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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 $\mathbf{ACA}_0$.

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis.

In this note, we give some applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel. We compare quantitatively the projective logarithmic capacity with the complex Monge-Ampere capacity on complex projective space and we deduce that the set of zero logarithmic capacity is of Monge-Ampere capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capacity. Finally we deduce that there is an analogous of classical Evans's theorem.

In 1904, Pavel Florensky who have been influenced by Cantor's ideas, wrote “On the symbols of infinity.” In this paper he says that Georg Cantor's transfinite mathematics is an example of symbolic vision of God, but not in a direct way. Cantor's idea from the “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” is that “the absolute can only be acknowledged, but never known.” The absolutely infinite sequence of numbers thus seems to him to be an appropriate symbol of the Absolute. Symbol, as Pavel Florensky wrote in his memoirs, was the most important concept in his philosophy throughout his life. Symbol has a distinctive ontological modus of existence and its function is to be a reference for the higher being, namely God. It could also be associated with Nicolaus' of Cusa concept of minimax. I analyze the meaning of symbol in Georg Cantor and Pavel Florensky's works and juxtapose it with the understanding of the symbol by later Florensky and other interpreters. I also examine the views of theologian Christian Tapp, who researched Cantor's interest in theology. He understands symbol as a minimal in the Cantor's theory. Johanna Van der Ween and Leon Horsten represent Georg Cantor's conception in the context of European philosophers, whom Cantor did read. The main problem of this paper is how symbol and absolute infinity could be connected, and whether the meaning of symbol implies understanding of the higher being or it is not necessarily part of the concept.

In recent work Philip Welch has proven the existence of ‘ineffable liars’ for Hartry Field’s theory of truth. These are offered as liar-like sentences that escape classification in Field’s transfinite hierarchy of determinateness operators. In this article I present a slightly more general characterization of the ineffability phenomenon, and discuss its philosophical significance. I show the ineffable sentences to be less ‘liar-like’ than they appear in Welch’s presentation. I also point to some open technical problems whose resolution would greatly clarify the philosophical issues raised by the ineffability phenomenon.

The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an e0-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.

It is well-known that the Ford-Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-Terminating example converges to a limit flow, but this limit flow need not be a maximum flow. Hence, one may pass to the limit and begin the algorithm again. In this way, we may view the Ford-Fulkerson algorithm as a transfinite algorithm. We analyze the transfinite running-Time of the Ford-Fulkerson algorithm using ordinal numbers, and prove that the worst case running-Time is ω Θ (, SCOPUS: ar.j, info:eu-repo/semantics/published

We formalise second-order ZF set theory in the dependent type theory of Coq. Assuming excluded middle, we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categoricity of extended axiomatisations fixing the number of Grothendieck universes. These results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and set-theoretic transfinite recursion. Following Aczel’s sets-as-trees interpretation, we give a concise construction of an intensional model of second-order ZF with a weakened replacement axiom. Whereas this construction depends on no additional logical axioms, we obtain intensional and extensional models with full replacement assuming a description operator for trees and a weak form of proof irrelevance. In fact, these assumptions yield large models with n Grothendieck universes for every number n.

We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap’s revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb’s Probabilistic Convention T, which says that the probability of φ equals the probability that φ is true. published

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which leads to a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity. The whole undertaking, which takes into account major stages of the research in large cardinals theory, tries to present a defensible argumentation against an ontology of infinity actually rooted in the notion of subjectivity within the world. This means that rather than talking of a general ontology of infinity in the ideal platonic or in the aristotelian sense of potentiality, even in the alternative sense of an ontology of the event in A. Badiou’s sense, one can argue from a subjective point of view about the impossibility of defining cardinalities greater than the first uncountable one $$\aleph _{1}$$ that would correspond to a distinct existence in real world terms or would be supported by a mathematical intuition in terms of reciprocity with experience. The argumentation from the particular standpoint includes also certain comments on the delimitative character of Godel’s constructive universe L and the influence of the constructive approach in narrowing the breadth of an ‘ontology’ of infinity.

