bluuebunny/crossref_metadata_2025_split huggingface dataset:
Review of von Juhos 1930: The problem of mathematical probability (1931g)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The object of this monograph is to present an analysis of the notions and fundamental principles underlying the calculus of mathematical probabilities, and in particular to purge them of all empirical impurities. To attain this goal (that is, a purely logical construction), the physical events (drawing of balls, etc.) considered in the calculus of probabilities are thought of as being replaced by “arbitrary acts of choice”, which are supposed to be “absolutely random events that are not spatio-temporally conditioned (hence unreal)” and therefore “purely logical constructs”. Mathematical probability is ultimately defined as a fraction associated with certain proposen Aussagefunktionen zugeordneter Bruch, namlich das Verhaltnis der Anzahl derjenigen Argumente, welche die betreffende Aussagefunktion zu einem wahren Satz machen, zur Anzahl derjenigen, welche sie zu einem sinnvollen Satz machen. In einem Anhang wird auf das Verhii.ltnis zum physikalischen Wahrscheinlichkeitsbegriff eingegangen.
A special case of the decision problem for theoretical logic (1932a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In connection with the method used to prove the completeness of the functional calculus (1930), one can develop a procedure that, for each formula with a normal form of the form permits us to decide whether it is satisfiable. First, one reduces the problem in a well known way to formulas of the form zuriick.
A property of the realizations of the propositional calculus ( 1932c)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In answer to a question that Menger asked me in a letter, one can demonstrate the following theorem:
Let any set6 of objects p, q, r, ... be given on which there are defined a unary operation ∼p and a binary operation p ::) q that do not lead outside the set 6.
Review of Skolem 1931 : On some propositional functions 1n arithmetic (1932d)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The author considers those propositional functions and statements that are built up by means of the five primitive logical operations (not, or, and, all, there exists) from relations of the form A = B and A &lt; B, where A and Bare number-theoretic functions that result from (arbitrarily iterated) application of the three operations of addition, multiplication by a rational number, and formation of the next smaller whole number.
Review of Church 1932 : A set of postulates for the foundation of logic (1932k)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
A formal system for logic and mathematics is presented that is supposed to avoid the antinomies, without use of the theory of types or reliance upon the axiomatic construction of set theory, but in another way, namely, by restricting the principle of the excluded middle. For example, while in this system the set of all sets that do not contain themselves (or the corresponding propositional function) can be formed, it cannot be proved that it either does or does not contain itself. Moreover, the proposition that a hypothesis from which a contradiction follows is false does not hold generally. It has not yet been established, according to the author, whether all antinomies (for example, that of Burali-Forti) are actually avoided in his system and whether it suffices for the construction of mathematics; but in any case he hopes to be able to achieve these goals by making a few modifications. In its present form the system contains eight primitive symbols (in addition to the variables), thirty-five axioms and five rules of inference. In conformity with the view that propositional functions can be neither true nor false (that is, be meaningless) for certain values of the variables, the notion of universality appears as a two-place operator II(a,b), with the meaning that, for all x for which a(x) holds, b(x) also holds, so that “for all” always relates only to a restricted domain a of individuals.
Review of Skolem 1932: On symmetric general solutions in the calculus of [Boolean] identities (1932n)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Let be an equation in the calculus of classes with undetermined coefficients ae and unknowns Xk- A general solution of (1) is said to be symmetric with respect to a group G of permutations, of the ae and Xi, that transform (1) into itself if for each S of G there is a permutation T of the Ui such that S and T, if applied simultaneously to the system of equations (2), will transform it into itself. If, in addition, Xi = gi{ae,xk} holds for all solutions Xi of (1) and if, furthermore, the permutation T corresponding to some S permutes the Ui in the same way as that in which S permutes the Xi, then the solution is said to be normal. It is shown that: 1. An equation with constant coefficients (without ae) has a symmetric general solution with respect to G if and only if it remains solvable when the Xi are equated within each system of transitivity [orbit] into which G splits the Xi- 2. If undetermined coefficients ae occur, let EAAJ A{xk} be the Boolean expansion of (1) in terms of the ae (the A being the constituents) and let GA be the subgroup of G that maps the A into themselves.
Review of Dingler 1931 : Philosophy of logic and arithmetic (19320)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
This book is principally concerned with the essential nature of mathematical calculi. In the author’s view, certain “basic capabilities”, not further analyzable, are required for the construction and manipulation of those calculi; they are opposed, as “active logic”, to the “passive logic” of the calculus itself (corresponding approximately to Hilbert’s distinction between metamathematics and mathematics). The nature and manner of use of these basic capabilities (especially in the construction of arithmetic) are set forth, and various epistemological questions are considered from this perspective. As a concrete application of the general discussion, there follows an investigation of a formal system of arithmetic, which is supposed to furnish the consistency proof sought by Hilbert, but which, in truth, could at best demonstrate the trivial fact that the elementary rules of computation (distributive law, and so on) are consistent.
