyclical collective choice rules. In particular, we examine the consequences of adding anonymity to weak Pareto, thereby complementing earlier results on acyclical social choice. Both finite and countably infinite populations are considered. As established in contributions by Donald Brown and by Jeffrey Banks, acyclical social choice is closely linked to prefilters in the presence of the weak Pareto principle. We introduce the notion of a conditional prefilter and use it to generalize their results. (...) In the finite-population case, adding anonymity implies that the collection of decisive sets is a special case of a conditional prefilter. We then identify the decisiveness structure that results from adding anonymity to the weak Pareto principle. Moving to infinite populations, we obtain the decisive set that consists of the entire population as a possibility, along with a new class of prefilters that we refer to as symmetric free Fréchet prefilters. The choice of the term Fréchet prefilter is motivated by the observation that they share a defining property with the well-known Fréchet filter—namely, that all sets in the requisite collection are such that their complement is finite. (shrink)

We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets (...) of reals of every size, but none of size, and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. (shrink)

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.

Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf. With a notion of satisfaction and -isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game (...) is associated a tree which is a tree of size κ with no κ-branches -branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf κ. The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models. (shrink)

We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this by (...) proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀ Π 1 b -separation of bounded arithmetic theories S 2 i from T 2 j is equivalent to the existence of a model of S 2 i which does not have a Δ 0 b -elementary extension to a model of T 2 j . More specific, let M ⊨ Ω 1 nst denote that there is a nonstandard element c in M such that the function n ↦ 2 log c is total in M . Let BLΣ 1 b be the bounded collection schema for Σ 1 b -formulas. We obtain the following preservation results: the ∀ Π 1 b -separation of S 2 i from T 2 j is equivalent to the existence of 1. a model of S 2 i +Ω 1 nst which is 1 b -closed w.r.t. T 2 j , 2. a countable model of S 2 i + BLΣ 1 b without weak end extensions to models of T 2 j. (shrink)

We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the (...) countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely $\Delta^1_2$ reductions, and hence to the infinite time computable reductions. (shrink)

The paper develops a theory of branching spatiotemporal histories that accommodates indeterminism and the insights of general relativity. A model of this theory can be viewed as a collection of overlapping histories, where histories are defined as maximal consistent subsets of the model's base set. Subsequently, generalized manifolds are constructed on the theory's models, and the manifold topology is introduced. The set of histories in a model turns out to be identical with the set of maximal subsets of the (...) model's base set with respect to being Hausdorff and downward closed. Further postulates ensure that the topology is connected, locally Euclidean, and satisfies the countable sub-cover condition. (shrink)

We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces . We prove that any classical probability space has a causally closed extension; any finite classical probability space with positive rational probabilities on the atoms of the event algebra can be extended to a causally up-to-three-closed finite space; and any classical probability space can be extended to a space in which all correlations between events that are logically independent modulo measure zero event (...) have a countably infinite common-cause system. Collectively, these results show that it is surprisingly easy to find Reichenbach-style ‘explanations' for correlations, underlining doubts as to whether this approach can yield a philosophically relevant account of causality. 1 Introduction2 Basic Definitions and Results in the Literature3 Causal Completability the Easy Way: ‘Splitting the Atom’4 Causal Completability of Classical Probability Spaces: The General Case5 Infinite Statistical Common-Cause Systems for Arbitrary Pairs6 ConclusionAppendix A. (shrink)

We give alternative proofs of results due to Paris and Wilkie concerning the existence of end extensions of countable models of $B\Sigma_{1}$, that is, the theory of $\Sigma_{1}$ collection.