We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.

1. The Physical Church-Turing Thesis. Physicists often interpret the Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system . . . (...) No physically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. (shrink)

This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options (...) available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments. (shrink)

Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detailed investigation of iterated Sacks indestructibility.

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms (...) is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for Hγ0+ with respect to Γ0 and for Hγ1+ with respect to Γ1 with γ0 = γΓ0≠γΓ1 = γ1 is in principle possible, and we present several natural models o... (shrink)

We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.

In the human cognitive economy there are four grades of epistemic involvement. Knowledge partitions into distinct sorts, each in turn subject to gradations. This gives a fourwise partition on ignorance, which exhibits somewhat different coinstantiation possibilities. The elements of these partitions interact with one another in complex and sometimes cognitively fruitful ways. The first grade of knowledge I call “anselmian” to echo the famous declaration credo ut intelligam, that is, “I believe in order that I may come to know”. As (...) construed here, one knows in this anselmian way that E = mc2 just in case one knows that sentence expresses a true statement, but without having to understand the proposition it expresses. Most epistemologists ignore the significance of this grade of epistemic involvement. In a second grade of epistemic involvement, knowing that E = mc2 is knowing what that sentence means and understanding the proposition it express. This is knowledge in the propositional or semantic sense, and is the dominant target of epistemological investigation. Tacit and implicit knowledge occupies another tier. A typical example would be something that someone has “known all along” but, until now, hasn’t had occasion to put her mind to it or formulate in words. TI-knowledge remains a minority interest in today’s epistemology. Operating at a fourth grade of epistemic involvement is what I call “impact”-knowledge, which is the knowledge of a matter at its deepest and most widespread. An example, to be discussed below, is the knowledge that was generated by the Wiles proof of Fermat’s last theorem. Its true importance lies not only, or even mainly, in its verification of a commonly accepted fact about numbers, but rather in its enrichment of the mathematics of elliptical curves and the promise it holds for greater advancement into the mathematical unknown. Knowledge of this fourth grade has yet to find a seat in the parliaments of epistemology. Knowledge of the anselmian sort is independent of the other three. Tacit and implicit knowledge is incompatible with anselmian and semantic knowledge but coinstantiable with impact-knowledge. Semantic knowledge is incompatible with tacit and implicit knowledge but coinstantiable with the others. Impact-knowledge is pairwise coinstantiable with the others. Below I will bring the ignorance partitions into such alignment as they have with these ones. In doing so, I’ll propose a naturalized causal response epistemology designed to give these interactive distinctions the theoretical air they need to breathe. (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.

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, (...) specifically, reverse Easton iterations of increasingly directed closed partial orders. (shrink)

We establish the hierarchy among twelve equivalence relations on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class ModL of all L-structures of size κ. It turns out that, concerning the number of different similarities and (...) the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages. (shrink)

In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate (...) in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal , for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

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)

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)

In 2002, Yorioka introduced the σ-ideal ${{\mathcal {I}}_f}$ for strictly increasing functions f from ω into ω to analyze the cofinality of the strong measure zero ideal. For each f, we study the cardinal coefficients (the additivity, covering number, uniformity and cofinality) of ${{\mathcal {I}}_f}$.

We describe a method of forcing against weak theories of arithmetic and its applications in propositional proof complexity.

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

El estudio de los "cardinales grandes" es uno de los principales temas de investigación de la teoría de conjuntos y de la teoría de modelos que ha contribuido con el desarrollo de dichas disciplinas. Existe una gran variedad de tales cardinales, por ejemplo cardinales inaccesibles, débilmente compactos, Ramsey, medibles, supercompactos, etc. Tres valiosos teoremas clásicos sobre cardinales medibles son los siguientes: (i) compacidad débil, (ii) Si κ es un cardinal medible, entonces κ es un cardinal inaccesible y existen κ cardinales (...) inaccesibles menores que κ , y (iii) Si existe un cardinal medible, entonces el axioma de constructibilidad (V=L) es falso. El objetivo de este artículo es presentar una demostración de cada uno de estos tres teoremas en el contexto de la teoría de modelos usando ideas del texto de Chang y Keisler (Model Theory). Tales demostraciones tienen en común el uso del método de construcción de modelos llamado ultraproductos, de lógicas infinitarias o fragmentos de la lógica de segundo orden, y del axioma de elección. Cardinales grandes y/o ultraproductos son importantes en teoría de conjuntos, teoría de modelos, análisis matemático, teoría de la medida, probabilidades, topología, análisis funcional, física, teoría de números, finanzas, etc. (shrink)

We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

We study the relationship between various properties of the real numbers and weak choice principles.

