Spinoza’s bold claim that there exists only a single infinite substance entails that finite things pose a deep challenge: How can Spinoza account for their finitude and their plurality? Taking finite bodies as a test case for finite modes in general I articulate the necessary conditions for the existence of finite things. The key to my argument is the recognition that Spinoza’s account of finite bodies reflects both Cartesian and Hobbesian influences. This recognition leads to the surprising realization there must (...) be more to finite bodies than their finitude, a claim that goes well beyond the basic substance-monism claim, namely, that anything that is, is in God. This leads to the conclusion, which may seem paradoxical, that finite bodies have both an infinite as well as a finite aspect to them. Finite bodies, I argue, both actively partially determine all the other finite bodies, thereby partially causing their existence insofar as they are finite, as well as are determined by the totality of other bodies. I articulate precisely what this infinite aspect is and how it is distinct from the general substance-monism dictum. (shrink)

Science is continually faced with describing that which is beyond. This book, through contributions from nine prominent scholars, tackles that challenge.

For every$n\in \omega \setminus \{0,1\}$we introduce the following weak choice principle:$\operatorname {nC}_{<\aleph _0}^-:$For every infinite family$\mathcal {F}$of finite sets of size at least n there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$with a selection function$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$such that$f(F)\in [F]^n$for all$F\in \mathcal {G}$.Moreover, we consider the following choice principle:$\operatorname {KWF}^-:$For every infinite family$\mathcal {F}$of finite sets of size at least$2$there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$with a Kinna–Wagner selection function. That is, there is a function$g\colon \mathcal (...) {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$with$\emptyset \not =f(F)\subsetneq F$for every$F\in \mathcal {G}$.We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace$\mathcal {F}$by a linearly ordered or a well-ordered family. (shrink)

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...) model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it. (shrink)

In Navigating Everyday Life, Peter Adams explores the moments when everyday experience seems to open up spaces beyond what we normally experience. Adams draws on two philosophical concepts: finitude, the things that bind a person to a situation, and transcendence, the things that lie beyond these boundaries.

On the question of whether the universe should be infinite or finite, there has been throughout the history of physics a struggle between materialism and idealism, between dialectics and metaphysics. Materialism asserts that the universe is infinite, while idealism advocates finitude. At every stage in the history of physics, these two philosophical lines have engaged in fierce struggle. Although developments in physics always demonstrate the failure of the finite universe doctrine, with every new advance in science the idealists distort and (...) take advantage of the latest results of physics to "prove" with varying sleights of hand that the universe is finite, serving the reactionary rule of the moribund exploiting classes. In the early part of this century after the rise of quantum theory and relativity theory, physics arrived at a new stage of development. After General Relativity was announced in 1916, a lot of people used it and similar theories of gravity to produce all sorts of models of the universe. The "finite universe" point of view became even more fashionable. Lenin pointed out that "That certain schools of the new physics have various dealings with Machism and other variants of modern idealism, is a fact not to be doubted for a moment." It is clear from reading all sorts of foreign literature that the schools of physics promoting a finite universe are linked up with all sorts of idealist philosophy, including theology. (shrink)

We write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality. In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

Preface to the fifth edition -- A world of difference -- A universe charged with the grandeur of God : Christian theism -- The clockwork universe : deism -- The silence of finite space : naturalism -- Zero point : nihilism -- Beyond nihilism : existentialism -- Journey to the east : eastern pantheistic monism -- A separate universe : the New Age spirituality without religion -- The vanished horizon : postmodernism -- A view from the Middle East : Islamic (...) theism -- The examined life. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice (...) of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

