My chief aim has been to convey the thought that the application of model theoretic techniques to natural languages needn't force a distortion of intentional phenomena. I hope that at least I have succeeded in accomplishing this.

I will give a brief overview of Saharon Shelah’s work in mathematical logic. I will focus on three transformative contributions Shelah has made: stability theory, proper forcing and PCF theory. The first is in model theory and the other two are in set theory.

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics (...) is small. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic (...) terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property (...) is an $\mathsf{E}$-formula. Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic. (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)

The position of the spindle determines the position of the cleavage plane, and is thus crucial for cell division. Although spindle positioning has been extensively studied, the underlying forces ultimately responsible for moving the spindle remain poorly understood. A recent pioneering study by Garzon‐Coral et al. uses magnetic tweezers to perform the first direct measurements of the forces involved in positioning the mitotic spindle. Combining this with molecular perturbations and geometrical effects, they use their data to argue that the forces (...) that keep the spindle in its proper position for cell division arise from astral microtubules growing and pushing against the cell's cortex. Here, we review these ground‐breaking experiments, the various biomechanical models for spindle positioning that they seek to differentiate, and discuss new questions raised by these measurements. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies (...) all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

We present a framework for model theoretic forcing in a non first order context, and present some applications of this framework to Banach space theory.

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order (...) logic , logic of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail. (shrink)

BackgroundCurrent policies regulating the use of animals for scientific purposes are based on balancing between potential gain of knowledge and suffering of animals used in experimentation. The balancing process is complicated, on the one hand by plurality of views on our duties towards animals, and on the other hand by more recent discussions on uncertainty in the probability of reaching the final aim of the research and problems of translational failure.MethodsThe study combines ethical analysis based on a literature review with (...) neuropsychiatry-related preclinical research as a case study.ResultsBased on the analysis and the case study we show that neuropsychiatry-related preclinical research is an especially interesting case from an ethical perspective. The 3R principles are used to minimize the negative consequences for the animals used in research. However, neuropsychiatric research is characterized by specific challenges in assessing the probability of success of reaching the final aim, due to our limited mechanistic knowledge of human neuropsychiatric illness. Consequently, the translational value of the currently used animal models may be difficult to prove, which undermines the validity of these models and complicated the ethical assessment.ConclusionsWe conclude that a combined approach that deals with both science and the ethical dimensions is necessary to address the problems of validity and translation in neuropsychiatry-related preclinical research. We suggest this approach to comprise first, improved experimental methods, e.g. by using systematic reviews, second, a more patients-based approach that leads to models that reflect interindividual variation better, and third, more interdisciplinary cooperation. (shrink)

This paper presents a modified random network model to illustrate how groups can form in the absence of evolutionary forces, assuming groups are collections of entities at any level of organization. This model is inspired by the Zero Force Evolutionary Law, which states that there is always a tendency for diversity and complexity to increase in any evolutionary system containing variation and heredity. That is, in the absence of evolutionary forces, the expectation is a continual increase in diversity (...) and complexity at any level of biological hierarchy. I show that, when modeled, this expectation of increasing variation results not only in the formation of groups, but also in a higher probability of group formation than is found in a model that is purely random. (shrink)

In this article we show that Morse-Kelley class theory provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document} of MK, the main result being that the Definability Lemma can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver’s (...) Theorem does not hold for class forcings. (shrink)

This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians.

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class (...) class='Hi'>forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T.

The impact of our desires and preferences upon our ordinary, everyday beliefs is well-documented [Gilovich, T. . How we know what isn’t so: The fallibility of human reason in everyday life. New York: The Free Press.]. The influence of such motivational factors on delusions, which are instances of pathological misbelief, has tended however to be neglected by certain prevailing models of delusion formation and maintenance. This paper explores a distinction between two general classes of theoretical explanation for delusions; the motivational (...) and the deficit. Motivational approaches view delusions as extreme instances of self-deception; as defensive attempts to relieve pain and distress. Deficit approaches, in contrast, view delusions as the consequence of defects in the normal functioning of belief mechanisms, underpinned by neuroanatomical or neurophysiological abnormalities. It is argued that although there are good reasons to be sceptical of motivational theories , recent experiments confirm that motives are important causal forces where delusions are concerned. It is therefore concluded that the most comprehensive account of delusions will involve a theoretical unification of both motivational and deficit approaches. (shrink)

