bluuebunny/crossref_metadata_2025_split huggingface dataset:
Über die metrische Einbettbarkeit der Quadrupel des R3 in Kugelflachen (1933b)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In Beantwortung einer im 37. Kolloquium (Klan/er 1999) aufgeworfenen Frage gilt: Ein metrisches Quadrupel, das mit vier Punkten des R3 kongruent ist, ist, wenn es nicht mit vier Punkten der Ebene kongruent ist, auch kongruent mit vier Punkten einer Kugelftiiche, in welcher als Abstand je zweier Punkte die Lange des kurzesten sie au/ ihr verbindenden Bogens erkliirt ist. Zurn Beweise sei T ein Tetraeder im R3, dessen sechs Kantenlangen a1, a2, ••• , ae den Abstanden der vier gegebenen Punkte gleich seien und von dem wir I annehmen konnen, daB es nicht in einer Ebene liegt. R heif3e der Radius der T umgeschriebenen Kugel. Wir setzen
dann ist also ai,x die Lange der Sehne eines Kreisbogens der Lange ai auf einem Kreis vom Radius .!:.; fiir x = 0 hat ai x den Grenzwert ai. Ein (eventuell ebenes) Tetraeder im R3, dessen sechs Kanten die Langen ai,x besitzen, wollen wir, falls ein solches Tetraeder existiert, mit T(x) bezeichn(!n. Dabei setzen wir T(0) = T. Den reziproken Radius der T(x) um-geschriebenen Kugel nennen wir f(x); es ist dann also f(0) = liegt T(x) in einer Ebene, so ist f(x) = 0.
On Wald’s axiomatization of the notion of betweenness (1933c)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In his axiomatization of the notion of betweenness (1931 and 1932), Wald proves that three points a, b and e stand in a relation defined for all triples of points in all metric spaces and satisfying certain conditions if and only if the relation ab + be = ae holds for the distances between them, that is, if and only if b lies between a and e in Menger’s sense. Among the conditions imposed on the relation there appears that of invariance under congruence, that is, if three points a, b and e satisfy the relation in question, then any three points a’, b’ and e’ congruent to them are also to stand in that relation to one another.
On the axiomatization of the relations of connection in elementary geometry (1933d)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
One should investigate the system of all those propositions about lattices that in normal form contain no existential quantifiers. The notions of point and line, which are definable by use of existential quantifiers (for example, a point is an element for which there does not exist a non-empty element that is a proper part of it), are not definable in this more restricted system. Investigations in this direction are to be found in Wernick 1929.
An interpretation of the intuitionistic propositional calculus (1933/)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
One can interpret1 Heyting’s propositional calculus by means of the notions of the ordinary propositional calculus and the notion ‘pis provable’ (written Bp) if one adopts for that notion the following system 6 of axioms:
Bemerkung iiber projektive Abbildungen (1999g)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Jede eineindeutige Abbildung &lt;/) der reellen projektiven Ebene E in sich, welche Gerade in Gerade Über/ii.hrt, ist eine Kollineation. Bezeichnet man fiir jeden Kegelschnitt der Ebene als die zugehorige Kegelschnittumgebung die Menge aller Punkte p von E, fiir welche jede p enthaltende Gerade genau zwei Punkte mit dem Kegelschnitt gemein hat, so ist das System Jt aller Kegelschnittumgebungen ein unbegrenzt feines Überdeckungssystem1 von E.
Remark concerning projective mappings (1933g)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Every one-to-one mapping of the real projectz”ve plane E into itself that carries straight lines into straight lines is a collineation. If for each conic section in the plane one designates as the corresponding conic-section neighborhood the set of all points p of E for which every straight line containing p has exactly two points in common with the conic section, then the system .
