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We generalise sheaf models of intuitionistic logic to univalent type theory over a small category with a Grothendieck topology. We use in a crucial way that we have constructive models of univalence, that can then be relativized to any presheaf models, and these sheaf models are obtained by localisation for a left exact modality. We provide first an abstract notion of descent data which can be thought of as a higher version of the notion of prenucleus on frames, from which can be generated a nucleus (left exact modality) by transfinite iteration. We then provide several examples., Simplified the definition of lex operation, simplified the encoding of the homotopy limit and remark that the homotopy descent data is a lex modality without using higher inductive types

A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.

Introduction.The generation of polynomial power laws of motion for the synthesis of cam mechanisms is complicated by the need to determine the coefficients of power polynomials. The study objective is to discover a rational capability of generating рower law swith arbitrary terms number under s with an rbitrary number of terms under the synthesis of cam mechanisms. Materials and Methods.A unified formula for determining the values of coefficients of power polynomials with any number of integers and/or non-integer exponents is derived through the so-called transfinite mathematical induction. Results.A unified formula for determining the values of coefficients, which gives correct results for any number of even and/or odd exponents, is presented. The correctness of the derived formula is validated by the results on the multiple checks for different numbers, even and odd values of the exponents of quinquinomial and hexanomial power functions. Discussion and Conclusions. A unified formula for determining the values of coefficients of power polynomials makes it possible to rationally define the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems. This provides a rational determination of the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems, and simple verification of the accuracy of the results obtained. The functions are particularly suitable for the synthesis of polydyne cams, as well as cams, since one polynomial can be used throughout the entire geometric mechanism cycle.

It is widely accepted that classical logic is trivialized in the presence of a transparent truth-predicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. (Journal of Philosophical Logic, 1–28, 2019) and Pailos (The Review of Symbolic Logic, Forthcoming) recovers classical logic, either in the sense that every classical (meta)inferential validity is valid at some point in the hierarchy (as is stressed in Barrio et al. (Journal of Philosophical Logic, 1–28, 2019)), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities with classical logic. Each of these logics is consistent with transparent truth—as is shown in Pailos (The Review of Symbolic Logic, Forthcoming)—, and this suggests that, contrary to standard opinions, transparent truth is after all consistent with classical logic. However, Scambler (Journal of Philosophical Logic, 49, 351–370, 2020) presents a major challenge to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. We embrace Scambler’s challenge and develop a new logic based on these hierarchies. This logic recovers both every classical validity and every classical antivalidity. Moreover, we will follow the same strategy and show that contingencies need also be taken into account, and that none of the logics so far presented is enough to capture classical contingencies. Then, we will develop a multi-standard approach to elaborate a new logic that captures not only every classical validity, but also every classical antivalidity and contingency. As a€truth-predicate can be added to this logic, this result can be interpreted as showing that, despite the claims that are extremely widely accepted, classical logic does not trivialize in the context of transparent truth.

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.

We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, $\Sigma_{2}$-codes and $\Pi_{3}$-reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have $\tmop{On}$ length tapes; 2) determine which admissible lengths of tapes for transfinite time machines with long tapes allow the machine to address each of their cells - a question raised by B. Rin; 3) characterise exactly the strength and behaviour of transfinitely acting Blum-Shub-Smale machines using a $\tmop{Liminf}$ rule on their registers - thereby establishing there is a universal such machine. This is in contradistinction to the machine using a `continuity' rule which fails to be universal.

We define and discuss transfinite asymptotic notions of smoothability, type, and equal norm type. We prove distinctness of these notions for a proper class of ordinals and that each class is an ideal. We also extend some results of Godefroy, Kalton, and Lancien to operators and ordinals greater than zero regarding the equivalence of equal norm asymptotic type and uniform renormings with power type smoothness. Finally, we discuss an extension of a non-linear result for quasi-reflexive, asymptotically $p$-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type $p$.

