bluuebunny/crossref_metadata_2025 huggingface dataset:
Notes on the Invisible Women in Ralph Ellison’s Invisible Man
Authors: Tate Claudia
Published: November 2023

Questions about t he female characters in Ralph Ellison’s Invisible Man seem to elicit two types of response: The initial one is “What women?” since women clearly occupy peripheral roles in the novel. And then after Mary Rambo and the other female characters—that is, the old slave woman, the magnificent blonde, the rich sophisticate Emma, the anonymous seductress, and finally the prophetic and pathetic Sybil— are recalled, the second response is something like “Oh, those stereotypes” (Sylvander, 77-79). Both replies are virtually automatic and both are legitimate, given the factual details of the narrative. But we must not be misled by what can be seen with a quick glance; we must not neglect what lies hidden behind the mask and proclaim that the mask is the face. Instead, we must remember Ellison’s own witty admonition that the rind is not the heart and look for the concealed truth which lies beneath the stereotyped exteriors of his female characters. In his essay “Twentieth-Century Fiction,” Ellison contends that stereotypes, though indisputably one-dimensional and therefore oversimplified, frequently hide complex aspects of human character. Moreover, he adds that “the Negro has been more willing perhaps than any other artist to start with the stereotype, accept it as true, and then seek out the human truth which it hides” (S&A, 43). Perhaps this is also an appropriate procedure to follow when examining the female characters in Invisible Man; that is, “start with the [female] stereotype[s], accept [them] as true, and then seek out the human truth which [they] hide.” Perhaps by following this example, we will not be attempting merely to define female humanity but to recognize, as Ellison suggests, broader aspects of the humanity of all of us.
Luminosity from the Lower Frequencies
Authors: Forrest Leon
Published: November 2023

I should like to discuss certain intellectual, cultural, and historical influences upon Ralph Ellison’s sense of the hero’s character in process and the structure of the major chapters throughout his monumental novel, Invisible Man. Several influences come to mind: Kenneth Burke, Lord Raglan, Dostoevski, and Faulkner, as well as the artistic and jazzlike rendering of folkloric sources. From the literary critic Burke, Ellison came to see the possibility of using a formula to structure a chapter. Burke held that a pattern could be employed to achieve character-in-process progression through the formula of purpose, passion, and perception: each chapter begins with a purpose for the hero, but then much of the action of the middle section involves a struggle, or passion, over this purpose, or quest. Out of this mix or confrontation with others and the self, the hero comes away with a heightened perception, a keener awareness about his life, so that a metamorphosis, or rebirth, is implied. But these moments are stages of his processing into life, and the cycle, once completed, unleashes new problems and struggles.
“Segregation Is Per Se Inequality”
Authors: Yarbrough Tinsley E
Published: November 2023

While Judge Waring was dismantling South Carolina’s white primary, urging total racial integration, and in the process becoming something of a southern pariah, the U.S. Supreme Court had been steadily chipping away at segregation in transportation, housing, and higher education. In 1950, the Court decided two extremely significant cases. McLaurin v. Oklahoma held that a state could not admit a black student to its previously all-white graduate school, then segregate the black student from his white classmates within the institution. On the same day that the McLaurin case was decided, the Court handed down an even more important decision in Sweatt v. Painter, ordering Herman Sweatt admitted to the University of Texas law school even though the state had recently established a law school for blacks. Chief Justice Vinson had observed for the Sweatt Court that, in determining whether segregated facilities were equal, trial judges were to consider not only tangible factors, but also intangible factors-”those qualities which are incapable of objective measurement but which make for greatness in a law school.”

Judge Waring had followed the line of Supreme Court cases closely, avidly reading the Court’s opinions as well as briefs secured from Walter White, Thurgood Marshall, and other sources. He had been particularly impressed with an amicus curiae brief filed by the Federal Council of Churches of Christ in America and the council’s assertion that segregation itself “necessarily predicates inferiority rather than equality.”
What the Book Is About
Authors: Gordimer Nadine
Published: November 2023

The theme of my novel is human conflict between the desire to live a personal, private life, and the rival claim of social responsibility to one’s fellow men-human advancement. Through the personage of Rosa Burger, the question is posed and explored: What is a meaningful life? Is there a cause greater than the gift of life itself Can one fulfill oneself simply by earning a living, falling in love, marrying, and producing the next generation, with no concern for anyone outside the family circle?
Still Waiting for the Great Feminist Novel
Authors: Gardner Susan
Published: November 2023

In July 1979 , Nadine Gordimer’s novel Burser’s Daushter—the story of a young Afrikaner woman from a Communist family, focussing on her personal and ideological rebellion after her father’s death during life imprisonment-was banned for import and distribution in the author’s country, South Africa. The Publications Control Board’s accusations declared the book “undesirable” on every count under section 47(2) of the Publicatons Act of 1974.
Burger’s Daughter Lighting a Torch in the Heart of Darkness
Authors: Liscio Lorraine
Published: November 2023