We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on Pω (λ) with the partition property for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.

Extending techniques of Dowd and those of Poizat, we study computational complexity of in the case when is a generic oracle, where is a positive integer, and denotes the collection of all -query tautologies with respect to an oracle . We introduce the notion of ceiling-generic oracles, as a generalization of Dowd's notion of -generic oracles to arbitrary finitely testable arithmetical predicates. We study how existence of ceiling-generic oracles affects behavior of a generic oracle, by which we show that is (...) not a subset of is comeager in the Cantor space. Moreover, using ceiling-generic oracles, we present an alternative proof of the fact (Dowd) that the class of all -generic oracles has Lebesgue measure zero. (shrink)

We prove various results on the notion of ordinal ultrafilters introduced by J. Baumgartner. In particular, we show that this notion of ultrafilter complexity is independent of the more familiar Rudin-Keisler ordering.

El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...) ellas. (shrink)

At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...) from calculation gave rise to concerns as to whether such methods were meaningful, or appropriate to mathematics. The discovery of paradoxes stemming from overly naive use of set-theoretic language and methods led to even more pressing concerns as to whether the modern methods were even consistent. This led to heated debates in the early twentieth century and what is sometimes called the “crisis of foundations.” In lectures presented in 1922, David Hilbert launched his Beweistheorie, or Proof Theory, which aimed to justify the use of modern methods and settle the problem of foundations once and for all. This, Hilbert argued, could be achieved as follows. (shrink)

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...) make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

If the total state of the universe is encodable by a real number, Hardin and Taylor have proved that there is a solution to one version of the problem of induction, or at least a solution to a closely related epistemological problem. Is this philosophical application of the Hardin-Taylor result modest enough? The paper advances grounds for doubt. [A longer and more detailed sequel to this paper, 'Proving Induction', was published in the Australasian Journal of Logic in 2011.].

We define a class of productive σ-ideals of subsets of the Cantor space 2 ω and observe that both σ-ideals of meagre sets and of null sets are in this class. From every productive σ-ideal I we produce a σ-ideal I κ , of subsets of the generalized Cantor space 2 κ . In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. Then we investigate additivity, (...) covering number, uniformity and cofinality of I κ . For example, we show that non(I = non(I ω 1 ) = non(I ω 2 ). Our results generalizes those from [5]. (shrink)

In this paper, we describe an improvement of a calculation procedure of logic programs. The procedure proposed before is the combination of a replacement procedure of logical formulae and a transformation procedure of equations to disjunctive normal form, and it can calculate logical consequences of the completion of any given first-order logic program, which is equivalent to the FLP in two-valued logic, soundly and completely in three-valued logic. The new procedure is also the combination of them, but the transformation procedure (...) is improved to be able to calculate two-valued logical consequences of the FLP more than the old one. We prove that it can calculate logical consequences of a completed program, which is not equivalent to the completion of the FLP, soundly and completely in three-valued logic. (shrink)

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions (...) and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. (shrink)

The collection of branches (maximal linearly ordered sets of nodes) of the tree $^{ (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal--for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint (...) from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and o is the minimum cardinality of a maximal off-branch family. Results concerning o include: (in ZFC) a ≤ p, and (consistent with ZFC) o is not equal to any of the standard small cardinal invariants b,a,d, or c = 2 ω . Most of these consistency results use standard forcing notions--for example, $\mathfrak{b = a in the Cohen model. Many interesting open questions remain, though--for example, whether d ≤ o. (shrink)

We show: There are pairs of universes V1V2 and there is a notion of forcing PV1 such that the change mentioned in the title occurs when going from V1[G] to V2[G] for a P-generic filter G over V2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.