The paper studies the systems of paired contracts between agents of two complementary groups (workers and firms, students and universities, depositors and banks). Multiple contracts are allowed, as well as flexible contracts when the contract is concluded with some intensity. Agent preferences are described using choice functions. It is shown that if these choice functions satisfy the condition that we call conservativeness, then there are so-called stable systems of contracts, when it is unprofitable for any pair of counterparties to change the concluded contracts. The existence of a stable system of contracts is established using the transfinite process of sequential approximation, which generalizes the classical Gale−Shapley algorithm. As a result, we relieved of the finiteness conditions of the set of contracts. Such properties of stable systems as polarization and latticeness are also studied.

Using the E-infinity Cantorian spacetime formalism, i.e. transfinite platonic set theory, we show that Newton’s gravitational inverse square law strictly implies quantum entanglement and visa versa to the extent that they are derivable from each other. Noting that the same inverse square law is quasi generic for electrical and magnetic field the unifying role of the Cantorian spacetime of E-infinity becomes evident. The implication and generalization of the above will be explored but its ramifications within physics and beyond is not easily assessed at this early stage of what may amount to a paradigm shift prompted by the simple but very effective golden mean number system, which was suspected recently of being the rationale behind the fine tuning of the universe and the standard model of high energy elementary particle physics.

We show that pure monomorphisms are cofibrantly generated—generated from a set of morphisms by pushouts, transfinite composition, and retracts—in any locally finitely presentable additive category. In particular, this is true in any category of R-modules. On the other hand, the classes of all monomorphisms and regular monomorphisms in a locally finitely presentable additive category need not be well-behaved.

We generalize the famous Tarski result by showing that: if X is a complete lattice, and f : X → X is an order-preserving mapping, then for all points x ∈ X, the limit superior and the limit inferior of the (possibly transfinite) sequence of iterations x, f(x), f2(x)..., fβ(x),... are fixed points of f. These limits are the sharp fixed-point bounds between which sufficiently large transfinite iterations are located.

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient tool to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed, and each cubic cell is treated as an SBFE subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with a particular class of transition elements on the subdomains’ surfaces. Thus, a triangulation of these surfaces as executed in previous works is avoided, and consequently, the number of surface elements and degrees of freedom is reduced. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly.

We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.

A subset $S$ of a cardinal $\kappa$ is Ramsey if for every function $f:[S]^{, Comment: this is the final version uploaded to JSL

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter., Comment: 15 pages, 2 figures. Substantial changes in Section 3 were required to correct an erroneous argument in Section 4 of the original version

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma.

We present variants of Goodstein's theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits.

In Barrio et al. (Forthcoming) Barrio Pailos and Szmuc prove that there are systems of logic that agree with classical logic up to any finite meta-inferential level, and disagree with it thereafter. This article presents a generalized sense of meta-inference that extends into the transfinite, and proves analogous results to all transfinite orders.

In this article, as implied by the title, I intend to argue for the unattainability of Cantor’s Absolute at least in terms of the proof-theoretical means of set-theory and of the theory of large cardinals. For this reason a significant part of the article is a critical review of the progress of set-theory and of mathematical foundations toward resolving problems which to the one or the other degree are associated with the concept of infinity especially the one beyond that of the natural intuition of natural numbers. Naturally the review includes the foundation and development of the theory of large cardinals, especially after Cohen’s revolutionary introduction of the forcing method, insofar as it paved the way toward a transcendence of the delimitative character of Godel’s constructible universe L and further toward ever stronger infinity assumptions. Given that Cantor in his theory of transfinite numbers defined the mathematical absolute in ontological rather than concrete mathematical terms, I proceed in the last section to a philosophical discussion with certain prompts from phenomenology regarding the transposability of the formal conception of the infinite to the level of subjective constitution and argue in these terms on the infeasibility of acceding to an ‘absolute’ infinite cardinality. Of course the argumentation is strongly based on a view of the set-theoretical universe V of von Neumann’s cumulative hierarchy as a formal representative of the essential traits one would seek from a model of Cantor’s Absolute. It is also based on the conclusions reached from the whole discussion within the context of formal-theoretical structures as such.

