An alternative proof is given that to each LLM proof net there corresponds at least one LLM sequent proof. The construction is inspired by the sweepline technique from computational geometry and includes a treatment of the multiplicative constants and of proof boxes.

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to prove (...) the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely (...) universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

We prove the existence of p-ideals that are nonmeagre subsets of P(ω) under various set-theoretic assumptions.

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X1, . . . , Xn be independent Banach-valued random variables. Let I be a random variable independent of X1, . . . , Xn and uniformly distributed over {1, . . . , n}. Put ˜.

Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets $A \subseteq \mathbf{R}$ is the game determined????? Rules: The two players alternate moves starting with player I. Each move a n is legal iff it is a real number and $0 , and for $n > 1, a_n . The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff ∑ a n (...) exists and ∑ a n ∈ A. We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinancy and the determinancy of other well-known and much-studied games. (shrink)

R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that $\lambda , where α * is the Σ 1 -projectum of α and λ is the Σ 2 -cofinality of α. The theorem is also established for the metarecursive case, α = ω 1 , where (...) α * = λ = ω. (shrink)

We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L r ; p)'s and E(c 0 ; p)'s for r, p ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by (...) showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ is Borel bireducible to $\mathbb{R}^{\mathbb{N}}/c_{0}$. (shrink)

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h, r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

This thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use (...) of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.The abstract was taken directly from the thesis.Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, GermanyE-mail: [email protected]: https://doi.org/10.26083/tuprints-00026584. (shrink)

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ 1 is meagre yet there (...) are ℵ 1 rectifiable curves in R 3 whose union is not meagre. The consistency of this statement when the phrase "rectifiable curves" is replaced by "straight lines" remains open. (shrink)

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization of (...) approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. (shrink)

Let X be a separable Banach space and Q be a coanalytic subset of . We prove that the set of sequences in X which are weakly convergent to some eX and is a coanalytic subset of . The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 60–64], this result leads to the following dichotomy theorem: if X is a (...) class='Hi'>Banach space, is a regular method of summability and is a bounded sequence in X, then there exists a subsequence iL such that either there exists eX such that every subsequence iH of iL is weakly summable w.r.t. to e and QiH,e); or for every subsequence iH of iL and every eX with QiH,e)the sequence iH is not weakly summable to e w.r.t. . This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 232–234]. Both theorems obtain some considerable generalizations. (shrink)

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A, B, C such that A is congruent to B, B is congruent to C, and A is congruent to B ∪ C; this result was the precursor of the Banach–Tarski paradox. Later, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences. The pieces involved were (...) nonmeasurable. In the present paper, we consider the problem of which systems of congruences can be satisfied using open subsets of the sphere ; of course, these open sets cannot form a partition of the sphere, but they can be required to cover "most of" the sphere in the sense that their union is dense. Various versions of the problem arise, depending on whether one uses all isometries of the sphere or restricts oneself to a free group of rotations, or whether one omits the requirement that the open sets have dense union, and so on. While some cases of these problems are solved by simple geometrical dissections, others involve complicated iterative constructions and/or results from the theory of free groups. Many interesting questions remain open. (shrink)

It is argued that the so-called fitting attitude- or buck-passing pattern of analysis may be applied to personal values too if the analysans is fine-tuned in the following way: An object has personal value for a person a, if and only if there is reason to favour it for a’s sake. One benefit with it is its wide range: different kinds of values are analysable by the same general formula. Moreover, by situating the distinguishing quality in the attitude rather than (...) the reason part, the analysis admits that personal value is recognizable as a value not only by the person for whom it has personal value, but for everyone else too. We thereby avoid facing two completely different notions of value, viz., one pertaining to impersonal value, and another to personal value. The analysis also elucidates why we are justified in our concern for objects that are valuable for us; if value just is, as it is suggested, the existence of reasons for such a concern, the justification is immediately forthcoming. (shrink)

Chemical structures are among the trademarks of our profession, as surely chemical as flasks, beakers and distillation columns. When someone sees one of us busily scribbling formulas or structures, he or she has no trouble identifying a chemist. Yet these familiar objects, which accompany our work from start to end, from the initial doodlings (Fig. I) to the final polished artwork in a publication (Fig. II), are deceptively simple. They raise interesting and difficult questions about representation. It is the intent (...) of this article to reflect upon molecular graphics. (shrink)

This distinction has sometimes been read as purely epistemic, resting not on things, but on our knowledge of them: there is one world, not two, though it may be apprehended in two ways. But this view is patently at odds with the text. Knowledge and opinion are δυνάμεις, "faculties," to be distinguished and defined by their objects, no less than by the state of mind they produce, and Plato clearly states that the fallibility and unclearness of opinion is rooted in (...) the relative unreality of opinables. (shrink)

R T Allen; The Meaning of Life and Education, Journal of Philosophy of Education, Volume 25, Issue 1, 30 May 2006, Pages 47–58, https://doi.org/10.1111/j.1467-9.

