bluuebunny/crossref_metadata_2025_split huggingface dataset:
Self-verifying axiom systems, the incompleteness theorem and related reflection principles
Authors: Willard Dan E. 
Published: May 2006

AbstractWe will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the “Tangibility Reflection Principle”. We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further.
Logics of belief change without linearity
Authors: Cantwell John 
Published: May 2006

AbstractEver since [4]. systems of spheres have been considered to give an intuitive and elegant way to give a semantics for logics of theory- or belief- change. Several authors [5, 11] have considered giving up the rather strong assumption that systems of spheres be linearly ordered by inclusion. These more general structures are calledhypertheoriesafter [8]. It is shown that none of the proposed logics induced by these weaker structures are compact and thus cannot be given a strongly complete axiomatization in a finitary logic. Complete infinitary axiomatizations are given for several intuitive logics based on hypertheories that are not linearly ordered by inclusion.
A note on the isomorphism problem for SK[G]
Authors: Chatzidakis Zoé , Pappas Peter 
Published: May 2006

Let G be an infinite abelian p-group and let K be a field of characteristic ≠ p. Let K[G] be the set of all sums Σg∈Gagg where the ag are in K, and all but finitely many ag are 0. Then K[G] is a K-algebra, with multiplication induced by the group multiplication on G.If G is countable, then the isomorphism type of K[G] has been completely described by S. D. Berman [1]. If G is a direct sum of countable groups, one can also describe K[G], as K[⊕iGi] ≃ ⊗iK[Gi]. If K contains all pn-th roots of unity, then K[G] is isomorphic to the ring of continuous functions from a Boolean space X to the field K with the discrete topology. In that case, the group UK[G] of invertible elements of K[G] is isomorphic to the direct sum of ∣G∣ copies of K×. More generally, if K is of the second kind with respect to p (see below for the definition), the group UK[G] has a simple description.Consider the subgroup SK[G] of elements Σgagg which have order a power of p and such that Σg ag = 1. This group is of course much simpler than UK[G]. Classifying SK[G] up to isomorphism reduces to the case where G has no element of infinite height, see [7]. If G is a direct sum of cyclic groups then the isomorphism type of SK[G] has been completely determined, in [2, 3, 7, 8]. The aim of this note is to show that a similar result is in general not possible for uncountable G. We use an invariant Γ associated to abelian groups, and for any regular uncountable cardinal κ, exhibit 2κ groups G for which Γ(G) = Γ(SK[G]) are pairwise distinct. Our work is based on a construction of Shelah [9], who constructed 2κ non-isomorphic abelian p-groups of cardinality κ for κ an uncountable regular cardinal.
Minimality and completions of PA
Authors: Knight Julia F. 
Published: May 2006

The results in this paper say that natural upper bounds for sets of degrees associated with theories and models of arithmetic cannot be minimal. The basic new result says that for any completion T of PA, there is another completion S such that S<TT and Rep(S) = Rep(T). This immediately implies that deg(T) is not minimal over {deg(X): X ∈ Rep(T)}. As an application of the basic result, we obtain the fact that if is a non-standard model of TA (true arithmetic), then deg () cannot be minimal over {deg(X): X is arithmetical}. More generally, if is a non-standard model of an arbitrary completion T of PA, then deg() cannot be minimal over {deg(): X ∈ Rep(T)}. We vary the basic result, making S′ ≡TT′. As an application of the variant, we obtain the fact that if is a non-standard model of PA, then {deg(): ≅ } has no minimal element.The remainder of the present section gives a brief account of the background needed for the basic new result and the variant. These two results are proved in Section 2. The applications are given in Section 3, along with further background needed for the applications. One important source of ideas used in the present paper is a paper of Scott [9]. In addition, there are ideas taken from Tennenbaum [12], Feferman [3], Marker [6], [7], and Solovay. Chapter 19 of [1] gathers together most of this material. In fact, it contains all that is really essential. In one application, we appeal to Solovay's result on degrees of models of an arbitrary completion of PA, a result which is not completely proved in [1]. However, for the best application, which implies all the others, we use only some ideas from the proof of Solovay's theorem. These are given in lemmas that are proved in Section 3, where they are needed, or taken from [1], While the proof of Solovay's result requires an infinitely nested priority construction, our best application rests on nothing more than finite-injury constructions.
Simplified axioms for many-valued quantification theory
Authors: Turquette Atwell R. 
Published: May 2006