The Essay on Freedom represents a milestone in the philosophy of F. W. J. Schelling. Although Schelling already deviated from a strictly idealistic framework with his philosophy of nature, notably from that of Kant and Fichte, because he was seeking a moment of self-positing activity of the absolute subject in the object itself, it was only with The Essay on Freedom that he stepped out of that framework. The central point considered is an introduction of the second principle that was (...) complementary, but irreducible to the standard idealist principle of self consciousness. It was about the ground or nature in God. Only through that duality of principles, it was possible to grasp the positive notion of freedom as the possibility for good and evil. In such manner comprehended, the fact of human freedom in its entirety became relation with the whole of beings. The purpose of such explanation was an attempt to solve the problem that was spanning through Schellingʼs entire philosophy – the problem of the relation between finite and infinite. (shrink)

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

What are the assumptions and tasks hidden in contemporary calls to "overcome" the metaphysical tradition? Reflecting upon the internal contradictions of the notions of "tradition" and "finiteness," Dennis J. Schmidt offers novel insights into how philosophy must relate to its traditions if it is to retain a vital sense of the plurality of "edges" that constitute its finiteness. He does this through a close examination of issues found in the work of Hegel and Heidegger, two philosophers who made the ideas (...) of both tradition and finiteness the center of their concern.Schmidt begins by asking how Heidegger can claim to have destroyed metaphysics despite Hegel's claim to have perfected its possibilities. Systematically following the development of Heidegger's critique of Hegel, Schmidt generates a dialogue between them. The topic of that dialogue is the nature of finiteness as it is articulated in time, nothing, the dialectical and hermeneutical circles, and in the notions of experience, work, technology, history, and preSocratic thought.Beginning with Heidegger's critique of Hegel in Being and Time, Schmidt's strategy is to disclose the complexities of philosophical discourse about the finite by drawing out the proximities between Hegel and Heidegger. The dialogue that results presents novel portraits of both philosophers. It also reveals that Heidegger's early, unacknowledged failure to separate himself from the Hegelian dialectic is the motive behind many of the turns and decisions of his later career.In concluding, Schmidt offers an interpretation of the wider significance of the results of that dialogue, and connects his study to other contemporary discussions of postmodernism. He expands upon the idea of the plurality of edges opened by finiteness, arguing that philosophy only understands its own past and future once it recognizes the meaning of its own finiteness.Dennis J. Schmidt is Associate Professor of Philosophy at the State University of New York at Binghamton. The Ubiquity of the Finite is included in the series Studies in Contemporary German Social Thought, edited by Thomas McCarthy. (shrink)

In the present article, we consider the question of the primary elements in Plato’s Timaeus, the components of the whole universe reduced, by an extraordinarily elegant construction, to two right triangles. But how does he reconcile such a model with the infinite diversity of the universe? A large part of this study is devoted to Cornford’s explanation in his commentary of the Timaeus and its shortcomings, in order to finally propose a revised one, which we think to be entirely consistent (...) with Plato’s text. This analysis is an essential step for the understanding of the connection between the sensible world and the mathematical principles that underlies Timaeus’ cosmological account. (shrink)

Schutz’s references to literature and arts in his theoretical works are manifold. But literature and theory are both a certain kind of a finite province of meaning, that means they are not easily accessible from the paramount reality of everyday life. Now there is another kind of referring to literature: metaphorizing it. Using it, as may be said with Lakoff and Johnson, to understand and to experience one kind of thing in terms of another. Literally metapherein means “to carry over”. (...) Metaphorizing in this view is then a specific kind of border-crossing between different provinces of meaning. That poses two questions: 1. What means finiteness of those provinces of meaning, what kind of border crossings are possible? What is the ground for metaphorizing meaning? 2. Could this concept used for founding a theory of the constitution of the societal and of society, that overcomes the dichotomy of structure/agency? These questions will be answered with one example in view: Schutz’ report to Kaufmann of his first visit of Husserl describing his experience as feeling like Wilhelm Meister at the Society of the Tower. In a first step this metaphor is presented together with some crumbs of metaphor theory. In a second step these crumbs will be connected to Husserl’s concept of experience. After developing a short overview over Schutz’ “finite provinces of meaning,” the relation of experience, metaphors to the intersubjectivity of these provinces in their dependence from writing and printing is discussed. (shrink)

