In this article, we first offer a model of design knowledge types and their interrelationships in value sensitive design. Then we demonstrate that value sensitive design is a formative framework, which provides a shaping influence on practice, enables creative appropriation, and supports theory and method development.

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

This paper addresses the knowledge required for individuals to evaluate Information and Communications Technologies decisions that relate to the organization and management of cyberspace, and to hold accountable the parties responsible for those decisions, whether the responsible party is a government actor, market actor or private individual. The authors argue that the Open Systems Interconnection model, with certain modifications, should serve as a primary educational tool in helping individuals to gain the understanding of ICT necessary to protect public interests related (...) to cyberspace. (shrink)

Both unipolar and bipolar depression have been linked with impairments in executive functioning. In particular, mood symptom severity is associated with differences in common EF, a latent measure of general EF abilities. The relationship between mood disorders and EF is particularly salient in adolescence and young adulthood when the ongoing development of EF intersects with a higher risk of mood disorder onset. However, it remains unclear if common EF impairments have associations with specific symptom dimensions of mood pathology such as (...) blunted positive affect, mood instability, or physiological arousal, or if differences in common EF more broadly relate to what is shared across various symptom domains, such as general negative affect or distress. To address this question, bifactor models can be applied to simultaneously examine the shared and unique contributions of particular mood symptom dimensions. However, no studies to our knowledge have examined bifactor models of mood symptoms in relation to measures of common EF. This study examined associations between common EF and general vs. specific symptom dimensions using structural equation modeling in adolescents and young adults with varying severity of mood symptoms. A General Depression factor capturing shared variance across symptoms statistically predicted lower Common EF. Additionally, a factor specific to physiological arousal was associated with lower Common EF. Anhedonia-specific and Mania-specific factors were not significantly related to Common EF. Altogether, these results indicate that deficits in common EF are driven by, or reflect, general features of mood pathology that are shared across symptom dimensions but are also specifically associated with physiological arousal. (shrink)

In this article we evaluate the BankXX program from several perspectives. BankXX is a case-based legal argument program that retrieves cases and other legal knowledge pertinent to a legal argument through a combination of heuristic search and knowledge-based indexing. The program is described in detail in a companion article in Artificial Intelligence and Law 4: 1--71, 1996. Three perspectives are used to evaluate BankXX:(1) classical information retrieval measures of precision and recall applied against a hand-coded baseline; (2) knowledge-representation and case-based (...) reasoning, where the baseline is provided by the functionality of a well-known case-based argument program, HYPO (Ashley, 1990); and (3) search, in which the performance of BankXX run with various parameter settings, for instance, resource limits, is compared. In this article we report on an extensive series of experiments performed to evaluate the program. We also describe two additional experiments concerning(1) the program's search behavior; and (2) the use of a modified form of precision and recall based on case similarity. Finally we offer some general conclusions that might be drawn from these particular experiments. (shrink)

The BankXX system models the process of perusing and gathering information for argument as a heuristic best-first search for relevant cases, theories, and other domain-specific information. As BankXX searches its heterogeneous and highly interconnected network of domain knowledge, information is incrementally analyzed and amalgamated into a dozen desirable ingredients for argument (called argument pieces), such as citations to cases, applications of legal theories, and references to prototypical factual scenarios. At the conclusion of the search, BankXX outputs the set of argument (...) pieces filled with harvested material relevant to the input problem situation.This research explores the appropriateness of the search paradigm as a framework for harvesting and mining information needed to make legal arguments. In this article, we describe how legal research fits the heuristic search framework and detail how this model is used in BankXX. We describe the BankXX program with emphasis on its representation of legal knowledge and legal argument. We describe the heuristic search mechanism and evaluation functions that drive the program. We give an extended example of the processing of BankXX on the facts of an actual legal case in BankXX's application domain — the good faith question of Chapter 13 personal bankruptcy law. We discuss closely related research on legal knowledge representation and retrieval and the use of search for case retrieval or tasks related to argument creation. Finally we review what we believe are the contributions of this research to the understanding of the diverse disciplines it addresses. (shrink)

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective (...) to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in (...) addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model. (shrink)

Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ + is greater than κ + of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations.

Capitalism is the best. It's free enterprise. Barter. Gimbels, if I get really rank with the clerk, 'Well I don't like this', how I can resolve it? If it really gets ridiculous, I go, 'Frig it, man, I walk.' What can this guy do at Gimbels, even if he was the president of Gimbels? He can always reject me from that store, but I can always go to Macy's. He can't really hurt me. Communism is like one big phone company. (...) Government control, man. And if I get too rank with that phone company, where can I go? I'll end up like a schmuck with a dixie cup on a thread. (shrink)

Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe.

Assuming ${2^{{N_0}}}$ = N₁ and ${2^{{N_1}}}$ = N₂, we build a partial order that forces the existence of a well-order of H(ω₂) lightface definable over ⟨H(ω₂), Є⟩ and that preserves cardinal exponentiation and cofinalities.