Diskussion iiber koordinatenlose Differentialgeometrie (1999h)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Um die Aussage, da6 die Riemannschen Raume sich im Kleinen wie euklidische Raume verhalten ( da6 “fiir unendlich benachbarte Punkte Riemannscher Raume die eukildische Geometrie galet”) im Sinne des von Menger natenlosen Riemannscher (1990, vglRaume . Differentialgeometrie auch die 1992) euklidische zu entwickelten Geometrie prazisieren, gelte”) Programmes waren im die Sinne einer zur des koordi-metrivon schen Kennzeichnung der euklidischen Raume dienenden Determinanten p4 eines (“Volumsdeterminanten”) metrischen Raumes
Review of Lewis 1932 : Alternative systems of logic (1933k)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The author asserts the existence of different systems of logic (and indeed of the propositional calculus) only in the sense that we can take as primitive different logical notions, whence the propositions that hold in them obvi-ously differ also. The system of primitive notions is always a finite class K of properties ( truth values) such that to each statement one of these properties is assigned and that, among them, there is one property, E, such that every statement with property E can be asserted. The theorems of logic concern propositional functions /(P1, ... ,Pn ) whose truth values depend only on the truth values of the arguments Pi· For example: (1) if K consists of the two properties “true” and “false”, one obtains the system of Principia mathematica; (2) if K consists of the properties “certain”, “certainly false”, and “doubtful”, one obtains the three-valued system of Lukasiewicz; (3) if K consists of the properties “necessary”, “impossible”, “true but not necessary”, and “false but not impossible”, one obtains Lewis’ system of strict implication. Interpretations are given for the propositional functions of systems (2) and (3); for example, the following interpretation is given for p :J q in the three-valued system: no disadvantage is sustained in converting a wager on p into a wager on q ( apart from the degree of un-certainty).
Review of Skolem 1933: A combinatorial theorem with an application to a decision problem in logic (1934a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
This article concerns the following theorem, which was proved by F. P. Ramsey in his 1929: If n, µ and 11 are three arbitrary natural numbers, a natural number N can be determined such that, if all 11-tuples of the elements of a set M containing at least N elements are partitioned in any way into µ disjoint systems C1, C2, ... , Cµ,, there always exists an n-element subset of M such that all of the 11-tuples formed from elements of this subset belong to the same Ci. The author’s proof is simpler than Ramsey’s and also furnishes smaller values for N.
Review of Quine 1933: A theorem in the calculus of classes (1934b)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The following formula of the calculus of classes,
ab + ac + be = (a + b)(a + c)(b + c), is generalized to the theorem: If µ is a finite number and v :5 µ, then the sum of all possible products from any v ofµ classes is equal to the product of all possible sums from any µ - v + 1 of these µ classes. The proof is carried out purely formally in the symbolism of Principia mathematica, using only theorems that are formally proved in Principia. It is also shown that the theorem is false for v &gt; .
Introductory note to 1995a
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
Der Verfasser stellt ein aus 13 unabhangigen Postulaten bestehendes Axiomensystem mit folgenden Grundbegriffen auf: K (Klasse der Satze}, Q {Klasse der falschen Satze), D {Klasse der unmoglichen Satze}, a x b (Konjunktion), a’ {Negation), a= b {Gleichheit), wobei x und’ als Operationen aufzufassen sind, die aus Elementen von K wieder Elemente von K erzeugen. Satze der Form a e D sind inhaltliche Aussagen (nicht Elemente von K). Daher kann der axiomatische Unmoglichkeitsbegriff D nicht iteriert angewendet werden, worin der Hauptunterschied gegeniiber dem Lewisschen System liegt. Fiir Aussagen der Form a e Q, a e D und die aus solchen mittels Konjunktion und Negation aufgebauten definiert der Verfasser einen inhaltlichen Unmoglichkeitsbegriff (unmoglich ist, was auf eine Widerspruch fiihrt) und zeigt, daB die Lewisschen Axiome aus seinen folgen, wenn man in ihnen die Unmoglichkeitszeichen erster Stufe durch seinen axiomatischen und die zweiter Stufe durch seinen inhaltlichen Unmoglichkeitsbegriff ersetzt. Im selben Sinn folgen die Axiome des Verfassers aus den Lewisschen, woraus aber noch nicht geschlossen werden kann, daB die beiden Systeme auch in ihren Folgerungen iibereinstimmen, da ja fiir die Unmoglichkeitszeichen dritter und hoherer Stufe gar keine Übersetzung vorliegt.
Review of Huntington 1934: Independent postulates related to C. I. Lewis’ theory of strict implication (1935a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
The author sets up an axiom system, consisting of thirteen independent postulates, with the following primitive notions: K (class of propositions), Q (class of false propositions), D (class of impossible propositions), ax b (conjunction), a’ (negation), a= b (identity), where x and’ are interpreted as operations that, when applied to elements of K, again generate elements of K. Propositions of the form a E D are contentual statements (not elements of K) .. Therefore the axiomatic notion of impossibility, D, cannot be iteratively applied, and therein lies the principal distinction between this system and Lewis’. For statements of the form a E Q or a E D, and those built up from such statements by means of conjunction and negation, the author defines a notion of contentual impossibility (impossible is that which leads to a contradiction) and shows that Lewis’ axioms follow from his if in them we replace the impossibility signs of the first degree by his axiomatic notion of impossibility and those of the second degree by his contentual notion of impossibility. Correspondingly, the author’s axioms follow from those of Lewis, from which, however, we cannot yet conclude that the two systems agree also in what they entail, since no translation at all is at hand for the impossibility signs of third and higher degrees.