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers.

We study a variant ATL FB of the alternating-time temporal logic ATL with a non-standard, ‘finitely bounded’ semantics (FBS). FBS was originally defined as a game-theoretic semantics where players must commit to time limits when attempting to verify eventuality (respectively, to falsify safety) formulae. It turns out that FBS has a natural corresponding compositional semantics that essentially evaluates formulae only on finite initial segments of paths and imposes uniform bounds on all plays for the fulfilment of eventualities. The resulting version ATL FB differs in some essential features from the standard ATL, as it no longer has the finite model property, though the two logics are equivalent on finite models. We develop two tableau systems for ATL FB . The first one deals with infinite sets of formulae and may run in a transfinite sequence of steps, whereas the second one deals only with finite sets of formulae in an extended language allowing explicit symbolic indication of time limits in formulae. We prove soundness and completeness of the infinitary tableau system and prove that it is equivalent to the finitary one. We also show that the finitary tableau system provides an exponential-time decision procedure for the satisfiability problem of ATL FB and thus establishes its EXPTIME-completeness. Furthermore, we present an infinitary axiomatization for ATL FB and prove its soundness and completeness.

Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the $$\limsup $$ of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the $$\limsup $$ rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines (wITTMs). We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers.

The existence of a linear order on a group that is compatible with the group structure generally requires transfinite methods. However, this can be circumvented by concentrating on the cons...

We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for $$\varDelta _0$$ formulae are provable in $$\mathbf {I}\varvec{\Sigma }_n$$. It is shown to be larger than the proof-theoretic ordinal $$|\mathbf {I}\varvec{\Sigma }_n|$$ by power of base 2. We also show a similar result for the structural transfinite induction, defined with fundamental sequences.

We derive relations between the transfinite diameter of a locally circled subset K of the complexified sphere in $$\mathbb {C}^3$$ and notions of weighted transfinite diameter of the projection of K to $$\mathbb {C}^2$$ . Our method is based on connecting a Chebyshev transform of K on an Okounkov body of V to the classical Chebyshev transform of the projection of K to $$\mathbb {C}^2$$ .

We study the existence and uniqueness (i.e., order independence) of any arbitrary form of iterated elimination procedures in an abstract environment. By allowing for a transfinite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa et al. (OR Lett 9:85–89, 1990), we identify a fairly weak sufficient condition of Monotonicity* for the order independence of iterated elimination procedure. Monotonicity* requires a Monotonicity property along any elimination path. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. We introduce a notion of CD* games, which incorporates Jackson’s (Rev Econ Stud 59:757–775, 1992) idea of “boundedness,” and show the iterated elimination procedure is order independent in the class of CD* games. In finite games, we also formulate and show an “outcome” order-independence result suitable for Marx and Swinkles’s (Games Econ Behav 18:219–245, 1997) notion of nice weak dominance.

W-types and their categorical analogue, initial algebras for polynomial endofunctors, are an important tool in predicative systems to replace transfinite recursion on well-orderings. Current arguments to obtain W-types in quotient completions rely on assumptions, like Uniqueness of Identity Proofs, or on constructions that involve recursion into a universe, that limit their applicability to a specific setting. We present an argument, verified in Coq, that instead uses dependent W-types in the underlying type theory to construct W-types in the setoid model. The immediate advantage is to have a proof more type-theoretic in flavour, which directly uses recursion on the underlying W-type to prove initiality. Furthermore, taking place in intensional type theory and not requiring any recursion into a universe, it may be generalised to various categorical quotient completions, with the aim of finding a uniform construction of extensional W-types.

The quantum information introduced by quantum mechanics is equivalent to a certain generalization of classical information: from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The “qubit”, can be interpreted as that generalization of “bit”, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the transfinite ordinal number corresponding to the infinity series in question. The transfinite ordinal numbers can be defined as ambiguously corresponding “transfinite natural numbers” generalizing the natural numbers of Peano arithmetic to “Hilbert arithmetic” allowing for the unification of the foundations of mathematics and quantum mechanics.