In Nadine Gordimer’s First Novel, The Lying Days, the protagonist, Helen Shaw, lists the writers who shaped her literary heritage: Auden, T. S. Eliot, Donne, Hemingway, D. H. Lawrence, Chekhov, Smollett, and Pepys. She is puzzled to find that “in nothing that I read could I find anything that approximated to my own life; to our life on a gold mine in South Africa.” If she failed to recognize her world in European narratives, Helen felt equally alienated from the pictures of Africa portrayed by books she read: “What did the great rivers, the savage tribes, the jungles and the hunt for huge palm-eared elephants have to do with the sixty miles of Witwatersrand veld that was our Africa?” (96-97).
The completeness of the axioms of the functional calculus of logic1 (1930)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

Whitehead and Russell, as is well known, constructed logic and mathematics by initially taking certain evident propositions as axioms and deriving the theorems of logic and mathematics from these by means of some precisely formulated principles of inference in a purely formal way (that is, without making further use of the meaning of the symbols). Of course, when such a procedure is followed the question at once arises whether the initially postulated system of axioms and principles of inference is complete, that is, whether it actually suffices for the derivation of every logicomathematical proposition, or whether, perhaps, it is conceivable that there are true propositions (which may even be provable by means of other principles) that cannot be derived in the system under consideration. For the formulas of the propositional calculus the question has been settled affirmatively; that is, it has been shown2 that every correct formula of the propositional calculus does indeed follow from the axioms given in Principia mathematica. The same will be done here for a wider realm of formulas, namely those of the “restricted functional calculus” ;3 that is, we shall prove Theorem I.
Über die Vollstandigkeit des Logikkalkiils (1930a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

Bei der axiomatischen Begriindung der Logik, wie sie z. B. in den Principia Mathematica vorliegt, taucht die Frage auf, ob die an die Spitze gestellten Axiome ‘‘vollstandig” sind, d. h. wirklich dazu ausreichen, jeden richtigen Satz der Logik auf formalem Wege zu deduzieren. Dieses Problem ist bisher nur fiir die einfachsten logischen Satze, namlich die des Aussagenkalkiils, gelost. Die Antwort fallt positiv aus, d. h.: Jede richtige (allgemeingiiltige) Aussageformel folgt aus den Axiomen der Principia Mathematica. Vortragender zeigt, wie man diesen Satz auf die Formeln des engeren Funktionenkalkiils (Formeln ohne gebundene Funktionsvariable) ausdehnen kann.
Introductory note to 1931a, 1932e, f and g
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

The items considered in this note all concern events that took place during the second Conference on Epistemology of the Exact Sciences, a meeting organized by the Gesellschaft fur empirische Philosophie (Society for Empirical Philosophy) and held in Konigsberg on 5-7 September 1930. Most interesting among them is 1991a, which contains Gödel’s unexpected announcement, during a discussion session at the conference, of the existence of formally undecidable propositions.
Diskussion zur Grundlegung der Mathematik (1931a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

Nach formalistischer Auffassung fiigt man zu den sinnvollen Satzen der Mathematik transfinite (Schein- )Aussagen hinzu, welche an sich keinen Sinn haben, sondern nur dazu dienen, das System zu einem abgerundeten zu machen, ebenso wie man in der Geometrie durch Einfiihrung der unendlich fernen Punkte zu einem abgerundeten System gelangt. Diese Auffassung setzt voraus, daB, wenn man zum System S der sinnvollen Satze das System T der transfiniten Satze und Axiome hinzufiigt und dann einen Satz aus S auf dem Umweg iiber Satze aus T beweist, dieser Satz auch inhaltlich richtig ist, daB also durch Hinzufiigung der transfiniten Axiome keine inhaltlich falschen Satze beweisbar werden. Diese Forderung pflegt man durch die der Widerspruchsfreiheit zu ersetzen. Ich mochte nun darauf hinweisen, daB diese beiden Forderungen keinesfalls ohne weiteres als aquivalent angesehen werden diirfen. Denn wenn in einem widerspruchsfreien formalen System A (etwa dem der klassischen Mathematik) ein sinnvoller Satz p mit Hilfe der transfiniten Axiome beweisbar ist, so folgt aus der Widerspruchsfreiheit von A bloB, daf3 non-p innerhalb des Systems A formal nicht beweisbar ist. Trotzdem bleibt es denkbar, da6 man non-p durch irgendwelche inhaltliche (intuitionistische) Überlegungen einsehen konnte, die sich in A nicht formal darstellen lassen. In diesem Falle ware trotz der Widerspruchsfreiheit von A in A ein Satz beweisbar, dessen Falschheit man durch finite Betrachtungen einsehen konnte. Sobald man den Begriff “sinnvoller Satz” hinreichend eng fa6t {z. B. auf die finiten Zahlgleichungen beschrankt), kann etwas Derartiges allerdings nicht eintreten. Hingegen ware es z. B. durchaus moglich, daB man einen Satz der Form (Ex)F(x), wo F I eine finite Eigenschaft natiirlicher Zahlen ist (die Negation des Goldbachschen Satzes hat z. B. diese Form), mit den transfiniten Mitteln der klassischen Mathematik beweisen und andererseits durch inhaltliche Überlegungen einsehen konnte, daB alle Zahlen die Eigenschaft non-F haben, und zwar bleibt dies, worauf ich eben hinweisen mochte, auch dann noch moglich, wenn man die Widerspruchsfreiheit des formalen Systems der klassischen Mathematik nachgewiesen hatte. Denn man kann von keinem formalen System mit Sicherheit behaupten, daB alle inhaltlichen Überlegungen in ihm darstellbar sind.
Discussion on providing a foundation for mathematics (1931a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

