scholarly journals On Accuracy and Coherence with Infinite Opinion Sets

2022 ◽  
pp. 1-37
Author(s):  
Mikayla Kelley
Keyword(s):  
De Finetti ◽  
Infinite Sets ◽  
Countably Infinite ◽  
Infinite Set

Abstract 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.

The point-continuous spectrum of second-order, ordinary differential operators

1979 ◽  
Vol 84 (3-4) ◽  
pp. 197-211 ◽  
Author(s):  
Christine Thurlow
Keyword(s):  
Boundary Condition ◽  
Continuous Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Ordinary Differential Operators ◽  
Isolated Points ◽  
Infinite Sets ◽  
Countably Infinite ◽  
Infinite Set

SynopsisGiven any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

Monotone but not positive subsets of the Cantor space

1987 ◽  
Vol 52 (3) ◽  
pp. 817-818 ◽  
Author(s):  
Randall Dougherty
Keyword(s):  
Partial Ordering ◽  
Background Information ◽  
Countable Union ◽  
Abstract Form ◽  
Cantor Space ◽  
Countable Intersection ◽  
Power Set ◽  
Countably Infinite ◽  
Infinite Set

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

How to Program an Infinite Abacus

1961 ◽  
Vol 4 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Finite Number ◽  
Recursive Function ◽  
Computable Function ◽  
Unlimited Supply ◽  
Countably Infinite ◽  
Infinite Set

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

scholarly journals Shape invariant rational extensions and potentials related to exceptional polynomials

2015 ◽  
Vol 30 (24) ◽  
pp. 1550146 ◽  
Author(s):  
S. Sree Ranjani ◽  
R. Sandhya ◽  
A. K. Kapoor
Keyword(s):  
Shape Invariant ◽  
Invariant Potential ◽  
Exceptional Polynomials ◽  
Infinite Sets ◽  
Infinite Set ◽  
Explicit Expressions

In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with [Formula: see text] exceptional polynomials as solutions. The method is simple and transparent and is elucidated using the radial oscillator and the trigonometric Pöschl–Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. Explicit expressions of the generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric Pöschl–Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.

A note on lemmas of Green and Kondo

1967 ◽  
Vol 63 (1) ◽  
pp. 83-85 ◽  
Author(s):  
A. O. Morris
Keyword(s):  
Symmetric Functions ◽  
Rational Numbers ◽  
Infinite Sets ◽  
Image Position ◽  
Countably Infinite

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown thatcan be expressed in the formwhere Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In additionwhere is the partition of n conjugate to (λ). In fact, Littlewood (5) showed thatwhere the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and.

The extensions of the modal logic K5

1985 ◽  
Vol 50 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Michael C. Nagle ◽  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Abstract Characterization ◽  
Propositional Modal Logic ◽  
Normal Logic ◽  
Countably Infinite ◽  
Infinite Set

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

scholarly journals First-order modal logic in the necessary framework of objects

2016 ◽  
Vol 46 (4-5) ◽  
pp. 584-609 ◽  
Author(s):  
Peter Fritz
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Consequence Relations ◽  
Possible World ◽  
Logical Constants ◽  
First Order ◽  
Timothy Williamson ◽  
Infinite Sets ◽  
Infinite Set ◽  
First Order Modal Logic

AbstractI consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

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