The basic aim of the present paper is to simplify the axioms for many-valued quantification theory which were developed by Rosser and Turquette in Many-valued logics ([10], pp. 33–34 and pp. 63–64). The simplification which is achieved results not only from a reduction in the number of axiom schemes and rules of inference, but also by obtaining greater formal similarity to some elegant axioms for standard 2-valued quantification theory. This result will not be accomplished by establishing the formal interdeducibility between our simplified axioms and the Rosser-Turquette set. On the contrary, it will be shown that these two sets of axioms are not formally interdeducible.What will be shown is that whenever the Rosser-Turquette axioms are used to define a plausible set of statements, this set of statements can be defined using our simplified axioms. The exact meaning of “plausible” will become clear from what follows, so for the moment it will be sufficient to remark that a set of statements is plausible if and only if each statement of the set is analytic in the sense that the statements take designated truth-values exclusively. In general, a Rosser-Turquette set of axioms would be used to define a set of plausible statements, and in such a case our simplified axioms could be used to achieve the same result.
Mathematical significance of consistency proofs
Authors: Kreisel G. 
Published: May 2006

The principal aim of the present paper is to sketch some mathematical applications of the work of the Hilbert school in the foundations of mathematics. Most of them depend on theε-theorems of Hilbert-Bernays II [6], or of Ackermann [1], but instead the work of Gentzen [5] or Schütte [25], which does not use theε-symbol, could have been applied.The avowed purpose of all this work is described in the introduction to volume I of Hilbert-Bernays: the consistency of the usual principles of proof is to be established by means of finitist methods, or at least, by means of methods which are more “evident” or more “constructive” than the principles under discussion. This formulation seems to have several defects:(1) Since the notion of constructive proof is vague, the whole formulation of the program is vague; and though an exact formulation constitutes, of course, an interesting problem for the logician, because of this vagueness the mathematician does not find the program attractive.(2) The formulation does not cover too well the actual substance of the material contained in [6]; e.g. theε-theorems for the predicate calculus go far beyond establishing mere consistency of the predicate calculus. It is significant that it is precisely these theorems which lead to the more interesting applications, e.g. the use of theε-theoremswithequality in the solution of Hilbert's 17th problem given below.
The classical and the ω-complete arithmetic
Authors: Grzegorczyk A. , Mostowski A. , Ryll-Nardzewski C. 
Published: May 2006

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:(1) 0, 1, and the number variables are nfs.
Extensions of some theorems of Anderson and Belnap
Authors: Rose Alan 
Published: May 2006

The object of this note is to extend to Post's m-valued propositional calculus1 the axiomatization by Anderson and Belnap2 of the 2-valued propositional calculus.Notation and definitions. We shall use the symbols ∼ and v for the primitives of Post and shall otherwise, except as stated below, use the same notation and definitions as in the previous paper.
An unsolvable problem in number theory
Authors: Putnam Hilary 
Published: May 2006

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form
A Novel Lipoxygenase in Pea Roots. Its Function in Wounding and Biotic Stress
Authors: Veronico Pasqua , Giannino Donato , Melillo M. Teresa , Leone Antonella , Reyes Aurelio , Kennedy Malcolm W. , Bleve-Zacheo Teresa 
Published: May 2006

Abstract
The genome of pea (Pisum sativum) contains genes encoding a family of distinct lipoxygenases (LOX). Among these, LOXN2 showed eight exons encoding a 93.7-kD enzyme, harboring two C-terminal deletions and an unusual arginine/threonine-tyrosine motif in the domain considered to control the substrate specificity. LOXN2, when overexpressed in yeast, exhibited normal enzyme activity with an optimum at pH 4.5, and a dual positional specificity by releasing a 3:1 ratio of C-9 and C-13 oxidized products. The predicted LOXN2 structure lacked a loop present in soybean (Glycine max) LOX1, in a position consistent with control of the degree of substrate access to the catalytic site and for LOXN2's dual positional specificity. The LOXN2 gene was tightly conserved in the Progress 9 and MG103738 genotypes, respectively, susceptible and resistant to the root cyst nematode Heterodera goettingiana. LOXN2 transcription was monitored in roots after mechanical injury and during nematode infection. The message peaked at 3 and 24 h after wounding in both genotypes and was more abundant in the resistant than in the susceptible pea. In nematode-infected roots, transcription of several LOX genes was triggered except LOXN2, which was repressed in both genotypes. In situ hybridization revealed that LOXN2 message was widespread in the cortex and endodermis of healthy roots, but specifically localized at high level in the cells bordering the nematode-induced syncytia of infected roots. However, LOXN2 transcript signal was particularly intense in collapsing syncytia of MG103738 roots, suggesting LOXN2 involvement in late mechanisms of host resistance.
The Influence of Political Platforms on Legislation in Indiana, 1901–1921
Authors: Berry Burton Y. 
Published: May 2006

