α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Moving In-Frame

2022 ◽  
pp. 217-234
Author(s):  
Andi Johnson ◽  
Richard Lessey ◽  
Rebeca Ramos O'Reilly ◽  
Jessica Shi
Keyword(s):  
Active Learning ◽  
Reflective Practice ◽  
Subject Area ◽  
Lessons Learned ◽  
The Other ◽  
Learning Engagement ◽  
Digital Education ◽  
The Subject ◽  
Further Development

The researchers explored the dual experience of individuals who are both taking dance and movement classes digitally at the same time as they are also teaching dance and movement classes digitally. By focusing on this duality through a series of interviews with practitioners, the researchers explore how the learner/educators do or do not adapt one set of skills into the other area of work and the lessons learned from this reflective practice. The results are analyzed and broken down into five sections: glitches of the practice, reflective practice, active learning, engagement, and reframing communication. Through further analysis, the researchers explore possibilities for shifting the mindset around digital education methods. The researchers then offer suggestions for further development in the field and where further research can expand on the subject area.

scholarly journals XI. A fourth memoir upon quantics

1859 ◽  
Vol 9 ◽  
pp. 165-165
Keyword(s):  
Canonical Form ◽  
The Other ◽  
The Subject ◽  
Further Development

The object of the present memoir is the further development of the theory of binary quantics; it should therefore have preceded so much of my third memoir, vol. cxlvii. (1857), p. 627, as relates to ternary quadrics and cubics. The paragraphs are numbered continuously with those of the former memoirs. The first three paragraphs, Nos. 62 to 64, relate to quantics of the general form (*≬ x, y ,..) m , and they are intended to complete the series of definitions and explanations given in Nos. 54 to 61 of my third memoir; Nos. 68 to 71, although introduced in reference to binary quantics, relate or may be considered as relating to quantics of the like general form. But with these exceptions the memoir relates to binary quantics of any order whatever: viz. Nos. 65 to 80 relate to the covariants and invailants of the degrees 2, 3, and 4; Nos. 81 and 82 (which are introduced somewhat parenthetically) contain the explanation of a process for the calculation of the invariant called the discriminant; Nos. 83 to 85 contain the definitions of the catalectieant, the lambdaic and the canonisant, which are functions occurring in Prof. Sylvester’s theory of the reduction of a binary quantic to its canonical form; and Nos. 86 to 91 contain the definitions of certain covariants or other derivatives connected with Bezout’s abbreviated method of elimination, due for the most part to Professor Sylvester, and which are called Bezoutiants, Cobezoutiants, &c. I have not in the present memoir in any wise considered the theories to which the catalecticant &c. and the other covariants and derivatives just referred to relate; the design is to point out and precisely define the different covariants or other derivatives which have hitherto presented themselves in theories relating to binary quantics, and so to complete, as far as may be, the explanation of the terminology of this part of the subject.

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

François Venant. Enige aanvullingen

1997 ◽  
Vol 111 (3) ◽  
pp. 163-176
Author(s):  
J. Bruyn
Keyword(s):  
The Other ◽  
Personal Style ◽  
Fresh Start ◽  
Untimely Death ◽  
The Subject ◽  
Further Development