When performing a line scan using optical coherence tomography (OCT), the distance between the successive scan lines is often large compared to the resolution along each scan line. If two sets of such line scans are acquired orthogonal to each other, intensity values are known along the lines of a square grid, but are unknown inside each square. To view these values as an image, intensities need to be interpolated at regularly spaced pixel positions. In this paper we evaluate three methods for interpolation from grid lines: linear, transfinite and weighted. The linear method does not preserve the known values along the grid lines. The transfinite method, known from mesh generation, preserves the known values but might cause artifacts further away from the grid lines. The weighted method, which we propose, is designed to combine the desired properties of the transfinite method close to grid lines and the stability of the linear method further away. An important parameter influencing the performance of the interpolation methods is the upsampling rate. We perform an extensive evaluation of the three interpolation methods across a range of upsampling rates. Our statistical analysis shows significant difference in the performance of the three methods. We find that the transfinite interpolation works well for small upsampling rates and the proposed weighted interpolation method performs very well for all upsampling rates typically used in practice. On the basis of these findings we propose an approach for combining two OCT scans, acquired such that the lines of the second scan are orthogonal to the first.

In this paper, the Hℤ-length of different groups is studied. By definition, this is the length of the Hℤ-localization tower or the length of the transfinite lower central series of Hℤ-localization. It is proved that, for a free noncyclic group, its Hℤ-length is ≥ ω+2. For a large class of ℤ[C]-modules M, where C is an infinite cyclic group, it is proved that the Hℤ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its Hℤ-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M⋊C is finitely presented and H2(M ⋊ C) is finite.

We consider the problem of retaining the motions of an abstract dynamic system in a given constraint set. Constructions from the programmed iteration method are extended to problems whose dynamics, in general, does not possess any topological properties. The weaker requirements are compensated by introducing transfinite iterations of the programmed absorption operator. The technique of fixed points of mappings in chain-complete partially ordered sets is used in the proofs. The proposed procedure produces a set where the retention problem is solved in the class of quasistrategies. The control interval is not assumed to be bounded.

In this thesis, we study Turing degrees in the context of classical recursion theory. What we are interested in is the partially ordered structures $\mathcal {D}_{\alpha }$ for ordinals $\alpha and $\mathcal {D}_{a}$ for notations $a\in \mathcal {O}$ with $|a|_{o}\geq \omega ^2$ .The dissertation is motivated by the $\Sigma _{1}$ -elementary substructure problem: Can one structure in the following structures $\mathcal {R}\subsetneqq \mathcal {D}_{2}\subsetneqq \dots \subsetneqq \mathcal {D}_{\omega }\subsetneqq \mathcal {D}_{\omega +1}\subsetneqq \dots \subsetneqq \mathcal {D(\leq \textbf {0}')}$ be a $\Sigma _{1}$ -elementary substructure of another? For finite levels of the Ershov hierarchy, Cai, Shore, and Slaman [Journal of Mathematical Logic, vol. 12 (2012), p. 1250005] showed that $\mathcal {D}_{n}\npreceq _{1}\mathcal {D}_{m}$ for any $n < m$ . We consider the problem for transfinite levels of the Ershov hierarchy and show that $\mathcal {D}_{\omega }\npreceq _{1}\mathcal {D}_{\omega +1}$ . The techniques in Chapters 2 and 3 are motivated by two remarkable theorems, Sacks Density Theorem and the d.r.e. Nondensity Theorem.In Chapter 1, we first briefly review the background of the research areas involved in this thesis, and then review some basic definitions and classical theorems. We also summarize our results in Chapter 2 to Chapter 4. In Chapter 2, we show that for any $\omega $ -r.e. set D and r.e. set B with $D, there is an $\omega +1$ -r.e. set A such that $D. In Chapter 3, we show that for some notation a with $|a|_{o}=\omega ^{2}$ , there is an incomplete $\omega +1$ -r.e. set A such that there are no a-r.e. sets U with $A. In Chapter 4, we generalize above results to higher levels (up to $\varepsilon _{0}$ ). We investigate Lachlan sets and minimal degrees on transfinite levels and show that for any notation a, there exists a $\Delta ^{0}_{2}$ -set A such that A is of minimal degree and $A\not \equiv _T U$ for all a-r.e. sets U.Abstract prepared by Cheng Peng.E-mail: pengcheng@nankai.edu.cn

Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees.In this thesis, we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics.The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees (answering a question of Slaman and Steel), that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions.This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss.The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about whether it is provable in $\mathsf {ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of $\mathsf {ZF}$ (which is incompatible with $\mathsf {ZFC}$ ), this holds for all partial orders of height two, but not for all partial orders of height three. Our proof also yields an analogous result for Borel partial orders and Borel embeddings in $\mathsf {ZF}$ , which shows that the Borel version of Sacks’ question has a negative answer.We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research.Abstract prepared by Patrick Lutz.E-mail: pglutz@berkeley.edu

We introduce two concepts in this work: polynomial mortar and transfinite mortar, and apply them to curved nonconforming sliding meshes. It is shown that, on curved meshes, polynomial mortar always carries geometric errors while transfinite mortar has no such errors, which makes the latter superior to the former in almost every aspect. For example, the latter has better accuracies and introduces smaller numerical disturbances. The proposed sliding-mesh method utilizing transfinite mortar is conservative, arbitrarily high-order accurate in space, retains the order of accuracy of a time marching scheme, does not change flow characteristics, and has minimum sliding-speed effects. It is by far the most accurate and most thoroughly studied sliding-mesh method. This method can be extended and applied to a wide range of flow problems, such as propellers, wind turbines, stirred tanks, etc.

Given an unshielded continuum K ⊂ C , the core decomposition D K L C of K with respect to local connectedness is known to exist [2]. Such a decomposition D K L C is monotone, locally connected under quotient topology, and refines every other monotone decomposition D ′ which is locally connected under quotient topology. Let ∼ be the closed equivalence whose classes form the decomposition D K L C . Then ∼ contains a symmetric closed relation RK which requires (x, y) ∈ RK if and only if x and y lie in a prime end impression of C ∖ K . We also say that the relation RK respects prime end impressions. From Akin’s viewpoint of dynamical systems, the equivalence ∼ may be obtained as one of the limit relations of RK, through transfinite process. We will propose a direct approach to realize this limit relation in a concrete construction. More or less, such an approach builds the elements of D K L C “from below ” . If K is an unshielded disconnected compactum, the core decomposition D K F S of K with respect to the finitely suslinian property has been obtained in [3]. In this case, we consider a symmetric closed relation that “respects limit continua” and show that the same approach also works, in constructing D K F S from below.

There are two well-known ways of doing arithmetic with ordinal numbers: the "ordinary" addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the "natural" (or Hessenberg) addition and multiplication (denoted $\oplus$ and $\otimes$), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted $\times$), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote $\alpha^{\times\beta}$. (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we will denote this $\alpha^{\otimes\beta}$. We show that $\alpha^{\otimes(\beta\oplus\gamma)} = (\alpha^{\otimes\beta}) \otimes(\alpha^{\otimes\gamma})$ and that $\alpha^{\otimes(\beta\times\gamma)}=(\alpha^{\otimes\beta})^{\otimes\gamma}$; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a "natural exponentiation" satisfying reasonable algebraic laws., Comment: 18 pages, 3 tables

An analog of Polya’s theorem on the estimate of the transfinite diameter for a class of multivalued analytic functions with finitely many branch points and of the corresponding class of admissible compact sets located on the associated (with this function) two-sheeted Stahl–Riemann surface is obtained.

In the paper it is proved that there exists a continuous surjective mapping F : X → Y , where X and Y are separable complete metric spaces of transfinite dimension ind less than or equal to α d ∈ ω + ∪ { ∞ } and α r ∈ ω + ∪ { ∞ } , respectively, such that, for each continuous surjective mapping f : X → Y , where X and Y are compact metric spaces of transfinite dimension ind less than or equal to α d and α r , respectively, there are isometries i : X → X and j : Y → Y satisfying the relation F ∘ i = j ∘ f . This result remains true if we suppose that, for each mapping f : X → Y , Y coincides with a fixed separable complete metric space Y and that the corresponding mapping j : Y → Y is the identity.