Numerically the political platform touches about six-tenths of one per cent of the legislation introduced in an average legislature. For instance, in the 1911 Indiana legislature, there were 1,111 bills introduced, while only six bills originated in the political platform of the controlling, in this case the Democratic, party. This revelation, although startling at first, is somewhat deceiving. In the first place, the political platform does not attempt, nor do the party leaders desire it, to touch upon all phases of legislation. There are various kinds of legislation, such as the relocation of county seats, weed laws and individual relief laws, which are too local or trivial to be included in the state platform, and there are other types, such as anti-liquor legislation, which the platform avoids because of their magnified importance.Although the percentage given is correct, this six-tenths of one per cent is about twenty-five per cent of the important legislation considered. In this study of the twenty year period in Indiana, beginning with the inauguration of Governor Durbin in 1901 and through the successive administrations and ending with the first legislature under Governor McCray in 1921, we are interested in those platform planks alone which positively pledge the party to legislation.
Village Government in New England
Authors: Bates Frank G. 
Published: May 2006

Throughout the United States there are provided special, though simple, forms of municipal government for villages under the various names of village, borough or town. Oddly enough the notable exceptions to this general practice are to be found in New England, the region which includes three of the most densely populated states in the Union. Moreover, it is in the two most densely populated of these that the least progress has been made toward the development of an orderly system of village government.At one time the whole area of New England, except certain unorganized tracts in the north, was under town government and so continued until the growth of urban conditions led to city incorporation. In every case incorporation was by special act, and such is the method employed in every case to the present time. The readiness of the town government to assume the functions proper to an urban community not only retarded city incorporation, but prevented, in large measure, the growth of special forms of government for villages. It is not unusual to find towns which include within their limits many square miles of rural territory, providing all or part of its population with fire protection, water, sewers, lights, side-walks, parks and libraries. But there has grown up, sometimes under general laws though more commonly as a result of special legislation, a heterogeneous collection of municipal governments and taxing authorities variously denominated villages, fire, water, lighting, sewer, highway and improvement districts, producing a confusion comparable only to the conditions in England before the municipal reforms of recent date.
Governmental Reorganization in Illinois
Authors: Fairlie John A. 
Published: May 2006

In its broad outlines the state government of Illinois resembles that of most of the American States—but with the most recent tendencies as yet only slightly developed. The present state constitution, adopted in 1870, illustrates the political ideas prevalent in the middle of the nineteenth century—the election of all classes of public officials, the disintegration of the executive branch of the government, and the distrust of the legislature, and the provisions of this constitution are stereotyped by an amending article under which alterations have proved almost impossible. There has, however, been an increasing development of administrative authorities, created by statute, and added one to another with almost no attempt at systematic organization. From 1909 to 1913, there were not less than 34 new state offices, boards and commissions established, and the total number of such executive agencies is now approximately 130.Under the present arrangements, there is no correlation between related offices, and little or no effective supervision; there is no budget system and no adequate accounting system. As a result the public administration is inefficient and wasteful; it fails to furnish the general assembly with satisfactory advice on legislation; and there is no clearly defined responsibility for the conduct of public affairs; appropriations and expenditures have been rapidly increasing, the appropriations in 1913 for two years amounting to nearly $38,000,000.
Social Order And Political Authority
Authors: Dickinson John 
Published: May 2006