Über die Waldsche Axiomatik des Zwischenbegriffes (1999c)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In seiner Axiomatik des Zwischenbegriffes (1991 und 1992) beweist
Wald, daf3 drei Punkte I a, b, e in einer gewissen Bedingungen geniigenden
Relation aller Punktetripel aller metrischen Raume dann und nur dann stehen, wenn fiir ihre Abstande die Beziehung ab + be = ae gilt, d. h. wenn b im Mengerschen Sinne zwischen a und e liegt. Unter den der Relation auferlegten Bedingungen tritt dabei die Kongruenzinvarianz auf, d. h. wenn drei Punkte a, b, e in der betreffenden Relation stehen, so sollen auch je drei mit ihnen kongruente Punkte a’, b’, e’ in der betreffenden Relation zu einander stehen.
Introductory note to 1933e
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Gödel’s paper 1999e, given at Menger’s colloquium in Vienna on 28 June 1932, deals with the relationship between classical and intuitionistic first-order arithmetic and shows how to interpret the former within the latter by a simple translation.
Zur intuitionistischen Arithmetik und Zahlentheorie (1933e)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
LaBt man den Grundbegriffen des Heytingschen Aussagenkalkiils1 die gleich bezeichneten des klassischen und der “Absurditat” (,) die Negation (∼) entsprechen, so erscheint der intuitionistische Aussagenkalkiil H als echtes Teilsystem des gewohnlichen A. Bei anderer Zuordnung (Übersetzung) der Begriffe ist jedoch umgekehrt der klassische ein Teilsystem des intuitionistischen. Denn es gilt: Jede nur aus Konjunktionen (/\) und Negationen (,) aufgebaute in A giiltige Formel ist auch in H beweisbar.
On intuitionistic arithmetic and number theory (1933e)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
If to the primitive notions of Heyting’s propositional calculus1 we let correspond those notions of the classical propositional calculus that are denoted by the same sign and if to absurdity (,) we let correspond negation (∼), then the intuitionistic propositional calculus H turns out to be a proper subsystem of the ordinary propositional calculus A. With another correlation (translation) of the notions, however, the classical propositional calculus is, conversely, a subsystem of the intuitionistic one.
Eine Interpretation des intuitionistischen Aussagenkalkiils (1933!)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Man kann den Heytingschen Aussagenkalkiil mittels der Begriffe des gewohnlichen Aussagenkalkiils und des Begriffes “p ist beweisbar” {bezeichnet mit Bp) interpretieren,1 wenn man fiir den letzteren das folgende Axiomensystem annimmt:
Aufierdem sind fiir die Begriffe -&gt;, ∼, . , V die Axiome und Schlufiregeln des gewohnlichen Aussagenkalkiils anzunehmen, ferner die neue Schlufi.. regel: Aus A darf auf BA geschlossen werden.
Zurn Entscheidungsproblem des logischen Funktionenkalkiils (1999i)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In Ergebnisse eines mathematischen Kolloquiums (1992a) habe ich ein Verfahren kurz skizziert, nach dem man fiir jede Formel des engeren Funk-tionenkalkiils,1 die in der Normalform nur zwei (und zwar benachbarte) Allzeichen enthalt, entscheiden kann, ob sie erfiillbar2 ist. Denselben Fall des Entscheidungsproblems hat dann L. Kalmar nach der gleichen Methode ausfiihrlich behandelt.3 In dem friiher von P. Bernays, M. Schonfinkel und W. Ackermann behandelten Fall eines Allzeichens hat sich heraus-gestellt, da6 solche Formeln, wenn sie iiberhaupt erfiillbar sind, schon in einem endlichen Individuenbereich erfiillbar sind. Das Ziel der folgenden Untersuchung ist, dies auch fiir den Fall zweier Allzeichen nachzuweisen. Zweitens soil gezeigt werden, da6 die Losung des nachst komplizierteren Falles (drei Allzeichen) bereits mit der Losung des ganzen Entscheidungs-problems aquivalent ware (vgl. S. 441).