Let (L; f) be a finite simple Ockham algebra and let (X;g) be its dual space. We first prove that every connected component of X is either a singleton or a generalised crown (i.e. an ordered set that is connected, has length 1, and all vertices of which have the same degree). The representation of a generalised crown by a square (0,1)-matrix in which all line sums are equal is used throughout, and a complete description of X, including the number (...) of connected components and the degree of the vertices, is given. We then examine the converse problem of when a generalised crown can be made into a connected component of (X; g). We also determine the number of non-isomorphic finite simple Ockham algebras that belong properly to a given subvariety P 2n,0. Finally, we show that the number of fixed points of (L; f) is 0,1, or 2 according to the nature of X. (shrink)

If we are material beings living in a material world -- and all the scientific evidence suggests that we are -- then we must find existential meaning, if there is such a thing, in this physical world. We must cast our lot with the natural rather than the supernatural. Many Westerners with spiritual inclinations are attracted to Buddhism -- almost as a kind of moral-mental hygiene. But, as Owen Flanagan points out in The Bodhisattva's Brain, Buddhism is hardly naturalistic. In (...) _The Bodhisattva's Brain_, Flanagan argues that it is possible to discover in Buddhism a rich, empirically responsible philosophy that could point us to one path of human flourishing. Some claim that neuroscience is in the process of validating Buddhism empirically, but Flanagan's naturalized Buddhism does not reduce itself to a brain scan showing happiness patterns. "Buddhism naturalized," as Flanagan constructs it, offers instead a fully naturalistic and comprehensive philosophy, compatible with the rest of knowledge -- a way of conceiving of the human predicament, of thinking about meaning for finite material beings living in a material world. (shrink)

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz (...) and the Maltsev hierarchies. (shrink)

Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of (...) learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work. (shrink)

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames. It was established in [ Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in the two-sorted classical predicate (...) calculusCPC 2. Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB n P m + characterized by the class of all posets of the finite height m and the finite branching n. A finite axiomatization of the predicate logicLP m + characterized by the class of all posets of the height m is known from [Yokota 1989]. We prove thatLB n P m + =,B n being the propositional axiom of the branching n. (shrink)

This work investigates the finite-time control problem for a nonlinear four-tank cross-coupled liquid level system by the port-controlled Hamiltonian model. A fixed-free methodology is exhibited which can be used to simplify the controller design procedure. To get an adjustable convergent gain of the finite-time control, a feasible technique named damping normalization is proposed. A novel parameter autotuning algorithm is given to clarify the principle of choosing parameters of the PCH method. Furthermore, a finite-time controller is designed by a state-error desired (...) Hamiltonian function, and the relationship between the settling time and a parameter is given, which can be applied in practical engineering easily to adjust the settling time according to the industrial need. Finally, simulation and experimental results verify the effectiveness of the proposed algorithm. (shrink)

We show that the lattice L 20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (J). We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π 2 necessary and sufficient condition for embedding a finite pseudolattice into J.

Finite Transcendence: Existential Exile and the Myth of Home introduces and situates “existential exile” as an experience of the fundamental finitude of human existence and demonstrates how a particular way of responding in faith may enable one to find home in exile. Using the literary and philosophical oeuvre of Albert Camus as a model, this book demonstrates the manner in which mythic literature can both present and engage the condition of exile toward its possible transcendence.

According to finite frequentism, the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B. I present fifteen arguments against this position.

In this paper, we investigate the Lindenbaum algebra ℒ of the theory T fin = Th of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra /ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions (...) characterize uniquely the algebra ℒ; moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra /ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature. (shrink)

We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).