International audience; We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

The lower limb movement is the basic human movement, involving complex issues such as balance and walking. This research is conducted to realize the the real-time recognition and regulation of human lower limb skeletal muscle movement the deep learning concept. The article proposes a regularized transfinite learning machine, BP neural network and support vector machine regression algorithm to construct models for knee joint angle estimation. Three different deep learning algorithms are evaluated for the estimation of knee joint angle of lower limbs by using surface electromyography (sEMG). This paper systematically studies the human lower limb skeletal muscle motion simulation model based on deep learning. A large number of experiments and triple cross validation is performed and the outcomes are obtained using various indicators like root mean square error, correlation coefficient and training time. The root mean square error and correlation coefficient of regularized transfinite learning machine and support vector machine are less comparative to the BP neural network. Also, the training time of regularized transfinite learning machine model is the shortest, which is higher by two orders of magnitude than BP neural network model, about 1% of the neural network. The regularized transfinite learning machine algorithm provides feasible and promising outcomes establishing its application capability for real-time recognition and control of human lower limb skeletal muscle movement in complex environment. The comparative investigation is done in terms of training time and it was found that the training time of proposed RELM model is the shortest, which is higher by two orders of magnitude than BP neural network model. The outcomes obtained for the proposed RELM based method are optimal among the other counterparts and it has vast applicability for real-time recognition and regulation of lower limb motion.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than , the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

In [2], Downey and Greenberg use the ordinals below ε0 to bound the number of mind-changes of computable approximations. This gives rise to a new transfinite hierarchy in the c.e. degrees; the totally α-c.a. degrees. This hierarchy is significant because it unifies the combinatorics of many constructions as well as giving natural definability results in the c.e. Turing degrees. We study the structure of this hierarchy; in particular we investigate collapse in upper cones. We give a proof in which we build a c.e. set using a strategy tree to show there is no uniform way to find a maximal totally ω^2-c.a. degree above a given totally ω-c.a. degree. Then we discuss extensions of this result.

Step-indexed logical relations are an extremely useful technique for building operational-semantics-based models and program logics for realistic, richly-typed programming languages. They have proven to be indispensable for modeling features like higher-order state , which many languages support but which were difficult to accommodate using traditional denotational models. However, the conventional wisdom is that, because they only support reasoning about finite traces of computation, (unary) step-indexed models are only good for proving safety properties like “well-typed programs don’t go wrong”. There has consequently been very little work on using step-indexing to establish liveness properties, in particular termination. In this paper, we show that step-indexing can in fact be used to prove termination of well-typed programs—even in the presence of dynamically-allocated, shared, mutable, higher-order state—so long as one’s type system enforces disciplined use of such state. Specifically, we consider a language with asynchronous channels, inspired by promises in JavaScript, in which higher-order state is used to implement communication, and linearity is used to ensure termination. The key to our approach is to generalize from natural number step-indexing to transfinite step-indexing , which enables us to compute termination bounds for program expressions in a compositional way. Although transfinite step-indexing has been proposed previously, we are the first to apply this technique to reasoning about termination in the presence of higher-order state.

Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back Berman, Witt Nystr\"om and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-Noetherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken., Comment: 99 pages. Final version, to appear in Advances Math

Hierarchical octree‐meshes constitute an extremely efficient approach to representing complicated three‐dimensional structures. This is due to their capability to refine meshes locally and adjust the cell size over several orders of magnitude very effectively. As octree‐meshes consist solely of cuboid cells, the mapping of the geometry is trivial. On the other hand, exploiting these advantages in the context of finite‐element‐based modeling strategies is not straightforward due to the presence of hanging nodes. Recently, it has been demonstrated that the scaled boundary finite element method (SBFEM) provides a promising tool for creating discretizations based on such octree‐meshes. In this approach, each cell of the mesh is treated as a three‐dimensional SBFEM subdomain whose boundary is discretized using two‐dimensional finite elements. However, in previous work, an additional triangulation was still required to create conforming meshes on some of the subdomains' faces. In the current contribution, we improve this technique by combining the SBFEM with particular transition elements based on transfinite mappings. The combined approach requires only quadrilateral elements and circumvents any hanging nodes from the beginning. Furthermore, the utilization of transfinite mappings allows for local p‐ and h‐refinement and even combining different classes of shape functions without introducing additional constraints.