On the decision problem for the functional calculus of logic (1933i)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The introductory note to 1999i, as well as to related items, can be found on page 226, immediately preceding 1992a.In Ergebnisse eines mathematischen K olloquiums ( 1992a) I briefly sketched a procedure by which one can decide, for each formula of the restricted functional calculus1 that in normal form contains only two ( and in fact adjacent) universal quantifiers, whether it is satisfiable.2 L. Kalmar then dealt with that case of the decision problem in detail by using the same method.3 In the case of one universal quantifier, a case which had been treated earlier by P. Bernays, M. Schonfinkel and W. Ackermann, it turned out that such formulas, if they are satisfiable at all, are satisfiable already in a finite domain of individuals. The goal of the investigation that follows is to prove this also for the case of two universal quantifiers. Secondly, it will be shown that solving the next more complicated case ( three universal quantifiers) would already be equivalent to solving the full decision problem (see below, page 322).
On undecidable propositions of formal mathematical systems (1934)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
A formal mathematical system is a system of symbols together with rules for employing them. The individual symbols are called undefined terms. Formulas are finite sequences of the undefined terms. There shall be defined a class of formulas called meaningful formulas, and a class of meaningful formulas called axioms. There may be a finite or infinite number of axioms.
Introductory note to 1994b
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In this review of my earliest article but one, Gödel’s last sentence can be puzzling. Given just µ classes, and v &gt; µ, how can Gödel speak of a product of v of them? The answer lies in the formalism of Principia mathematica. Definitions are couched in terms of variables whose values are not restricted to the ranges for which the definitions were wanted, and meanings thus accrue to the waste cases that made no intuitive sense. They are the “don’t-cares”, but evidently I cared a little, and Gödel indulged me.
Introductory note to 1934c and 1935
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
These are reviews by Gödel of Skolem’s papers, 1933a and 1934, on non-standard models of arithmetic. As Gödel remarks, Skolem’s two papers are essentially one. This work of Skolem’s was one of six or seven fundamental and wide-ranging contributions which he made to logic in the years 1919-1934. Skolem’s contributions were honored then, but they look still better as time goes on.
Review of Notcutt 1934 : A set of axioms for the theory of ded uction (1934!)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The author is of the opinion that in the usual presentations of formal systems certain rules necessary for their manipulation are tacitly assumed (for example, that the formulas are to be read from left to right), and he sets up an axiom system for the propositional calculus that is supposed to be free from this flaw, that is, one in which “nothing is left to the intelligence of the reader, where everything assumed is explicitly stated”. A second part of the work is concerned with the (ramified) theory of types. What are called “inter-typical variables” are introduced, that is, variables that simultaneously range over all possible types. If contradictions are to be avoided, these variables may never occur bound. The “inter-typical variables” are supposed to make the axiom of reducibility superfluous.
Diskussionsbemerkung (1936)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In Wirklichkeit hangt fiir jeden einzelnen Wirtschafter die Nachfrage auch von seinem Einkommen ab und dieses wieder vom Preis der Produktionsmittel. Man kann ein dementsprechendes Gleichungssystem formulieren und auf seine Losbarkeit hin untersuchen.
Über die Lange von Beweisen (1936a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Unter der Liinge eines Beweises in einem formalen System S moge die Anzahl der Formeln, aus denen er besteht, verstanden werden. 1st ferner jede natiirliche Zahl in bestimmter Weise durch ein Symbol (Zahlzeichen) in S darstellt (z. B. durch ein Symbol der Fonn 1 + 1 +···+ 1) so moge eine Funktion &lt;I&gt;( x) berechenbar in S heiBen, falls es zu jedem Zahlzeichen m ein Zahlzeichen n gibt, so daB t/&gt;(m) = n in S beweisbar ist. Insbesondere sind z. B. alle rekursiv definierten Funktionen schon in der klassischen Arithmetik (d. h. im System 81 der unten definierten Folge) berechenbar. Sei nun Si das System der Logik i-ter Stufe, wobei die natiirlichen Zahlen als Individuen betrachtet werden. D. h. genauer: Si soll Variable und Quantifikatoren fiir natiirliche Zahlen, fiir Klassen natiirlicher Zahlen, fiir Klassen von Klassen natiirlicher Zahlen u. s. w. bis Klassen von i-tem Typus samt den zugehorigen logischen Axiomen enthalten, aber keine Variable von hoherem Typus.