The first of these papers carried a warning against conceiving social order in an absolute sense. It does not mean suppression of all forms of competition and defeat, but only the protection of selected human interests at the expense of others. Even taking order in this limited sense, it was found that custom and voluntary adjustment furnishan incomplete apparatus for its furtherance. The effectiveness of custom was seen to be connected with the auxiliary activity of an organ of leadership to serve as a focus for the radiation of custom, and a pivot for change. Voluntary adjustment, while capable of resolving many interest-conflicts beyond the reach of custom, was also found to rely largely on the initiatory activity of an organ of mediation. Finally, we have thus far considered social order primarily from the point of view of the prevention or adjustment of direct conflicts of interests between individuals.The functioning of a social group requires more than that its members should refrain from what other members resent as invasions of their interests. If the advantages of group life are to be reaped, it is necessary that at many points the group should, aswe say, act as a unit, which means that each member should so shape his conduct with reference to the conduct of others that the acts of all, when thus geared together, will produce an aggregate or group result. This gearing together of the acts of separate individuals is what is commonly called “coöperation.” Coöperation has value only as it furthers interests which belong to individuals.
The Investigative Function of Congress
Authors: Galloway George B. 
Published: May 2006

In the winter and spring of 1923–24 a blizzard of congressional investigations swept the national capital. Party lines were tied up, statesmen were snowed under, and Washington officialdom was chilled by a storm of inquiries into the conduct of government departments. Scarcely a corner of the administration escaped inquisition. The subjects of inquiry ranged from the naval oil reserve leases at Teapot Dome and Elk Hill to the administration of the Veterans' Bureau under Director Forbes of dubious fame. Upwards of two-score inquests were instituted by Congress, through its committees, into the official behavior of the executive branch of the national government. In the end, and as a direct result, two cabinet members lost their portfolios.Such wide and effective use by Congress of its committees for the purpose of inquiry and examination of administrative conduct raised certain questions in the minds of students of government. What were the constitutional limits of the investigative function of Congress? What was the relation of this function to the law-making function and to impeachment? Was punishment by impeachment a satisfactory remedy for official misconduct, or had it become obsolete as too cumbersome? Did the vote of the Senate calling for Secretary Denby's resignation usher in a new process in this country? Should cabinet officials be required to appear on the floor of Congress and answer questions? Would this be preferable to investigating their actions long after they have been performed? Does the steady increase of federal powers necessitate greater supervision by Congress over the President and administrative officers? Have students of politics overlooked a function of Congress frequently and forcibly exercised for the control of administration?
Greece's Experiment with Proportional Representation
Authors: Polyzoides Adamantios Th. 
Published: May 2006

On November 7 Greece held its first general election under the system of proportional representation, using a modified form of the Belgian system. This innovation was imposed on the country despite the most strenuous opposition by all of the old parties, the majority of the press, and the bulk of public opinion, and its adoption was a clear victory of the minority parties, assisted by the Military League and the then dictator General Kondylis.The arguments of the established parties in favor of the old plurality system ran on lines too familiar to require extensive statement here. The former system, according to its supporters, usually assures the election of large majorities, one way or the other, and enables Parliament to give the country what we call a strong government, such as Greece needed at the time of the election. Great Britain and the United States were offered as the outstanding examples of the efficiency of the two-party system, which is best served by the old-fashioned electoral method of absolute plurality. Naturally enough, Belgium was cited as the worst exponent of the evils of proportional representation.
Revolutionary Communism in the United States
Authors: Watkins Gordon S. 
Published: May 2006

On October 15, 1919, a press dispatch from Buffalo, New York, indicated that the primary election returns in that city gave an average of about three hundred votes to the three candidates whose names appeared on the ballot as representatives of the Communist party, the protagonist of the soviet form of government in the United States. The total number of votes cast was 54,000, which shows that the Communist vote was insignificant, numerically. Newspapers ridiculed the diminutive radical vote. All phenomena have a genesis, and it is the fact that a Communist ballot was cast rather than the quantitative character of the vote that has significance for the student of political philosophy and action. To those who saw the revolutionary left wing of the old Socialist party organize the Communist party and the Communist Labor party in Chicago during the first week of September, 1919, this initial appearance of the revolutionary Communists in American political life presages important developments. The event was heralded by revolutionists in this country and in Europe as the emergence of a new era in the political and economic life of the United States. The widespread dissemination of ultra-radical propaganda in connection with recent strikes is further evidence of the revolutionary purposes of Communists in America.Optimistic prophecies are prevalent to the effect that bolshevism will find no fertile soil in the United States, since American workmen are too prosperous to become susceptible to revolutionary political and industrial philosophy. Similar predictions were voiced even on the floor of the Socialist emergency convention and the Communist Labor party convention last September.