Review of Kalmar 1934 : On a theorem of Lowenheim (1935c)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
A simpler proof is given for Lowenheim’s theorem that for each firstorder expression a binary one 1.13 can be specified such that is valid if and only if 1.13 is (1.13 is said to be binary if it does not contain functional variables with more than two argument places).
Introductory note to 1936a
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
In this abstract, published in Ergebnisse eines mathematischen Kolloquiums, Gödel announced a “speed-up” theorem that holds when one switches from a weaker logic to a stronger one. (The term “logic” may be somewhat misleading since apparatus for proving facts about numbers is, of course, present.) Gödel had already shown that a logic Sn+l of a higher order could prove formulas that a logic Sn of a lower order could not prove. Now he considered the question of formulas that can be proved in both the weaker and the stronger logics, and his theorem stated that, if the length of a proof is defined to be the number of lines in it, there are formulas that can be proved in both Sn and Sn+1 but that have a proof in Sn+l much shorter than their shortest proof in Sn, Thus Sn+l achieves a speed-up over Sn, This speed-up can be by an arbitrary function computable in Sn, in the sense that, for any function &lt;I&gt; computable in Sn, there would be a formula A and a number k such that A can be proved in Sn+l in k lines but, although provable in Sn also, A cannot be proved in Sn in &lt;I&gt;( k) lines. The notion of computability in a formal system that is used here is what Gödel had previously called Entscheidungsdefinitheit in 1991. Interestingly, Gödel did not remark at first that the notion of computable function in Sn is independent of n, but this crucial observation was added in the page proof before publication.a Gödel later remarked, in the first paragraph of his paper 1946, that his notion of computability in a logic is equivalent to Turing computability, and he stressed that this is an absolute notion independent of the formalism chosen. Incidentally, it is worth mentioning here that Gödel does not bother to make a distinction between a function &lt;I&gt; and the syntactic object corresponding to its representation in Sn, It is true of course, by virtue of Gödel’s own incompleteness results, that two different representations of the same function may fail to be provably equivalent.
On the length of proofs (1936a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
By the length of a proof in a formal system S we shall understand the number of formulas of which the proof consists. Further, if every natural number is represented in Sin a definite way by a symbol (a numeral) (for example, by a symbol of the form 1 + 1 + 1), then a function cp(x) will be said to be computable in S in case for each numeral m there exists a numeral n such that ¢(m) = n is provable in S. In particular, all recursively defined functions, for example, are already computable in classical arithmetic (that is, in the system S1 of the sequence defined below).
Now let Si be the system of logic of the ith order, the natural numbers being taken as individuals. That is, more precisely: Si shall contain, together with the appropriate logical axioms, variables and quantifiers for natural numbers, for classes of natural numbers, for classes of classes of natural numbers, and so on, up to classes of the ith type, but no variable of a higher type. Then, as is known, there are propositions of Si that are provable in Si+1 but not in Si. If, on the other hand, we consider those formulas / that are provable in Si as well as in Si+1, then the following holds: For each function ¢ that is computable in Si there exist infinitely many formulas f such that, if k is the length of a shortest proof of f in Si and l is the length of a shortest proof of f in Si+1, then k &gt; ¢(l). If, for example, we set ¢(n) = 106n, it follows that there are infinitely many formulas whose shortest proof in Si is more than 106 times as long as their shortest proof in Si+1. Thus, passing to the logic of the next higher order has the effect, not only of making provable certain propositions that were not provable be/ore, but also of making it possible to shorten, by an extraordinary amount, infinitely many of the proofs already available.
Consistency proof for the generalized continuum hypothesis1 (1939a)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
If M is an arbitrary domain of things in which a binary relation ϵ is defined, call “propositional function over M” any expression ϕ containing (besides brackets) only the following symbols: 1.
Russell’s mathematical logic* (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
Mathematical logic, which is nothing else but a precise and complete formulation of formal logic, has two quite different aspects. On the one hand, it is a section of mathematics treating of classes, relations, combinations of symbols, etc., instead of numbers, functions, geometric figures, etc. On the other hand, it is a science prior to all others, which contains the ideas and principles underlying all sciences. It was in this second sense that mathematical logic was first conceived by Leibniz in his Characteristica universalis, of which it would have formed a central part. But it was almost two centuries after his death before his idea of a logical calculus really sufficient for the kind of reasoning occurring in the exact sciences was put into effect (in some form at least, if not the one Leibniz had in mind) by Frege and Peano. Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic. Peano, on the other hand, was more interested in its applications within mathematics and created an elegant and flexible symbolism, which permits expressing even the most complicated mathematical theorems in a perfectly precise and often very concise manner by single formulas.
