We discuss a problem asked by E. van Douwen and A. Miller [5] in various forcing models.

We describe a fairly general procedure for preserving I3 embeddings j: Vλ → Vλ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I3 embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency (...) was established assuming an I1 embedding. (shrink)

A complete Boolean algebra ${\mathbb{B}}$ satisfies property ${(\hbar)}$ iff each sequence x in ${\mathbb{B}}$ has a subsequence y such that the equality lim sup z n = lim sup y n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of (...) property ${(\hbar)}$ with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}$ is not a theorem of ZFC and that there is no cardinal ${\mathfrak{k}}$ , definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}$ is a theorem of ZFC. Also, we show that the set ${\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}$ is equal to ${[0, \mathfrak{h})}$ or ${[0, {\mathfrak h}]}$ and that both values are consistent, which, with the known equality ${{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}$ completes the picture. (shrink)

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a (...) higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper. (shrink)

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_2, \mu)}$$\end{document} -ITP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_3, \mu')}$$\end{document} -ITP hold, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu\geq \aleph_2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu'\geq \aleph_3}$$\end{document}. (shrink)

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF | p ∈ F}. We define a countably based MF space to be a space of the form MF for some (...) countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text]. (shrink)

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.

We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...) account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. (shrink)

Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of such that , the ideal generated by the non-stationary ideal over together with , is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis . We also show that in our model we can make -saturated, where S is the (...) set of all such that , the order type of x, is a regular cardinal and x is stationary in sup. Furthermore we construct a model where is κ+-saturated but GCH fails. We show that if SS is stationary in , then S can be split into λ many disjoint stationary subsets. (shrink)

In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size $2^{2^{\aleph }0}$ , the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size $2^{2^{\aleph }0}$ , and in fact these big chains occur in every infinite interval. We also (...) study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice. (shrink)

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA0, those in second order set theory extending NBG are. In this article, we establish the equivalence between Δ01-LFP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-LFP}}$$\end{document} and Δ01-FP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-FP}}$$\end{document}, which assert the existence of a least and of a fixed point, respectively, for positive elementary operators. Our proof also shows (...) the equivalence between ID1 and ID^1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\it ID}_1}$$\end{document}, both of which are defined in the standard way but with the starting theory PA replaced by ZFC. (shrink)

En este artículo se examinan algunas implicaciones del naturalismo matemático de P. Maddy como una concepción filosófica que permite superar las dificultades del ficcionalismo y el realismo fisicalista en matemáticas. Aparte de esto, la mayor virtud de tal concepción parece ser que resuelve el problema que plantea para la aplicabilidad de la matemática el no asumir la tesis de indispensabilidad de Quine sin comprometerse con su holismo confirmacional. A continuación, sobre la base de dificultades intrínsecas al programa de Maddy, exploramos (...) un camino naturalista mejor motivado, el enfoque cognitivista corporeizado, y sugerimos que él permite explicar de manera adecuada la postulación de ciertos cardinales grandes, en particular, la aceptación de conjuntos no-construibles. (shrink)

This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).

We define , a square principle in the context of , and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have hold when κ is supercompact, whereas □κ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on is also considered, this one based on a family of club subsets of . Finally, a new square principle for (...) cardinals, denoted , is introduced. This principle is proved consistent with κ being supercompact. It is shown to yield a non-reflection result similar to that given by □κ. (shrink)

. Remarkably, despite the tremendous success of axiomatic set-theory in mathematics, logic and meta-mathematics, e.g., model-theory, two philosophical worries about axiomatic set-theory as the adequate catch of the set-concept keep haunting it. Having dealt with one worry in a previous paper in this journal, we now fulfil a promise made there, namely to deal with the second worry. The second worry is the Skolem Paradox and its ensuing Skolemite skepticism. We present a comparatively novel and simple analysis of the argument (...) of the Skolemite skeptic, which will reveal a general assumption concerning the meaning of the set-concept (we call it Connexion M). We argue that the Skolemite skeptics argument is a petitio principii and that consequently we find ourselves in a dialectical situation of stalemate.Few (if any) working set-theoreticians feel a tension – let alone see a paradox – between, on the one hand, what the Löwenheim–Skolem theorems and related results seem to be telling us about the set-concept, and, on the other hand, their uncompromising and successful use of the set-concept and their continuing enthusiasm about it, in other words: their lack of skepticism about the set-concept. Further, most (if not all) working settheoreticians have a relaxed attitude towards the ubiquitous undecidability phenomenon in set-theory, rather than a worrying one. We argue these are genuine philosophical problems about the practice of set-theory. We propound solutions, which crucially involve a renunciation of Connexion M. This breaks the dialectical situation of stalemate against the Skolemite skeptic. (shrink)

We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques employed will (...) be the reflection principles and Gandy forcing. (shrink)

This article attempts a subjectively based approach, in fact one phenomenologically motivated, toward some key concepts of forcing theory, primarily the concepts of a generic set and its global properties and the absoluteness of certain fundamental relations in the extension to a forcing model M[G]. By virtue of this motivation and referring both to the original and current formulation of forcing I revisit certain set-theoretical notions serving as underpinnings of the theory and try to establish their deeper subjectively founded content (...) and also their influence in reaching relative consistency results by the forcing method. In this perspective, the present approach may be seen as offering an alternative view of the consistency results of K. Gödel and P. Cohen in mathematical foundations reaching a subjective level that may be taken as ultimately conditioning the non-decidability of key infinity statements on the level of formal theory. (shrink)

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and (...) also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process. (shrink)

We pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to their bounding and dominating numbers, the traditional scheme for gaps. Many open questions remain.

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable (...) Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory. (shrink)

By an ω1 we mean a tree of power ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech-Kunen tree if it has κ branches for some κ strictly between ω1 and $2^{\omega _1 }$ . In Sect. 1, we construct a model ofCH plus $2^{\omega _1 } > \omega _2$ , in which there exists a Kurepa tree with not (...) Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ω1 over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees. (shrink)

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In $\S1$ we prove that Loeb spaces are compact under various assumptions, and in $\S2$ we prove that Loeb spaces are not compact under various other assumptions. The results in $\S1$ and $\S2$ give a quite complete answer to a question of D. Ross in [9], [11] and [12].

K.J. Devlin has extended Jensen's construction of a model ofZFC andCH without Souslin trees to a model without Kurepa trees either. We modify the construction again to obtain a model with these properties, but in addition, without Kurepa trees inccc-generic extensions. We use a partially defined ◊-sequence, given by a fine structure lemma. We also show that the usual collapse ofκ Mahlo toω 2 will give a model without Kurepa trees not only in the model itself, but also inccc-extensions.

We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.

Elementary submodels of some initial segment of the set-theoretic universe are useful in order to prove certain theorems in general topology as well as in algebra. As an illustration we give proofs of two theorems due to Arkhangelskii concerning cardinal invariants of compact spaces.

We prove that the automorphism group ofP(ω)/fin remains simple if ℵ2 Cohen reals are added to a model of ZFC+CH.

Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of (...) solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

We describe a fairly general procedure for preserving I3 embeddings j: V λ → V λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I3 embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in (...) which consistency was established assuming an I1 embedding. (shrink)

In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.

For the classA of uncountable Archimedian real closed fields we show that the statement “TheL <ω-theory ofA is complete” is independent of ZFC. In particular we have the following results:Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory of the real numbers ℝ (...) (by a quantifier-elimination procedure).Formally: is incomplete. is complete and decidable. (shrink)

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) (...) questions, and improves results, of Faber, Laver and McKenzie. (shrink)

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)