Recently, John Bell has proposed that a specific conditional logic, C, be considered as a serious candidate for formally representing and faithfully capturing various (possibly all) formalized notions of nonmonotonic inference. The purpose of the present paper is to develop evaluative criteria for critically assessing such claims. Inference patterns are described in terms of the presence or absence of residual classical monotonicity and intrinsic nonmonotonicity. The concept of a faithful representation is then developed for a formalism purported to encode a (...) pattern of nonmonotonic inference already captured by another. In the main body of the paper these evaluative criteria are applied to assess (negatively) whether C or any conditional logic provides a faithful representation for nonmonotonic patterns of inference captured by inference operators and relations modeling the dynamics of belief change. (shrink)

It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again.

Sets are often taken to be collections, or at least akin to them. In contrast, this paper argues that. although we cannot be sure what sets are, what we can be entirely sure of is that they are not collections of any kind. The central argument will be that being an element of a set and being a member in a collection are governed by quite different axioms. For this purpose, a brief logical investigation into how set theory and collection (...) theory are related is offered. The latter part of the paper concerns attempts to modify the `sets are collections' credo by use of idealization and abstraction, as well as the Fregean notion of sets as the extensions of concepts. These are all shown to be either unmotivated or unable to provide the desired support. We finish on a more positive note with some ideas on what can be said of sets. The main thesis here is that sets are points in a set structure, a set structure is a model of a set theory, and set theories constitute a family of formal and informal theories, loosely defined by their axioms. (shrink)

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta (...) may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and co-existential images. As a byproduct, we conclude that co-existential images of pseudo-arcs are pseudo-arcs. This is of interest because the corresponding statement for confluent maps is still open, and co-existential maps are often—but not always—confluent. (shrink)

We study some properties of the quotient forcing notions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{tr(I)} = \wp(2^{< \omega})/tr(i)}$$\end{document} and PI = B(2ω)/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2ω. We show that the remainder forcing RI = Qtr(I)/PI is σ-closed in these cases. We also study the cardinal invariant of the continuum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, the distributivity (...) number of the quotient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Dense(\mathbb{Q})/nwd}$$\end{document}, in order to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\wp(\mathbb{Q})/nwd}$$\end{document} collapses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59–80, 2004). (shrink)

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal was (...) established by Bartoszyński and Kada. (shrink)

We will prove that there exists a model of ZFC+"c = ω 2 " in which every $M \subseteq \mathbb{R}$ of cardinality less than continuum c is meager, and such that for every $X \subseteq \mathbb{R}$ of cardinality c there exists a continuous function f: R → R with f[X] = [0, 1]. In particular in this model there is no magic set, i.e., a set $M \subseteq \mathbb{R}$ such that the equation f[M] = g[M] implies f = g for (...) every continuous nowhere constant functions f, g: R → R. (shrink)

Two of Hilary Putnam's model-theoretic arguments against metaphysical realism are examined in detail. One of them is developed as an extension of a model-theoretic argument against mathematical realism based on considerations concerning the so-called Skolem-Paradox in set theory. This argument against mathematical realism is also treated explicitly. The article concentrates on the fine structure of the arguments because most commentators have concentrated on the major premisses of Putnam's argument and especially on his treatment of metaphysical realism. It is shown that (...) the validity of Putnam's arguments is doubtful and that realists are by no means forced to accept the theses Putnam ascribes to them. It is concluded that Putnam fails to give convincing arguments for rejecting mathematical or metaphysical realism. Furthermore, Putnam's internal realism is discussed critically. (shrink)

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper (...) classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems. (shrink)

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 study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number ${\mathfrak{b}}$ , the intersection number of the class of all meager ideals is equal to ${\mathfrak{h}}$ and the intersection number of the class of all F σ ideals is between ${\mathfrak{h}}$ and ${\mathfrak{b}}$ , consistently different from both.