What is Cantor’s continuum problem? (1947)
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 question, of course, could arise only after the concept of “number” had been extended to infinite sets; hence it might be doubted if this extension can be effected in a uniquely determined manner and if, therefore, the statement of the problem in the simple terms used above is justified. Closer examination, however, shows that Cantor’s definition of infinite numbers really has this character of uniqueness, and that in a very striking manner. For whatever “number” as applied to infinite sets may mean, we certainly want it to have the property that the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space). From this, however, it follows at once that two sets (at least two sets of changeable objects of the space-time world) will have the same cardinal number if their elements can be brought into a one-to-one correspondence, which is Cantor’s definition of equality be tween numbers.
An example of a new type of cosmological solutions of Einstein’s field equations of gravitation (1949)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
All cosmological solutions with non-vanishing density of matter known at present have the common property that, in a certain sense, they contain an “absolute” time coordinate, owing to the fact that there exists a one parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that the non-existence of such a system of three spaces is equivalent with a rotation of matter relative to the compass of inertia.
On a hitherto unutilized extension of the finitary standpoint (1958)
Authors: Gudel Kurt , Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van 
Published: November 2023

Abstract
P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is. in Hilbert’s sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory.
Introduction: Toward A History Of Texts
Authors: Davidson Cathy N 
Published: November 2023

Abstract
Throughout the first decade of the nineteenth century, Ethan Allen Greenwood, a rather pedantic young diarist, each day recorded both the weather and the title of the book he was reading. He sometimes observed that a particular work was “instructive” or “entertaining” and occasionally noted the library from which the volume was borrowed—the Adelphi Fraternity Library, the Social Friends Library, or the unnamed circulating library he joined in 1806. His meticulous account of his activities and expenses—whether living at home in Worcester County, Massachusetts, or at Dartmouth College or later in Boston, or whether traveling around the countryside as an itinerant painter—provides posterity with an unusually comprehensive portrait of “Ethan Allen Greenwood, His Life and Times.” Looking over the record that he left, we can well imagine that we know this serious and sober, parsimonious and abstemious young man Franklinesquely working his way toward fame and fortune.
Literacy, Education, and the Reader
Authors: Davidson Cathy N 
Published: November 2023

Abstract
From Bunyan and ballads to Brydone and Mrs. Radcliffe, the readers in the new Republic were obviously embarked on a pilgrimage of their own and, by the turn of the century, were already well on their way. But could Dolly the dairymaid and Jonathan the hired man really read The Mysteries of Udolpho (1794)? That question is basic to the present discussion of early American culture. It was crucial then, too, as is attested to by John Adams’s often quoted claim that America was the most literate nation on earth. “A native American who cannot read and write,” Adams boasted, “is as rare as a comet or an earthquake.” Numerous travelers of the time, visiting from France or En-gland, concurred with this appraisal, yet Lyle H. Wright estimates that in Adams’s day only some 1.5 million individuals, or somewhat less than half of the population, were literate. The discrepancy between the two claims is striking. Nor is it entirely explained away by Kenneth A. Lockridge’s suggestion that Adams’s “universal literacy” did exist but only in John Adams’s America—an America that defined itself as New England, elite, urban, white, male.
Introduction
Authors: Read Kay Almere , González Jason J 
Published: November 2023

Abstract
Climbing the stairs, I emerged from the bustling, crammed subway station into the subdued light of a summer day in Mexico City’s rainy season. The day before at this particular underground stop, a shrine had been dedicated to the Virgencita (Veer’-hen-seeta), the “Little Virgin” (Figure 1). The month before, she had made herself known through a simple water spot that took her shape on the concrete floor of the Hidalgo Metro stop, one of the busiest sub­ way stations in the city’s center.
Murder and Martyrs in Anglo-Saxon England
Authors: Cormack Margaret 
Published: November 2023

Abstract
The arenas of Rome left an indelible imprint on Christianity. Christians believed that the martyrs who voluntarily gave up their lives for Christ were rewarded by joining Him immediately in heaven. Since they had proved their faith by making the ultimate sacrifice, the Last Judgment would find them among the judges, not the judged. In the meantime, they were able to aid those still on earth in all manner of ways, from curing sickness or changing the weather to afflicting tyrants with loathsome diseases. Their relics were sources of healing and power and became objects of pilgrimage.