AbstractSince J. G. van Gelder was able to identify a number of works by François Venant (1591/92-1636) in 1938 (note 2) and Kurt Bauch and Astrid Tümpel added to these one painting and a drawing (notes 14 and 3), the artist has been known as one of the so-called Pre-Rembrandtists. Together with his contemporaries Claes Cornelisz. Moeyaert (c. 1590/91-1655) and Jacob Pynas (1592/93-after 1650) he was one of the younger artists of this group. Its style was dominated by Pictcr Lastman (1583-1633) and Jan Pynas (1581/82-1633), both of whom underwent the influence of Adam Elsheimer during their stay in Rome. Venant married a younger sister of Lastman in 1625. The latter's influence on his work had however set in well before that year. Jacob's Dream, signed and dated 161(7?) (note 10, fig. 2) testifies to this, as well as showing traces of Elsheimer's influence, possibly transmitted by Jan Pynas (notes 12 and 13, fig. 3). A somewhat later signed work, David's parting from Jonathan (note 5, fig.1), closely follows Lastman's version of the subject of 1620 (note 6) though the grouping of the two figures may be taken as typical of Venant's personal style. In an unsigned picture of Gideon's Scacrifice, which may also be dated to the early 1620s (note 14, fig. 4), the artist once more makes use of motifs from various works by Lastman. Two undated drawings would seem to represent a slightly later stage in the artists's development. The Baptism of the Eunuch (notes 16 and 18, fig. 5) betrays the attempt to emulate Lastman's pictures on the subject, especially one of 1623 (note 17), by enhancing the dramatic actions in the scene, and so does Gideon's Sacrifice (note 20, figs. 6 and 8), which seems to be based on Lastman's Sacrifice of Monoah of 1627 (note 21, fig.7). To these works, spanning a period from 1617 (?) to the late '20s, may be added two more, another drawing and a painting. The drawing of Daniel at Belshazzar's Feast was formerly attributed to Lastman (notes 25-33, figs. and 10). While the technique, notably the use of wash, differs from that in the drawings mentioned above, the similarities to these in linear rhythm and conception are such that they may all be attributed to the same hand. The technical differences may be accounted for by assuming a slightly later date and, more particularly, a different purpose; whereas the other drawings were in all likelihood self-contained products, Belshazzar's Feast appears to be a sketch for a painting. The last phase of Venant's career seems to be represented by the largest painting known to us and the only one on canvas, Elisha Refusing Naäman's Gifts (note 34, fig. 11). It shows the artist disengaging himself from Lastman at last, possibly after the latter's death in 1633. While the composition is still reminiscent of his carlier work, here Venant seems to have made a fresh start by allowing study from life to play a more important role than before. The landscape differs radically from earlier backgrounds and may well have been influenced by Barholomeus Breenbergh, who returned from Italy around 1630 and whose influence may also be detected in the heavy wash that marks the Belshazzar drawing. The artist's further development was cut short by his untimely death, probably of the plague, in 1636.

The uniform regular set theorem in α-recursion theory

1978 ◽  
Vol 43 (2) ◽  
pp. 270-279 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
Typical Situation ◽  
Recursively Enumerable ◽  
Constructible Sets ◽  
Admissible Ordinals ◽  
Special Case

Several new features arise in the generalization of recursion theory on ω to recursion theory on admissible ordinals α, thus making α-recursion theory an interesting theory. One of these is the appearance of irregular sets. A subset A of α is called regular (over α), if we have for all β < α that A ∩ B ∈ Lα, otherwise A is called irregular (over α). So in the special case of ordinary recursion theory (α = ω) every subset of α is regular, but if α is not a cardinal of L we find constructible sets A ⊆ α which are irregular. The notion of regularity becomes essential, if we deal with α-recursively enumerable (α-r.e.) sets in priority constructions (α-r.e. is defined as Σ1 over Lα). The typical situation occurring there is that an α-r.e. set A is enumerated during some construction in which one tries to satisfy certain requirements. Often this construction succeeds only if we can insure that every initial segment A ∩ β of A is completely enumerated at some stage before α. This calls for making sure that A is regular because due to the admissibility of α an α-r.e. set A is regular iff for every (or equivalently for one) enumeration f of A (f is an enumeration of A iff f: α → A is α-recursive, total, 1-1 and onto) we have that is the image of the set σ under f).

scholarly journals On Legendre's functions P n ( θ ), when n is great and θ has any value

1916 ◽  
Vol 92 (642) ◽  
pp. 433-437
Keyword(s):  
Approximate Formula ◽  
The Other ◽  
Mathematical Methods ◽  
Other Hand ◽  
Great Generality ◽  
Associated Functions ◽  
Complete Series ◽  
The Subject ◽  
Comparison Of The Results ◽  
Further Development

As is well known, an approximate formula for Legendre's function P n (θ), when n , is very large, was given Laplace. The subject has been treated with great generality by Hobson, who has developed the complete series proceeding by descending powers of n , not only for P n but also for the "associated functions." The generality aimed at by Hobson requires the Use of advanced mathematical methods. I have thought that a simpler derivation, sufficient for practical purposes and more within the reach of physicists with a smaller mathematical equipment, may be useful. It had, indeed, been worked out independently. The series, of which Laplace's expression constitutes the first term, is arithmetically useful only when nθ is at least moderately large. On the other hand, when θ is small, p n tends to identity itself with the Bessel's function J 0 ( nθ ), is was first remarked by Mehler. A further development of this approximation is here proposed. Finally, a comparison of the results of the two methods of approximation with the numbers calculated by A. Lodge for n = 20 is exhibited.