Hausdorff's paradoxical decomposition of a sphere with countably many points removed actually produced a partition of this set into three pieces A,B,C such that A is congruent to B, B is congruent to C, and A is congruent to B ∪ C. While refining the Banach–Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences: the only nontrivial restriction is that the system should not (...) require any set to be congruent to its complement. Later, J. F. Adams showed that this restriction can be removed if one allows arbitrary isometries of the sphere to witness the congruences. The purpose of this paper is to characterize those systems of congruences which can be satisfied by partitions of the sphere or related spaces into sets with the property of Baire. A paper of Dougherty and Foreman gives a proof that the Banach–Tarski paradox can be achieved using such sets, and gives versions of this result using open sets and related results about partitions of spaces into congruent sets. The same method is used here; it turns out that only one additional restriction on a system of congruences is needed to make it solvable using subsets of the sphere with the property of Baire with free rotations witnessing the congruences. Actually, the result applies to any complete metric space acted on in a sufficiently free way by a free group of homeomorphisms. We also characterize the systems solvable on the sphere using sets with the property of Baire but allowing all isometries. (shrink)

We use the compactness theorem of continuous logic to give a new proof that $$L^r([0,1]; {\mathbb {R}})$$ isometrically embeds into $$L^p([0,1]; {\mathbb {R}})$$ whenever $$1 \le p \le r \le 2$$. We will also give a proof for the complex case. This will involve a new characterization of complex $$L^p$$ spaces based on Banach lattices.

This article draws on scientific explanations of obesity to motivate the creation of a system of paternalistic public health interventions into the obesity epidemic. Libertarian paternalists argue that paternalism is warranted in light of the cognitive limits of human decision-making abilities. There are further, specific biological limits on our capacity to choose and maintain a healthy diet. These biological facts strengthen the general motivation for libertarian paternalism. As a consequence, the creation of a system of paternalistic public health interventions into (...) the obesity epidemic is warranted. (shrink)

Objectives—To assess perceptions of the informed consent process in patients undergoing urgent abdominal surgery.Design—A prospective observational study was carried out using structured questionnaire-based interviews. Patients who had undergone urgent abdominal surgery were interviewed in the postoperative period to ascertain their perceptions of the informed consent process. Replies were compared to responses obtained from a control group undergoing elective surgery, to identify factors common to the surgical process and those specific to urgent surgery. Patients' perceptions of received information were also compared (...) to the information perceived to have been provided by the consent obtainers. Setting—Gastrointestinal surgical service of a university teaching hospital.Patients—Seventy-four consecutive patients undergoing urgent abdominal surgery and 80 control patients undergoing elective surgery. Main measurements—Principal outcome measures were patients' perceptions of factors interfering with the ability to give informed consent, assessment of the quality of informed consent and the degree of discussion of the expected outcomes. Results—Forty-nine of the seventy-four (66%) patients undergoing urgent surgery perceived that pain did not affect their ability to give informed consent. Twenty-seven reported an adverse effect of analgesia on the ability to give informed consent. Only 22% of patients undergoing urgent surgery perceived that there had been any discussion of potential side effects and complications of surgery. Conclusion–The majority of patients in this series with acute intra-abdominal surgical conditions perceive that they retain the ability to give informed consent for surgery. There is a need for improved discussion of therapeutic options and likely outcomes. (shrink)

Before talking about the vision of Orikhov's essence of the relationship between the church and the state, one must first determine what the church is about - Catholic or Orthodox. After all, the thinker lived in Poland when there were still strong, even parity positions of both denominations. He himself was brought up in a family where his father was Catholic and his mother was Orthodox. This was reflected in his mentality: he repeatedly publicly stated the benefits of certain tenets (...) of the Orthodox faith. But most of all he settled on the problem of relations between the Catholic Church and the state. (shrink)

For all the time and effort expended on empowerment and participation ‘programmes’, many fail each year. This paper argues that the cause is a faulty view of communication widespread among managers and their teachers: the conduit, transmission model. It frustrates participation and is an ideology of management control. It rests on untenable beliefs about meaning and how language relates to the world. The paper proposes a new model of communication in terms of ‘communicating other/wise’ and offers examples of how it (...) can be practised in management education and by managers aiming to bring empowered and participatory workplaces into being. (shrink)

In the coverage of the problems of war and peace in Catholicism, researchers sometimes omit the regional component of their manifestation. We are talking about local traditions in Catholicism, which have their own specific approach to solving this problem and sometimes may differ significantly from the "conventional" position. The study of this field is necessary because in the delineated field the position of those theologians who were on the "outskirts" of Catholicism in the intercultural dialogue, on the one hand, led (...) to a deviation from the humanistic orientation about Catholicism. the war. An example would be the concept of a "holy war" of the Crusades, which had the anti-Islamic rhetoric generated not only in the West but also in the East. (shrink)