We give complete proofs of the results of Galvin and Nagy on a problem of Erdős and Hajnal concerning certain combinatorial games of transfinite length.

To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bezier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bezier rotation surfaces are constructed using a generalized Bezier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bezier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bezier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bezier rotation surfaces provide a valuable way for the design of rotation surfaces.

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space $X$ with a scattered algebraic $1$-wild set $\mathbf{aw}(X)$, then all transfinite fundamental groupoid operations are also well-defined., 17 pages, 2 figures

We present a new approach to deductive program verification based on auxiliary programs called ghost monitors. This technique is useful when the syntactic structure of the target program is not well suited for verification, for example, when an essentially recursive algorithm is implemented in an iterative fashion. Our approach consists in implementing, specifying, and verifying an auxiliary program that monitors the execution of the target program, in such a way that the correctness of the monitor entails the correctness of the target. The ghost monitor maintains the necessary data and invariants to facilitate the proof. It can be implemented and verified in any suitable framework, which does not have to be related to the language of the target programs. This technique is also applicable when we want to establish relational properties between two target programs written in different languages and having different syntactic structure. We then show how ghost monitors can be used to specify and prove fine-grained properties about the infinite behaviors of target programs. Since this cannot be easily done using existing verification frameworks, we introduce a dedicated language for ghost monitors, with an original construction to catch and handle divergent executions. The soundness of the underlying program logic is established using a particular flavor of transfinite games. This language and its soundness are formalized and mechanically checked., Proceedings of the ACM on Programming Languages, 4 (POPL), ISSN:2475-1421

One way to deal with the Liar paradox is the paracomplete approach to theories of truth that gives up proofs by contradiction and the Law of the Excluded Middle. This allows one to reject both the Liar sentence and its negation. The simplest paracomplete theory of truth is KFS due to Saul Kripke. At face value, this theory suffers from the problem that it cannot say anything about the Liar paradox, so a de-fender of this theory cannot explain their rejection of the Liar sentence within the language of KFS. This was one of the motivations for Hartry Field to extend KFS with a conditional that is not definable within KFS. With the help of this conditional, Field defines a determinateness operator that can be used to explain one’s rejection of the Liar sentence within the object language of his theory. The determinateness operator can be transfinitely iterated to create stronger notions of determinate-ness required to explain the rejection of paradoxical sentences involving the determinateness operator. In this paper, we show that Field’s complex extension of KFS is not required in order to express rejection of paradoxical sentences like the Liar sentence. Instead one can work with a transfinite hierarchy of determinateness operators that are definable in KFS. After defining this hierarchy of determinateness operators, we compare their properties with the transfinitely iterable determinateness operator due to Field.

The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the corresponding forcing extensions are forced and forced statements are true -- is equivalent over G\"odel-Bernays set theory GBC to the principle of elementary transfinite recursion $\text{ETR}_{\text{Ord}}$ for class recursions of length $\text{Ord}$. It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal{L}_{\text{Ord},\omega}(\in,A)$, allowing any class parameter $A$; to the existence of truth predicates for the language $\mathcal{L}_{\text{Ord},\text{Ord}}(\in,A)$; to the existence of $\text{Ord}$-iterated truth predicates for first-order set theory $\mathcal{L}_{\omega,\omega}(\in,A)$; to the assertion that every separative class partial order $\mathbb{P}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$. Unlike set forcing, if every class forcing notion $\mathbb{P}$ has a forcing relation merely for atomic formulas, then every such $\mathbb{P}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between GBC and Kelley-Morse set theory KM., Comment: 34 pages. Commentary concerning this paper can be made at http://jdh.hamkins.org/class-forcing-theorem

A positive zero-sum stochastic game with countable state and action spaces is shown to have a value if, at every state, at least one player has a finite action space. The proof uses transfinite algorithms to calculate the upper and lower values of the game. We also investigate the existence of ( $$\epsilon $$ -)optimal strategies in the classes of stationary and Markov strategies.

We consider a dynamic approach to games in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite sequential games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies. For infinite games in extensive form we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are $$\Delta ^0_2$$-sets.