We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the (...) set theoretical level of the theories. It is argued that this fundamental level can help solving some difficulties of the physical dualities. (shrink)

In this article we introduce and study hyperclass-forcing in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between β-models ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} of MK∗∗ and transitive models M+ of SetMK∗∗ which will allow us to go from ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} to M+ and vice versa. So instead of forcing with a hyperclass in MK∗∗ we can force over the corresponding SetMK∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK∗∗ model M+ can be forced to look like Lκ∗[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L_{\kappa ^*}[X]$$ \end{document}, where κ∗ is the height of M+, κ strongly inaccessible in M+ and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK∗∗, which will in turn be an extension of the original ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document}. Our main result combines hyperclass forcing with coding methods of Beller et al. and Friedman to show that every β-model of MK∗∗ can be extended to a minimal such model of MK∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic. (shrink)

This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to (...) introduce models as new entities. In conclusion, I argue for a view that takes both sets and models to be fundamental entities in set theory. (shrink)

The process of abstraction and concretisation is a label used for an explicative theory of scientific model-construction. In scientific theorising this process enters at various levels. We could identify two principal levels of abstraction that are useful to our understanding of theory-application. The first level is that of selecting a small number of variables and parameters abstracted from the universe of discourse and used to characterise the general laws of a theory. In classical mechanics, for example, (...) we select position and momentum and establish a relation amongst the two variables, which we call Newton’s 2nd law. The specification of the unspecified elements of scientific laws, e.g. the force function in Newton’s 2nd law, is what would establish the link between the assertions of the theory and physical systems. In order to unravel how and with what conceptual resources scientific models are constructed, how they function and how they relate to theory, we need a view of theory-application that can accommodate our constructions of representation models. For this we need to expand our understanding of the process of abstraction to also explicate the process of specifying force functions etc. This is the second principal level at which abstraction enters in our theorising and in which I focus. In this paper, I attempt to elaborate a general analysis of the process of abstraction and concretisation involved in scientific- model construction, and argue why it provides an explication of the construction of models of the nuclear structure. (shrink)

I argue against the conception of scientific models advocated by the proponents of the Semantic View of scientific theories. Part of the paper is devoted to clarifying the important features of the scientific modeling view that the Semantic conception entails. The liquid drop model of nuclear structure is analyzed in conjunction with the particular auxiliary hypothesis that is the guiding force behind its construction and it is argued that it does not meet the necessary features to render it a (...) model of the theory, as the Semantic View demands. Given that this model is indicative of how quantum mechanics is applied in the domain of nuclear physics, I claim that the Semantic View does not adequately account for scientific models. (shrink)

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε .Analogously to the usual forcing and (...) the usual generic extension for FA-models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA-set theory can be developed in a similar way to that for FA-set theory.Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (shrink)

Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest (...) ZF model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it. (shrink)