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...) G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ⩽κ not only does not collapse κ+ but also preserves the strength of κ. This provides a general theory covering the known cases of tree iterations which preserve large cardinals [3], Friedman and Halilović [5], Friedman and Honzik [6], Friedman and Magidor [8], Friedman and Zdomskyy [10], Honzik [12]).

I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of (...) the paper is concerned with the Hyperuniverse Programme, whose aim is to discover an optimal mathematical principle for expressing the maximality of the set-theoretic universe in height and width. (shrink)

In this article, we report on eight grand challenges for value sensitive design, which were developed at a one-week workshop, Value Sensitive Design: Charting the Next Decade, Lorentz Center, Leiden, The Netherlands, November 14–18, 2016. A grand challenge is a substantial problem, opportunity, or question that motives sustained research and design activity. The eight grand challenges are: Accounting for Power, Evaluating Value Sensitive Design, Framing and Prioritizing Values, Professional and Industry Appropriation, Tech policy, Values and Human Emotions, Value Sensitive Design (...) and Intelligent Algorithms, and Value Tensions. Each grand challenge consists of a discussion of its importance and a set of tractable key questions. (shrink)

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^\kappa $$\end{document}, κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah ), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is κκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa ^\kappa $$\end{document}-bounding, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,>\,$$\end{document}cov_λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\_\lambda $$\end{document}), arXiv:0904.0817, Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper :439–456, 2016). (shrink)

In thinking and talking about rights, including property rights, it seems natural to put the argument in either moral or legal terms. From the former viewpoint, rights are part of a description of what actions are right or wrong. The fact that I have a right to do something is an argument, although not necessarily a sufficient argument, that someone who prevents me from doing it is acting wrongly.

The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...) to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]. (shrink)

The main result in this paper is a method for obtaining derivation trees from sentences of certain formal grammars. No parsing algorithm was previously known to exist for these grammars.Applied to Montague's PTQ the method produces all parses that could correspond to different meanings. The technique directly addresses scope and reference and provides a framework for examining these phenomena. The solution for PTQ is implemented in an efficient and useful computer program.

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $, there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

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

In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:Vλ+1≺Vλ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j : V_{\lambda+1} {\prec} V_{\lambda+1}}$$\end{document} with the failure of GCH (...) at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}. (shrink)

We show that if M is a countable transitive model of $\text {ZF}$ and if $a,b$ are reals not in M, then there is a G generic over M such that $b \in L[a,G]$. We then present several applications such as the following: if J is any countable transitive model of $\text {ZFC}$ and $M \not \subseteq J$ is another countable transitive model of $\text {ZFC}$ of the same ordinal height $\alpha $, then there is a forcing extension N of (...) J such that $M \cup N$ is not included in any transitive model of $\text {ZFC}$ of height $\alpha $. Also, assuming $0^{\#}$ exists, letting S be the set of reals generic over L, although S is disjoint from the Turing cone above $0^{\#}$, we have that for any non-constructible real a, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees. (shrink)

Since 1991 the Franciscan Francis of Marchia, master of theology at the University of Paris (fl. 1320), has begun receiving his due attention as an exciting and innovative thinker. This volume examines his doctrines in cosmology, physics, metaphysics, ethics, and politics.

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening (...) of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

In this panel, we explore the future of value sensitive design (VSD). The stakes are high. Many in public and private sectors and in civil society are gradually realizing that taking our values seriously implies that we have to ensure that values effectively inform the design of technology which, in turn, shapes people’s lives. Value sensitive design offers a highly developed set of theory, tools, and methods to systematically do so.

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni-Friedman :77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where a is a cardinal at most κ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ⁺⁺, the maximum possible) and [1] (for α = κ⁺, after collapsing κ⁺⁺) . In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of (...) Mitchell order α ) , [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails. (shrink)

The fact that “natural” theories, i.e. theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. However, one easily establishes the existence of theories with incomparable logical strengths using self-reference . As a result, PA+Con is not the least theory whose strength is greater than that of PA. But still we can ask: is there a sense in which (...) PA+Con is the least “natural” theory whose strength is greater than that of PA? In this paper we exhibit natural theories in strength strictly between PA and PA+Con by introducing a notion of slow consistency. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...) the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

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

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the (...) generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

In this paper we review the most common forms of reflection and introduce a new form which we call sharp-generated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH#. IMH# is an analogue of the IMH :591–600, 2006)) which is compatible with the existence of large cardinals.

The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.

In this paper we review the most common forms of reflection and introduce a new form which we call sharp‐generated reflection. We argue that sharp‐generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp‐maximality with the corresponding hypothesis. The statement is an analogue of the (Inner Model Hypothesis, introduced in ) which is compatible with the existence of large cardinals.

The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, instead of the (...) cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals. (shrink)

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs is far from complete.In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete (...) under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of ℒω1ω, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids. (shrink)

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with.