Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even ℝ may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ℝ, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises (...) the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ℝ if and only if the axiom of countable choice holds for families of subsets of ℝ, and every metric space has a unique equation image-completion. (shrink)

This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational (...) value of sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some technical properties in relation to other axioms of the system. (shrink)

I criticise a paper by Peter Forrest in which he argues that a principle of unrestricted countable fusion has paradoxical consequences. I argue that the paradoxical consequences that he exhibits may be due to his Whiteheadean assumptions about the nature of spacetime rather than to the principle of unrestricted countable fusion.

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown (...) that no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning (...) can be reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.

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 give a positive answer to a problem posed by G. Hofer-Szabo and M. Redei regarding the existence of infinite common cause systems. An example of a countably infinite CCS is presented, as well as the proof that no CCSs of greater cardinality exist.

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.

We study a strengthening of the notion of a perfectly meager set. We say that a subset A of a perfect Polish space X is countably perfectly meager in X, if for every sequence of perfect subsets $\{P_n: n \in \mathbb N\}$ of X, there exists an $F_\sigma $ -set F in X such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each n. We give various characterizations and examples of countably perfectly meager sets. We prove (...) that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński. (shrink)

To what extent are countability distinctions subject to systematic semantic variation? Could there be a language with no countability distinctions—in particular, one where all nouns are count? I argue that the answer is no: even in a language where all NPs have the core morphosyntactic properties of English count NPs, such as combining with numerals directly and showing singular/plural contrasts, countability distinctions still emerge on close inspection. I divide these distinctions into those related to sums and those related to parts. (...) In the Sahaptian language Nez Perce, evidence can be found for both types of distinction, in spite of the absence of anything like a traditional mass–count division in noun morphosyntax. I propose an extension of the Nez Perce analysis to Yudja, analyzed by Lima as lacking any countability distinctions. More generally, I suggest that at least one countability distinction may be universal and that languages without any countability distinctions may be unlearnable. (shrink)

At a very fundamental level an individual (or a computer) can process only a finite amount of information in a finite time. We can therefore model the possibilities facing such an observer by a tree with only finitely many arcs leaving each node. There is a natural field of events associated with this tree, and we show that any finitely additive probability measure on this field will also be countably additive. Hence when considering the foundations of Bayesian statistics we may (...) as well assume countable additivity over a σ-field of events. (shrink)

It is sometimes alleged that arguments that probability functions should be countably additive show too much, and that they motivate uncountable additivity as well. I show this is false by giving two naturally motivated arguments for countable additivity that do not motivate uncountable additivity.

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)

We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } (...) \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11]. (shrink)

It is a common presupposition in ontology (metaphysics) that a so-called 'principle of individuation' amounts to a principle of counting. Against this presupposition I argue that the predicates 'x is the same individual as y' and 'x is one with y' are neither co-extensional nor co-intensional. Non-countable entities such as masses or stuffs (or the referents of nouns in classifier languages) also fulfill the requirements of individuality. I suggest that Leibniz' 'principle of the identity of indiscernibles' (PII) should be (...) taken as a principle of individuality, rather than as a principle of the individuality of countables. (shrink)

The main conclusion is this conditional: If the principle of reflection is a valid constraint on rational credences, then it is not rational to have a uniform credence distribution on a countable outcome space. The argument is a variation on some arguments that are already in the literature, but with crucial differences. The conditional can be used for either a modus ponens or a modus tollens; some reasons for thinking that the former is most reasonable are given.

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of (...) non-empty sets with at most n points are non-empty. (shrink)

Two countable filters on ω are incompatible if they have no common infinite pseudointersection. Letting α(P f ) denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on ω, we prove the consistency of t $.

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω , using ultrafilters on a countable index set, which we also assume to be ω . Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have (...) no maximal element, and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors. (shrink)

Let α be any countable admissible ordinal greater than ω. There is a transitive set A such that A is admissible, locally countable, On A = α, and A satisfies Σ 1 -separation. In fact, if B is any nonstandard model of $KP + \forall x \subseteq \omega$ (the hyperjump of x exists), the ordinal standard part of B is greater than ω, and every standard ordinal in B is countable in B, then HC B ∩ (standard (...) part of B) satisfies Σ 1 -separation. (shrink)

It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

In this paper we give a positive answer to a problem posed by Hofer-Szabó and Rédei (Int. J. Theor. Phys. 43:1819–1826, 2004) regarding the existence of infinite Reichenbachian common cause systems (RCCSs). An example of a countably infinite RCCS is presented. It is also determined that no RCCSs of greater cardinality exist.

It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

A family A ⊆ ℘(ω) is called countably splitting if for every countable $F \subseteq [\omega]^{\omega}$ , some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an $F_{\sigma}$ splitting family that does not contain a (...) closed splitting family. (shrink)

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a (...) submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the (...) invariant cuts of the model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated. (shrink)

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

By using some classical reasoning we show that any countably presented formal topology, namely, a formal topology with a countable axiom set, is spatial.

We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the $D_\phi-ranks$ are finite and definable.

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

We follow [8] in asking when a set of ordinals $X \subseteq \alpha$ is a countable union of sets in K, the core model. We show that, analogously to L, and X closed under the canonical Σ 1 Skolem function for K α can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

We study complexity of the index set of countably categorical theories and Ehrenfeucht theories in finite languages.

In this paper, we give a classification of ℵ0-categorical coloured linear orders, generalizing Rosenstein's characterization of ℵ0-categorical linear orderings. We show that they can all be built from coloured singletons by concatenation and ℚn-combinations . We give a method using coding trees to describe all structures in our list.

Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.

We can look at this model theoretically as follows. By the linearly ordered predicate calculus, we simply mean ordinary predicate calculus with equality and a special binary relation symbol <. It is required that in all interpretations, < be a linear ordering on the domain. Thus we have the usual completeness theorem provided we add the axioms that assert that < is a linear ordering.

Every Berman’s variety \ which is the subvariety of Ockham algebras defined by the equation \ and \) determines a finitary substitution invariant consequence relation \. A sequent system \ is introduced as an axiomatization of the consequence relation \. The system \ is characterized by a single finite frame \ under the frame semantics given for the formal language. By the duality between frames and algebras, \ can be viewed as a \-valued logic as it is characterized by a (...) distributive lattice of \ elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system \ is established for \. The Craig interpolation property of \ is shown proof-theoretically utilizing \. (shrink)

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.

In the paper all countable Boolean algebras with m distinguished. ideals having countably-categorical elementary theory are described and constructed. From the obtained characterization it follows that all countably-categorical elementary theories of Boolean algebras with distinguished ideals are finite-axiomatizable, decidable and, consequently, their countable models are strongly constructivizable.