The use of the word “possible” without qualification invariably leads us to the endless, or at least well beyond our finite powers of analysis. Yet, to qualify possibility may itself become an endless task, if only because of the multitude of the available modes of approach. Somewhere in the middle of the realm of possibility one must parachute down and start ordering the scattered riches for our finite purposes with our limited and only newly acquired perspectives. What we have glimpsed (...) of the realms of possibility before our descent may still be of use, but our immediate tasks will bind us to the sternly empirical gravitational pulls of what happens to have erupted in the historical fields of discovery. (shrink)

The tangle modality is a propositional connective that extends basic modal logic to a language that is expressively equivalent over certain classes of finite frames to the bisimulation-invariant fragments of both first-order and monadic second-order logic. This paper axiomatises several logics with tangle, including some that have the universal modality, and shows that they have the finite model property for Kripke frame semantics. The logics are specified by a variety of conditions on their validating frames, including local and global connectedness (...) properties. Some of the results have been used to obtain completeness theorems for interpretations of tangled modal logics in topological spaces. (shrink)

In this paper we relax the assumption that the logical constants of ordinary first-order logic be linearly ordered. As a consequence, we shall have formulas involving not only partially ordered quantifiers, but also partially ordered connectives. The resulting language, called the language of informational independence will be given an interpretation in terms of games of imperfect information. The II-logic will be seen to have some interesting properties: It is very natural to define in this logic two negations, weak negation as (...) failure to verify a sentence, and strong negation as the existence of a falsifying strategy: One can express in this logic complete problems of finite structures, like the non-connectedness and 3-colorability of finite graphs, the satisfiability problem for Boolean circuits built up from NAND-gates, etc. (shrink)

The instrumental classification of depression made possible by the Diagnostic and Statistical Manual and the widespread pharmacological approach to treatment in mainstream biopsychiatry has generated a cottage industry of criticism. This paper explores the potential shortcomings of the DSM/bio-psychiatric model and introduces the value of philosophical counseling—specifically by means of integrating the insights of Existentialism and Buddhism—as a way to overcome a number of diagnostic and methodological problems. Philosophical counseling, in this regard, is not overly concerned with the objective question (...) of “What we are?” as biophysical beingswith overt behaviors but with a more fundamental question, namely, “How we are?” that is, how do we experience our existence as finite, impermanent beings, how does this experience shape and determine depressive episodes, and how can we come to accept our own finitude and impermanence? (shrink)

The conception of subjectivity that dominates the Western philosophical tradition, particularly during the Enlightenment, sets up a simple dichotomy: either the subject is ultimately autonomous or it is merely a causally determined thing. Fichte and Freud challenge this model by formulating theories of subjectivity that transcend this opposition. Fichte conceives of the subject as based in absolute activity, but that activity is qualified by a check for which it is not ultimately responsible. Freud explains the behavior of the self in (...) terms of biological drives and social pressures, yet both forces are actively interpreted by the subject itself. The tensions that arise from these very different approaches show that both Fichte and Freud are trying to overcome this deeply imbedded dichotomy between freedom and determinism. Although some would respond to these tensions by trying to forge a Hegelian synthesis, such a resolution covers over the paradoxical nature of finite subjectivity. (shrink)

In this study, an accurate convergence time of the supertwisting algorithm is proposed to build up a framework for nonaffine nonlinear systems’ finite-time control. The convergence time of the STA is provided by calculating the solution of a differential equation instead of constructing Lyapunov function. Therefore, precise convergence time is presented instead of estimation of the upper bound of the algorithm’s reaching time. Regardless of affine or nonaffine nonlinear systems, supertwisting control provides a general solution based on virtual control law (...) skill ensuring the output of the systems converges to the origin point at exact time. Benchmark tests are simulated to demonstrate the effectiveness and efficiency of the algorithm. (shrink)

The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. (...) With each (consistent) derivation there is associated a Buridan-Volpin (orBV) structure [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski modelM, then from any normal derivationD inLK* of can be constructed aBV modelM D=[M, {r(x)}] of where each ranger(x) is finite. (shrink)

A cutset of H is a subset of ∪ H which meets every element of H.H has the finite cutset property if every cutset of H contains a finite one. We study this notion, and in particular how it is related to the compactness of H for the natural topology. MSC: 04A20, 54D30.

