Morley's Theorem Research Articles

An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.

On cardioids and Morley's theorem

Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.

Mathematical model creation for Reducing damages in basketball players

Introduction: Hand and wrist stability are critical for optimal upper extremity function. Instability occurs when injury or a pathological condition alters balance, typically in the basketball players and handball players. Injury also can occur whenever the ligaments are subjected to tensile forces that exceed their physiological capacities. For example, thumb is stable in extension when ulnar collateral ligament sprain or tear and gamekeepers thumb injury happen but it appears to be unstable inflexion. The goal was to decrease this injury to a region. Materials and Methods: Precise hand positioning on 5 Basketball is drawn for guides to teach proper shooting technique by using Morley theorem proof results (figure1). Lemma: if points z. y. z. y follow conditions: zy = zy = yz. yzy = zyz = π − 2α > π then will be located on a circle. Circleayzyz is as a Basketball and points are as Head of distal phalanges. If angle size between finger number 1 and finger number 2 and finger number 5 equals3α , this size between finger number 2 and finger number 3 and between finger number 3 and finger number 4 equals2α , It is same for between finger number 4 and finger number 5; as a result, force that comes from the Basketball brings the least damage in the fingers and thumb. Results: The alpha angle size is different in Athletes; but according to the theorem, it must be less thanπ. Our Studies have found alpha angle normal size (about15°) and relation between alpha angle size and Distance between fingers in basketball players follow: sin (π − α) +3 1 sin(π−α)3. When the alpha equals 15, the equation is 2.12 cm, by considering this result shooting range increase and damages is in lowest point. Conclusion Results also can be used in evaluations. ligament laxity maybe reduced by using Mathematical model.

Upward Morley's theorem downward

By a celebrated theorem of Morley, a theory T is ℵ1‐categorical if and only if it is κ‐categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem “to the finite”. In more detail, we are presenting conditions, implying that certain finite subsets of certain ℵ1‐categorical T have at most one n‐element model for each natural number (counting up to isomorphism, of course).

A note on Morley's triangle theorem

In this note, we offer a proof of a variant of Morley's triangle theorem, when the exterior angles of a triangle are trisected. We also offer a generalization of Morley's theorem when angles of an n-gon are n-sected.

The Morley tricorn a solid figure constructed from the diagram for Morley's theorem

Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1.Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.

On Morley's theorem, determinantal identities and the QR algorithm

Three results are shown in this contribution. First, the diagram used in the proof of Morley's theorem is shown to provide a template for the creation of a closed solid figure. Second, it is shown how a proposition governing a class of determinants can be used to produce a rich supply of identities involving, e.g., Fibonacci and related families of numbers. Finally, it is shown in the case of an arbitrary 2 × 2 symmetric matrix A, how to obtain explicit expressions for all the elements of all the matrices involved in implementing the determination of the eigenvalues of A by means of the QR algorithm.

Model theory without choice? Categoricity

AbstractWe prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

Technology Tips: Investigating Bricard's Proof of Morley's Theorem with The Geometer's Sketchpad

The Geometer's Sketchpad®(GSP), the dynamic geometry software in use in mathematics classrooms for many years, provides a means of investigating classic mathematical theorems. In this article, we will describe our use of GSP to examine Bricard's (1922) proof of Morley's theorem as presented by Coxeter (1969).

A short proof of Morley's theorem

A short proof of Morley's theorem

Uncountable dense categoricity in cats

AbstractWe prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley's theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω, then it has a unique complete model of density character λ for every λ ≥ ω.

Notes on Quasiminimality and Excellence

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

Model theory: an introduction

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

Some results on permutation group isomorphism and categoricity

AbstractWe extend Morley's Theorem to show that if a theory is κ-p-categorical for some uncountable cardinal κ, it is uncountably categorical. We then discuss ω-p-categoricity and provide examples to show that similar extensions for the Baldwin-Lachlan and Lachlan Theorems are not possible.

Generalizing Morley's Theorem

AbstractWe study the categoricity of the classes of (possibly slightly saturated) elementary submodels of a homogeneous structure.

81.48 Two short proofs of Morley’s Theorem

81.48 Two short proofs of Morley’s Theorem

Basic principles of mechanical theorem proving in elementary geometries

At the end of 1976 and the beginning of 1977, the author discovered a mechanical method for proving theorems in elementary geometries. This method can be applied to various unordered elementary geometries satisfying the Pascalian Axiom, or to theorems not involving the concept of ‘order’ (e.g., thatc is ‘between’a andb) in various elementary geometries. In Section 4 we give the detailed proofs of the basic principles underlying this method. In Sections 2 and 3 we present the theory of well-ordering of polynomials and a constructive theory of algebraic varieties. Our method is based on these theories, both of which are based on the work of J. F. Ritt. In Section 5 we use Morley's theorem and the Pascal-conic theorem discovered by the author to illustrate the computer implementation of the method.

Some model theory of modules. II. on stability and categoricity of flat modules

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

Morley's Theorem and a Converse

Morley's Theorem and a Converse

Morley's Theorem and a Converse

Morley's Theorem and a Converse