On Accuracy and Coherence with Infinite Opinion Sets

There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009, Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this paper, I establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, I establish the necessary results to extend the classic accuracy argument for probabilism to certain classes of infinite sets of propositions including countably infinite partitions.

ON THE DECIDABILITY OF THE INTERSECTION OF ALL TABULAR P N L-LOGICS

В (Попов 2018) проводилось исследование счетно-бесконечной иерархии табличных P N L-логик - логик P N L[3], P N L[4], P N L[5] и т. д. Центральный результат предлагаемой статьи: ∩i∈N P N L[i + 2] является разрешимой паранормальной логикой. In previous work (Попов 2018), we studied one countably infinite hierarchy of tabular P N L-logics (that is, P N L[3], P N L[4], P N L[5] and so on). The central result of the present article: ∩i∈N P N L[i + 2] is a decidable paranormal logic.

Finitely Supported Binary Relations between Infinite Atomic Sets

In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the T-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets ℘fs(Ak×Al), ℘fs(Ak×℘m(A)), ℘fs(℘n(A)×Ak) and ℘fs(℘n(A)×℘m(A)) do not contain uniformly supported infinite subsets. Moreover, the functions space ZAm does not contain a uniformly supported infinite subset whenever Z does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.

Airy Structures for Semisimple Lie Algebras

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $${\mathbb {C}}$$ C , and to some extent also over $${\mathbb {R}}$$ R , up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $$\mathfrak {sl}_2$$ sl 2 , $$\mathfrak {sp}_4$$ sp 4 and $$\mathfrak {sp}_{10}$$ sp 10 . Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.

Identification of an unbounded bi-periodic interface for the inverse fluid-solid interaction problem

This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem.

What is the Size of the Hilbert Hotel's Computer?

The Hilbert's Hotel is a hotel with countably infinitely many rooms. The size of its hypothetical computer was the pretext in order to think about whether it makes sense and what would be log_2(\aleph_0). Thus, at the road of this journey, this little paper demonstrates – surprisingly – that there exist countably infinite sets strictly smaller than \mathbb{N} (the natural numbers), with very elementary mathematics, so shockingly stating the inconsistency of the Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC)

Polynomial ring representations of endomorphisms of exterior powers

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.