The main purpose of this dissertation is to apply and develop new forcing techniques to obtain models where several cardinal characteristics are pairwise different as well as force many (even more, continuum many) different values of cardinal characteristics that are parametrized by reals. In particular, we look at cardinal characteristics associated with strong measure zero, Yorioka ideals, and localization and anti-localization cardinals.In this thesis we introduce the property “F-linked” of subsets of posets for a given free filter F on (...) the natural numbers, and define the properties “ $\mu $ -F-linked” and “ $\theta $ -F-Knaster” for posets in a natural way. We show that $\theta $ -F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. These kinds of posets led to the development of a general technique to construct $\theta $ - $\textrm {Fr}$ -Knaster posets (where $\textrm {Fr}$ is the Frechet ideal) via matrix iterations of ${<}\theta $ -ultrafilter-linked posets (restricted to some level of the matrix). The latter technique allows proving consistency results about Cichoń’s diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal characteristics associated with it are pairwise different. Another important application is to show that three strongly compact cardinals are enough to force that Cichoń’s diagram can be separated into 10 different values. Later on, it was shown by Goldstern, Kellner, Mejía, and Shelah that no large cardinals are needed for Cichoń’s maximum (J. Eur. Math. Soc. 24 (2022), no. 11, p. 3951–3967).On the other hand, we deal with certain types of tree forcings including Sacks forcing, and show that these increase the covering of the strong measure zero ideal $\mathcal {SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal characteristics of the continuum. Even more, Sacks forcing can be used to force that $\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$, which is the first consistency result where more than two cardinal characteristics associated with $\mathcal {SN}$ are pairwise different. To obtain another result in this direction, we provide bounds for $\operatorname {\mathrm{cof}}(\mathcal {SN})$, which generalizes Yorioka’s characterization of $\mathcal {SN}$ (J. Symbolic Logic 67.4 (2002), p. 1373–1384). As a consequence, we get the consistency of $\operatorname {\mathrm{add}}(\mathcal {SN})=\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$ with ZFC (via a matrix iteration forcing construction).We conclude this thesis by combining creature forcing approaches by Kellner and Shelah (Arch. Math. Logic 51.1–2 (2012), p. 49–70) and by Fischer, Goldstern, Kellner, and Shelah (Arch. Math. Logic 56.7–8 (2017), p. 1045–1103) to show that, under CH, there is a proper $\omega ^\omega $ -bounding poset with $\aleph _2$ -cc that forces continuum many pairwise different cardinal characteristics, parametrized by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localization and anti-localization cardinals, respectively. This answers several open questions from Klausner and Mejía (Arch. Math. Logic 61 (2022), pp. 653–683).Abstract prepared by Miguel Antonio Cardona-MontoyaE-mail: [email protected]: https://repositum.tuwien.at/handle/20.500.12708/19629. (shrink)

Victoria’s Model Litigant Guidelines aim to regulate the conduct of government parties in civil disputes in a manner that goes beyond the ethical duties of ordinary litigants. Despite the sheer number of disputes involving the Victorian government to which the regime applies, little academic attention has been given to Victoria’s MLGs. The article explores the nature and extent of the regulatory force exerted by the MLGs by applying regulatory theory to the MLG regime. Particular attention is given to (...) applying Ayers and Braithwaite’s theory of Responsive Regulation that includes situating the guidelines within their regulatory pyramid. The analysis finds that in the absence of a distinct regulator, the MLG regime is predominantly self-regulatory. The responsiveness of the MLGs appears to be restricted owing to an absence of public involvement in the MLG regime. The article also highlights that likely challenges to the MLG regime as a mode of regulation include the nature of the regulatory context, the decentralisation of government services, the lack of publicly available information on the government’s adherence to the MLGs and the content of the MLGs themselves. (shrink)

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard models of Ł∀.

In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. (...) The paper does not contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research. (shrink)

This history of physics focuses on the question, "How do bodies act on one another across space?" The variety of answers illustrates the function of fundamental analogies or models in physics as well as the role of so-called unobservable entities. Forces and Fields presents an in-depth look at the science of ancient Greece, and it examines the influence of antique philosophy on seventeenth-century thought. Additional topics embrace many elements of modern physics--the empirical basis of quantum mechanics, wave-particle duality and the (...) uncertainty principle, and the action-at-a-distance theory of Wheeler and Feynman. 1961 ed. (shrink)

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard models of Ł∀.

We give a sufficient condition for a countable model M of PA to be expandable to an ω-model of AST with absolute Ω-orderings. The condition is in terms of saturation schemes or, equivalently, in terms of the ability of the model to code sequences which have some kind of definition in (M, ω). We also show that a weaker scheme of saturation leads to the existence of wellorderings of the model with nice properties. Finally, we answer (...) affirmatively the question of whether the intersection of all β-expansions of a β-expandable model M is the set RA(M, ω)--the ramified analytical hierarchy over (M, ω). The results are based on forcing constructions. (shrink)

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 forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse (...) that seems to necessitate the addition of subsets 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 analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions. (shrink)