We consider intervals of the form $\left [ \frac {r}{s}, u \right ] $, where $r,s$ are positive integers with gcd($r,s$)=1 and $u$ is a real number, or of the form $[0, \smash {(\sqrt {a}{-}\sqrt {b})^2} ]$, where $a,b$ are positive integers. Thanks to a lemma of Chudnovsky, we give first a lower bound of the integer transfinite diameter of such intervals. Then, using the method of explicit auxiliary functions and our recursive algorithm, we explain how to get an upper bound for this quantity. We finish with some numerical examples. Secondly, we prove inequalities on the integer transfinite diameter of Farey intervals, i.e., intervals of the type $\bigl [ \frac {a}{q}, \frac {b}{s} \bigr ] $, where $|as-bq|=1$.

We develop the theory of APD profiles introduced by J. Dydak for ∞-pseudometric spaces ( [3] ). We connect them with transfinite asymptotic dimension defined by T. Radul ( [4] ). We give a characterization of spaces with transfinite asymptotic dimension at most ω + n for n ∈ ω and a sufficient condition for a space to have transfinite asymptotic dimension at most m ⋅ ω + n for m , n ∈ ω , using the language of APD profiles. This condition together with a result from [7] enables us to answer positively the question in [3, 5.15] .

By Solovay’s celebrated completeness result [31] on formal provability we know that the provability logic ${\textbf {GL}}$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result in [22] by considering a polymodal version ${\mathsf {GLP}}$ of ${\textbf {GL}}$ with modalities $[n]$ for each natural number n referring to ever increasing notions of provability. Modern treatments of ${\mathsf {GLP}}$ tend to interpret the $[n]$ provability notion as “provable in a base theory T together with all true $\Pi ^0_n$ formulas as oracles.” In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity $\Pi ^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $\Sigma ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $\alpha $ will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper. The paper proves soundness and completeness for the proposed interpretation for a wide class of theories, namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic.

We study the computational strength of resetting $\alpha$-register machines, a model of transfinite computability introduced by P. Koepke in \cite{K1}. Specifically, we prove the following strengthening of a result from \cite{C}: For an exponentially closed ordinal $\alpha$, we have $L_{\alpha}\models$ZF$^{-}$ if and only if COMP$^{\text{ITRM}}_{\alpha}=L_{\alpha+1}\cap\mathfrak{P}(\alpha)$, i.e. if and only if the set of $\alpha$-ITRM-computable subsets of $\alpha$ coincides with the set of subsets of $\alpha$ in $L_{\alpha+1}$. Moreover, we show that, if $\alpha$ is exponentially closed and $L_{\alpha}\not\models$ZF$^{-}$, then COMP$^{\text{ITRM}}_{\alpha}=L_{\beta(\alpha)}\cap\mathfrak{P}(\alpha)$, where $\beta(\alpha)$ is the supremum of the $\alpha$-ITRM-clockable ordinals, which coincides with the supremum of the $\alpha$-ITRM-computable ordinals. We also determine the set of subsets of $\alpha$ computable by an $\alpha$-ITRM with time bounded below $\delta$ when $\delta>\alpha$ is an exponentially closed ordinal smaller than the supremum of the $\alpha$-ITRM-clockable ordinals. Moreover, we obtain some sufficient and necessary conditions on ordinals $\alpha$ for which the $\alpha$-wITRM-clockable ordinals are bounded by $\alpha$.

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal $\alpha$ there exists an ordinal $\beta$ such that $1+\beta\cdot(\beta+\alpha)$ (ordinal arithmetic) admits an almost order preserving collapse into $\beta$. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta\cdot\alpha$ at the place of $\beta\cdot(\beta+\alpha)$. We will also characterize the principles that any set is contained in a countable coded $\omega$-model of arithmetical transfinite recursion resp. arithmetical comprehension., Comment: This is the accepted version of a paper published in The Journal of Symbolic Logic

We analyse Kreisel's notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church's thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman.