The slow-growing and the Graegorczyk hierarchies

1983 ◽  
Vol 48 (2) ◽  
pp. 399-408 ◽  
Author(s):  
E.A. Cichon ◽  
S.S. Wainer
Keyword(s):  
Initial Segment ◽  
Elementary Proof ◽  
The Other ◽  
Precise Description ◽  
Limit Ordinal ◽  
Rates Of Growth ◽  
Positive Integers ◽  
Slow Growing ◽  
Initial Choice ◽  
Crucial Parameter

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.

Semantics of the infinitistic rules of proof

1976 ◽  
Vol 41 (1) ◽  
pp. 121-138
Author(s):  
Krzysztof Rafal Apt
Keyword(s):  
Completeness Theorem ◽  
Recursion Theory ◽  
Point Of View ◽  
First Order ◽  
Order Language ◽  
The Subject ◽  
Order Arithmetic ◽  
Additional Rule ◽  
First Time ◽  
Further Development

This paper is devoted to the study of the infinitistic rules of proof i.e. those which admit an infinite number of premises. The best known of these rules is the ω-rule. Some properties of the ω-rule and its connection with the ω-models on the basis of the ω-completeness theorem gave impulse to the development of the theory of models for admissible fragments of the language . On the other hand the study of representability in second order arithmetic with the ω-rule added revealed for the first time an analogy between the notions of re-cursivity and hyperarithmeticity which had an important influence on the further development of generalized recursion theory.The consideration of the subject of infinitistic rules in complete generality seems to be reasonable for several reasons. It is not completely clear which properties of the ω-rule were essential for the development of the above-mentioned topics. It is also worthwhile to examine the proof power of infinitistic rules of proof and what distinguishes them from finitistic rules of proof.What seemed to us the appropriate point of view on this problem was the examination of the connection between the semantics and the syntax of the first order language equipped with an additional rule of proof.

scholarly journals Further researches on the extrusion of granules by trypanosomes and on their further development

1913 ◽  
Vol 86 (589) ◽  
pp. 377-389 ◽  
Keyword(s):  
Royal Society ◽  
The Other ◽  
Preliminary Note ◽  
The Subject ◽  
Further Development

In March, 1911, in the course of some work on trypanosomes carried out at the Wellcome Tropical Research Laboratories, Khartoum, the extrusion of certain granules from trypanosomes was observed by one of us (W. B. F.). The Director of the Laboratories, Dr. Andrew Balfour, was informed of these observations, and he himself shortly after observed a somewhat similar extrusion of granules from Spirochætes (spirochætosis of fowls), an account of which he published. In June, 1911, a preliminary note on the subject was communicated to the Royal Society by one of us (W. B. F.). Since then, a great deal of work has been done on the subject by us conjointly, but for the most part independently; by one of us (W. B. F.) at Khartoum and in London, by the other (H. S. R.) at Yei in the Lado Enclave.

Σ2 Induction and infinite injury priority argument, Part I: Maximal sets and the jump operator

1998 ◽  
Vol 63 (3) ◽  
pp. 797-814 ◽  
Author(s):  
C. T. Chong ◽  
Yue Yang
Keyword(s):  
Recursion Theory ◽  
Central Position ◽  
Jump Operator ◽  
Concerted Effort ◽  
Injury Type ◽  
Recursively Enumerable ◽  
The Subject ◽  
Intensive Development ◽  
Priority Argument ◽  
Fragments Of Peano Arithmetic

The study of recursion theory on models of fragments of Peano arithmetic has hitherto been concentrated on recursively enumerable (r.e.) sets and their degrees (with a few exceptions, such as that in [2] on minimal degrees). The reason for such a concerted effort is clear: priority arguments have occupied a central position in post Friedberg-Muchnik recursion theory, and after almost forty years of intensive development in the subject, they are still the essential tools on which investigations of r.e. sets and their degrees depend. There are two possible approaches to the study within fragments of arithmetic: To give a general analysis of strategies, and identify their proof-theoretic strengths (for example in [6] on infinite injury priority methods), or to consider specific theorems in recursion theory, and, if possible, pinpoint the exact levels of induction provably equivalent to the theorems. The work reported in this paper belongs to the second approach. More precisely, we single out two infinitary injury type constructions of r.e. sets—one concerning maximal sets and the other based on the notion of the jump operator—to be the topics of study.

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