The Hardy–Weinberg equilibrium has been argued by Sober, Stephens and others to represent the zero-force state for evolutionary biology understood as a theory of forces. I investigate what it means for a model to involve forces, developing an explicit account by defining what the zero-force state is in a general theoretical context. I use this account to show that Hardy–Weinberg equilibrium is not the zero-force state in biology even in the contexts in which it applies, and argue based (...) on this that drift should not be understood as an evolutionary force. (shrink)

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines (...) of development. (shrink)

In this article my primary intention is to engage in a discussion on the inherent constraints of models, taken as models of theories, that reaches beyond the epistemological level. Naturally the paper takes into account the ongoing debate between proponents of the syntactic and the semantic view of theories and that between proponents of the various versions of scientific realism, reaching down to the most fundamental, subjective level of discourse. In this approach, while allowing for a limited discussion of physical (...) and positive science models, I am primarily focused on the structure and ontology of mathematical models, in particular Cohen’s forcing models and to a lesser extent Gödel’s constructible universe, to the extent that these were designed to answer questions bearing on the scope, the capacity and ultimately the ontology of models themselves, therefore influencing in one or the other way the status of models in general. This status, it is argued, is largely defined by the way models subsume a set-theoretical structure whose constraints, reducible to an extra-linguistic level of discourse, may implicitly condition the epistemic status of models as representations of axiomatic theories. In the last section I deal extensively with the inner constraints of theories as subjectively originated in a less technical philosophically oriented discussion with certain prompts from phenomenology. (shrink)

Forcing with [κ] κ over a model of set theory with a strong partition cardinal, M. Spector produced a generic ultrafilter G on κ such that κ κ /G is not well-founded. Theorem. Let G be Spector-generic over a model M of $ZF + DC + \kappa \rightarrow (\kappa)^\kappa_\alpha, \kappa > \omega$ , for all $\alpha . 1) Every cardinal (well-ordered or not) of M is a cardinal of M[ G]. 2) If A ∈ M[ G] (...) is a well-ordered subset of M, then A ∈ M. Let Φ = κ κ /G. 3) There is an ultrafilter U on Φ such that every member of U has a subset of type Φ, and the intersection of any well-ordered subset of U is in U. 4) Φ satisfies Φ → (Φ) α β for all $\alpha and all ordinals β. 5) There is a linear order Φ' with property 3) above which is not "weakly compact", i.e., $\Phi' \nrightarrow (\Phi')^2$. (shrink)

The operation of fundamental forces in quantum field theory is explicated here as the exchange of particles, consistently with the standard methodology of particle physics. The particles involved are seen to bear little relation to any classical particle but, rather, comprise unified collections of compresent, conserved quantities indicated by propagators. The exchange particles, which supervene upon quantum fields, are neither more fundamental than fields nor replace them as has often previously been assumed in models of exchange forces. It is (...) argued that this explication of quantum field-theoretic exchange forces definitively improves upon its predecessors. (shrink)

Standard lore holds that magnetic forces are incapable of doing mechanical work. More precisely, the claim is that whenever it appears that a magnetic force is doing work, the work is actually being done by another force, with the magnetic force serving only as an indirect mediator. However, the most familiar instances of magnetic forces acting in everyday life, such as when bar magnets lift other bar magnets, appear to present manifest evidence of magnetic forces doing work. These sorts of (...) counterexamples are often dismissed as arising from quantum effects that lie outside the classical regime. In this paper, we show that quantum theory is not needed to account for these phenomena, and that classical electromagnetism admits a model of elementary magnetic dipoles on which magnetic forces can indeed do work. In order to develop this model, we revisit the foundational principles of the classical theory of electromagnetism, showcase the importance of constraints from relativity, examine the structure of the multipole expansion, and study the connection between the Lorentz force law and conservation of energy and momentum. (shrink)

We present a simple and completely model-theoretical proof of a strengthening of a theorem of Ajtai: The independence of the pigeonhole principle from IΔ0. With regard to strength, the theorem proved here corresponds to the complexity/proof-theoretical results of [10] and [14], but a different combinatorics is used. Techniques inspired by Razborov [11] replace those derived from Håstad [8]. This leads to a much shorter and very direct construction.