Review of Church 1935: A proof of freedom from contradiction (1936b)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The introductory note to 1936b, as well as to related items, can be found on page 256, immediately preceding 1932k.] By placing far-reaching restrictions on his earlier system, the author obtains a new one with only three primitive notions, namely: (1) the operation of “application” of a function to an argument, (2) “identity”, wiesen werden und ist andererseits so umfassend, da.B jede durch Rekursion (beliebig hoher Ordnung) definierte zahlentheoretische Funktion in ihm enthalten ist, d. h. zu jeder solchen Funktion &lt;I&gt; gibt es eine Formel F des Systems, so da.B, wenn n eine beliebige natiirliche Zahl und m = &lt;I&gt;( n) ist, die Formel F(n) = m beweisbar ist, wobei die natiirlichen Zahlen 1, 2, 3, ... usw. durch folgende Formeln ausgedriickt werden: &gt;./x · f(x), &gt;.Jx • /(f(x)), &gt;.Jx · /(1(/(x))) ... usw. (vgl. Gödel 199,le). Die Zahlen 1, 2 werden gleichzeitig als Wahrheitswerte “falsch”, ‘‘wahr” verwendet und die Operationen des Aussagenkalkiils als zahlentheoretische Funktionen 242 I durch Definition eingefiihrt. Die Formel 2 ist das einzige Axiom, und die Schluf3regeln (die der Verfasser “Konversionen” nennt) sind samtlich umkehrbar, so daf3 eine Formel dann als bewiesen gilt, wenn sie in 2 konvertiert werden kann. Der Alloperator tritt nicht als Grundbegriff auf, sondern wird vertreten durch gewisse niittels der drei Grundbegriffe aufgebaute Formeln II mit der Eigenschaft, daf3 II(F, G) dann und nur dann beweisbar ist, wenn G(x) aus F(x) vermittels gewisser von den Konversionen im allgemeinen verschiedenen Schluf3regeln abgeleitet werden kann.
The consistency of the generalized continuum hypothesis(1939)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
This lemma is proved by a generalization of Skolem’s method for constructing enumerable models. Since the axiom of choice is not used in the construction of the models, but holds in the models, the consistency of the axiom of choice is obtained as an incidental result.
The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory (1940)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In order to define Δ and to prove the above properties of it from the axioms of Σ, it is necessary first to develop abstract set theory to a certain extent from the axioms of Σ-This is done in Chapters II-IV. Although the definitions and theorems are mostly stated in logistic symbols, the theory developed is not to be considered as a formal system but as an axiomatic theory in which the meaning and the properties of the logical symbols are presupposed to be known. However, to everyone familiar with mathematical logic it will be clear that the proofs could be formalized, using only the rules of Hilbert’s “engerer Funktionenkalkul”.
Introductory note to 1944
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
This paper was written for The philosophy of Bertrand Russell, a volume of Paul Arthur Schilpp’s series the Library of living philosophers. In his letter of invitation of 18 November 1942, Schilpp proposed the title of the paper and also wrote: “In talking the matter over last night with Lord Russell in person, I learned that he too would not only very greatly appreciate your participation in this project, but that he considers you the scholar par excellence in this field.” Gudel sent in the manuscript on 17 May 1943. There followed a lengthy correspondence about stylistic editing proposed by Schilpp and Gudel’s own deliberation concerning revision. Before Gudel had submitted the final version, Russell had completed his reply to the other papers and decided that under the circumstances he would not reply to Gudel’s. When Gudel finally sent in the revised version on 28 September, he wrote to Russell attempting to change his mind about not replying. He undertook to disabuse Russell of his impression that what Gudel said would not be controversial, and emphasized his criticisms of Russell.
Introductory note to 1949 and 1952
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
These two papers represent Gudel’s main contribution to relativistic cosmology. In the 1920s and the 1930s, the Friedmann-Robertson Walker cosmological models had been introduced as the simplest solutions of the equations of Einstein’s general theory of relativity that were consistent with the observed red-shift of distant galaxies. These models were spatially homogeneous and isotropic, and were expanding but were non-rotating. Gudel was the first to consider models that were rotating. The possible rotation of the universe has a special significance in general relativity because one of the influences that led Einstein to the theory in 1915 was Mach’s principle. The exact formulation of the principle is rather obscure, but it is generally interpreted as denying the existence of absolute space. In other words, matter has inertia only relative to other matter in the universe. The principle is generally taken to imply that the local inertial frame defined by gyroscopes should be non-rotating with respect to the frame defined by distant galaxies.
Rotating universes in general relativity theory(1952)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In such a coordinate system, however, since parallel displacement (in its 176 origin) means constancy of the components, the angular velocity relative to the compass of inertia, in 0, is given by the same expressions as in Newtonian physics, i.e. the right-hand sides of (8) are its components. Evidently ω is the only vector the first 3 components of which, in the particular coordinate systems defined, coincide with the angular velocity computed as in Newtonian physics and the 4th component is 0.