In this theoretical work, we analyze general constructions for integral-based (transfinite, continuous) barycentric coordinates and consider a simple variational principle to arrive at a continuous version of the Laplace barycentric coordinates. We demonstrate how our approach leads to a general description of the integral-based barycentric coordinates and establish links with Dirichlet energy minimization problems for conical surfaces. Both the 2D and 3D cases are studied. An application to a surface generation problem is briefly considered.

This and the following two chapters discuss and criticize some existing systems of mathematics. This one is about classical mathematics as inspired by Georg Cantor, and especially its use of undefinable sets. The author refutes some arguments for actual infinity, and suggests that the picture the Cantorians provide of a realm of higher infinities is so unclear that there are good reasons to doubt they are even describing a possibility. More specifically, this chapter’s topics include the relationship between large cardinal axioms and theorems of arithmetic; Cantor’s two principles of generation for the transfinite ordinals; his proof that there are “more” real numbers than natural numbers; and the continuum hypothesis.

We show that for any $\omega $ -r.e. degree $\boldsymbol{d}$ and $n$ -r.e. degree $\boldsymbol{b}$ with $\boldsymbol{d}\lt \boldsymbol{b}$ , there is an $(\omega +1)$ -r.e. degree $\boldsymbol{a}$ strictly between $\boldsymbol{d}$ and $\boldsymbol{b}$ . We also show that there is a maximal incomplete $(\omega +1)$ -r.e. degree. As a corollary, $\mathcal{D}_{\omega }$ is not a $\Sigma _{1}$ -elementary substructure of $\mathcal{D}_{\omega +1}$ .

Beklemishev introduced an ordinal notation system for the Feferman-Schutte ordinal Γ 0 based on the autonomous expansion of provability algebras. In this paper we present the logic BC (for Bracket Calculus). The language of BC extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, BC is based on a self-contained signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. The presented logic is proven to be equivalent to RC Γ 0 , that is, to the strictly positive fragment of GLP Γ 0 . We then define a combinatorial statement based on BC and show it to be independent of the theory ATR 0 of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.

Lewis Carroll era escritor y matemático. La trama que el autor desarrolla en su magistral Alicia en el país de las maravillas se despliega en el marco de una jugada de naipes sin reglas, o con leyes que cambian sin causa alguna. La lectura de Alicia recrea un piso imaginario y cognitivo propicio para introducirnos en las reflexiones filosóficas de Quentin Meillassoux en su libro Después de la finitud, en el que el concepto de transfinito (infinito de infinito) lo lleva a afirmar categóricamente que las propias leyes que gobiernan nuestro mundo y sus cosas pueden ser de otra manera, sin razón alguna, abonando la idea central del libro: la contingencia radical como absoluto especulativo. El presente trabajo diseña su campo de relevancia en torno al intercambio de categorías, insumos y recursos analíticos entre la Crítica Jurídica y el Realismo Especulativo, específicamente centrado en la lectura de Después de la finitud, de Meillassoux., Facultad de Ciencias Jurídicas y Sociales

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and representable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen.

In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is $\Delta^0_1$-definable relative to the previous stages of the recursion. It is known that this principle is provable in $\mathbf{ACA}_0$. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still provable in $\mathbf{ACA}_0$.

In this paper, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with `tree-like' structure and exponential growth rate of neighborhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komj\'athy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees., Comment: 26 pages, 6 figures