We prove that for a positive integer m and a non-negative integer n, if the transfinite asymptotic dimension of an ∞-pseudometric space X is at most m ⋅ ω + n , then X has an integral APD profile ( n + 1 , α 1 , … , α m ) for some non-decreasing functions α 1 , … , α m . This answers a question of Orzechowski posed in 2020.

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $\Omega$ be a regular uncountable cardinal. Let $m, Comment: 120 pages. Changes this version: Corrected definition of k-maximal iteration tree (for MS-premice with superstrongs; see Footnote 11). Corrected small errors in section 2 and its application (see Footnote 16). Changed definition of putative tree (see Footnote 13). Small corrections/improvements in Thm 4.47, Cor 7.6, Lem 9.8, Sec 9.1.4. Updated bibliography. Other minor edits

In this paper, we study the relationship between complementary-finite asymptotic dimension and transfinite asymptotic dimension. We prove that for every metric space X, coasdim ( X ) ≤ γ implies trasdim ( X ) ≤ γ for every countable ordinal number γ. Therefore, the metric space with complementary-finite asymptotic dimension has asymptotic property C. Furthermore, we construct an example of a metric space satisfying asymptotic property C and finite decomposition complexity with arbitrary high asymptotic dimension growth, which shows that asymptotic property C and finite decomposition complexity need not imply sub-exponential dimension growth in general.

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions. Moreover, the length of the shortest such sequence is essentially the same as the value of certain natural ranks they defined on the Baire class 1 functions. They also introduced the notion of pseudouniform convergence to generate some classes of bounded Baire class 1 functions from others. The main aim of this paper is to generalize their results to Baire class $\xi$ functions. For our proofs to go through, it was essential to first obtain similar results for Baire class 1 functions defined on not necessary compact Polish spaces. Using these new classifications of bounded Baire class $\xi$ functions, one can define natural ranks on these classes. We show that these ranks essentially coincide with those defined by Elekes et. al.

Neste pequeno artigo, analiso como a intuição geométrica estava presente no desenvolvimento seminal da teoria cantoriana dos conjuntos. Deste fato, decorre que a noção de conjunto ou de número transfinito não era tratada por Cantor como algo que merecesse uma fundamentação lógica. Os paradoxos que surgiram na teoria de Cantor são fruto de tal descompromisso inicial, e as tentativas ulteriores de resolvê-los fizeram com que aspectos intuitivos e esperados sobre os conjuntos ou infinito se perdessem. Em especial, observa-se aqui as consequências “não geométricas” do axioma da construtibilidade de Gödel.

INSPIRED by a dynamist Naturphilosophie and looking for a mathematics of the natura naturans, the founders of modern mathematics in Germany made some lasting contributions in the attempt to go beyond perceptible space (“Anschauungsraum”). Hermann Grassmann’s extension theory, Johann Benedict Listing’s topology, Bernhard Riemann’s non-Euclidean manifold theory, Carl Gustav Jacob Jacobi’s approach to non-mechanistic theory and last but not least Georg Cantor’s transfinite set theory were all influenced by the tradition of Naturphilosophie. One central motivation for the new mathematics was to decipher the so-called “inner side of nature” which was thought to be invisible but nonetheless physical. As a result, mathematics was understood not only in terms of quantity but of morphogenetic quality and as a tool to recognize the intrinsic possibilities of structuring and organizing nature. This essay focuses on the philosophical viewpoints of Christian Samuel Weiss, Justus and Hermann Grassmann, Bernhard...

Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of Lω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ωω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank AD(•), introduced by P. Dodos, uses the transfinite Schreier families and has the property that AD(X) < ω1 if and only if X is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order l1 spreading models. We also compute this rank for several Banach spaces. In particular, it is shown that the class of Banach spaces \(\left( {X_0^{{\omega ^\xi }}} \right)\xi < {\omega _1}\), which each admit l1 and c0 spreading models hereditarily, and were introduced by S. A. Argyros, the first and third author, satisfy \(AD\left( {X_0^{{\omega ^\xi }}} \right) = {\omega ^\xi } + 1\). This answers some questions of Dodos.

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets.

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used and each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor. The fourth author’s research was partially supported by Generalitat Valenciana under project GV/2015/035.