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

The partial cooperation displayed by subjects in the Centipede Game deviates radically from the predictions of traditional game theory. Even standard, infinite population, evolutionary settings have failed to provide an explanation for this behavior. However, recent work in finite population evolutionary models has shown that such settings can produce radically different results from the standard models. This paper examines the evolution of partial cooperation in finite populations. The results reveal a new possible explanation that is not open to the standard (...) models and gives us reason to be cautious when employing these otherwise helpful idealizations. *Received January 2007; revised November 2007. †To contact the author, please write to: Department of Logic and Philosophy of Science, University of California, 3151 Social Science Plaza A, Irvine, CA 92697-5100; e-mail: [email protected]. (shrink)

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

The interesting and suggestive interpretation offered by A. J, Watt in this journal of Spinoza’s account of God’s causality to some extent anticipates the discussion of the topic which I am undertaking in a forthcoming book on Spinoza’s philosophy. To a greater extent it is, of course, anticipated by Stuart Hampshire in his study of Spinoza. I agree with Mr. Watt’s objections to some of the traditional interpretations of Spinoza’s doctrine and I think it is in fact immune from the (...) strictures of some critics who attempt to explain the succession of finite modes in Substance as a product of imaginal thinking, a view which must lead to a paralogism in Spinoza’s thought so gross that he could not have been unaware of it and could not have committed unless he had been. I shall not enter into this part of the discussion, as I have no wish to contest what Mr. Watt has written with respect to it, but shall confine my attention to his own interpretation. (shrink)

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to. In this paper, we examine the still open question of whether the prenex version of,, is equal to. We give new dependent choice‐based axiomatizations of the ‐consequences of and. Our dependent choice axiomatizations give new normal forms for the ‐consequences of and. We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to (...) show new results such as is finitely axiomatized and that there is a finitely axiomatized theory,, containing and contained in. On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for. (shrink)

We study a logic whose formulae are interpreted as properties of a finite set over some universe. The language is propositional, with two unary operators inclusion and extension, both taking a finite set as argument. We present a basic Hilbert-style axiomatisation, and study its completeness. The main results are syntactic and semantic characterisations of complete extensions of the logic.

The new result of this paper is that for θ-stable we have S1[θ] = D[θ, L, ∞]. S1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the followingMain (...) Theorem. Suppose that μ is regular satisfying μ ≥ |T|+, p is a finite type, and Δ is a set of formulas closed under Boolean operations. If either R[p, Δ, μ+] < ∞ or p is Δ-stable and μ satisfies “for every sequence {μi : i < |Δ| + ℵ0} of cardinals μi < μ we have that equation image holds”, then S[p, Δ, μ+] = D[p, Δ, μ+] = R[p, Δ, μ+].The S rank above is a localized version of Hrushovski's S1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of D and R, which is an old result of Shelah. A particular case of the theorem offers a new sufficient condition for the equality of S1 and D[·, L, ∞]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition. (shrink)

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the (...) recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu. (shrink)

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

The analogy between sequences of behavior and sequences of words is examined in detail and found to be both logically and empirically defensible. Linguistic distinctions are shown to be applicable to other forms of behavior. A behavioral grammar framework is developed and applied to the study of conversational behavior. The framework is shown to be superior to simpler finite state, associative models and the formal limits of the latter are discussed.

We study the strictness of the modal μ-calculus hierarchy over some restricted classes of transition systems. First, we prove that over transitive systems the hierarchy collapses to the alternationfree fragment. In order to do this the finite model theorem for transitive transition systems is proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment. Finally, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for (...) reflexive systems the same results holds for finite models. (shrink)