According to the formalist view one adjoins to the meaningful propositions of mathematics transfinite (pseudo-)assertions, which in themselves have no meaning, but serve only to round out the system, just as in geometry one rounds out a system by the introduction of points at infinity. This view presupposes that, if one adjoins to the system S of meaningful propositions the system T of transfinite propositions and axioms and then proves a theorem of S by making a detour through theorems of T, this theorem is also contentually correct, hence that through the adjunction of the transfinite axioms no contentually false theorems become provable. This requirement is customarily replaced by that of consistency. Now I would like to point out that one cannot, without further ado, regard these two demands as equivalent. For, if in a consistent formal system A (say that of classical mathematics) a meaningful proposition p is provable with the help of the transfinite axioms, there follows from the consistency of A only that not-p is not formally provable within the system A. Nonetheless it remains conceivable that one could ascertain not-p through some sort of contentual (intuitionistic) considerations that are not formally representable in A. In that case, despite the consistency of A, there would be provable in A a proposition whose falsity one could ascertain through finitary considerations.
Review of Betsch 1926: Fictions in mathematics (1931d)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

This comprehensive work, which was awarded a prize by the Vienna Academy, reaches in all directions far beyond the theme “fictions in mathematics”. On the one hand, it discusses fictions in general and, in connection with them, the philosophical problems of reality and of truth; on the other, it treats in detail the problems of the foundations of mathematics. The method of presentation is chiefly historical. As concerns the theme proper, that is, the question whether and in what sense there are fictions in mathematics, the author’s attitude is predominantly negative. With respect to genuine fictions (contradictory but expedient assumptions), his answer always comes out negative, and all of Vaihinger’s assertions in this regard are shown to be erroneous. Whether unreal fictions (assumptions not in accord with reality) can be admitted to occur in mathematics depends, as the author seeks to show, upon one’s attitude toward foundational questions, conventionalism and empiricism (in Mill’s sense) in the end allowing such fictions the widest scope. In conclusion it is pointed out that fictions in a quite different sense (as tentative assumptions or provisional definitions) play a large role in the historical development of mathematics.
Introductory note to 1992
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

In this short note on intuitionistic propositional logic (H),a Gödel shows that H cannot be viewed as a system of many-valued logicb.
Ein Spezialfall des Entscheidungsproblems der theoretischen Logik (1932a)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

Im Anschlufi an die fiir den Beweis der Vollstandigkeit des Funktionenkalkiils verwend te Methode (1990) kann man ein Verfahren entwickeln, welches fiir jede Formel mit einer Normalform der Gestalt: zu entscheiden gestattet, ob sie erfiillbar ist. Man fiihrt das Problem zunachst in bekannter Weise1 auf Formeln der Gestalt: problem is undecidable.
On completeness and consistency (1932b)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

The introductory note to Gödel 1932b, as well as to related items, can be found on page 126, immediately preceding 1930b.] Let Z be the formal system that we obtain by supplementing the Peano axioms with the schema of definition by recursion (on one variable) and the logical rules of the restricted functional calculus. Hence Z is to contain no variables other than variables for individuals (that is, natural numbers), and the principle of mathematical induction must therefore be formulated as a rule of inference.
Introductory note to 1992c
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

We easily impose a true-false dichotomy on the truth-functional formulas of propositional logic by assigning arbitrary truth values to all the propositional letters and then letting the truth tables determine the truth values of all compound formulas. But now let us imagine, instead of the familiar formulas, a well-ordered but perhaps indenumerable supply. It is still to be closed under the operations’∼’ and’:::,’, but there need no longer be minimal units such as letters. Can we specify a true-false dichotomy for these formulas, conformable still to the truth-functional constraints on ‘∼’ and ‘:::,’? Gödel does so by a transfinite recursion based on Lukasiewicz’s three axioms (Lukasiewicz and Tarski 1990) and modus ponens.
Review of Klein 1931 : On the theory of abstract connectives (1932h)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

The author investigates domains of individuals in which two operations, a U b and a n b, are defined and satisfy the axioms of the calculus of classes for union and intersection. The analog of complementation is introduced as a one-to-one symmetric mapping of the domain onto itself, by means of which the two connectives U and n are interchanged. In addition to the calculi of classes and of propositions, the operations max (a, b) and min (a, b), for example, or greatest common divisor and least common multiple, are considered as realizations of the axioms. After derivation of various simple theorems (for example, the uniqueness of the decomposition of elements of the domain into prime elements, if such a decomposition exists at all), what is called a “characteristic” mapping is introduced.
Review of Hoensbroech 1931: Relations between the content and extension of notions (1932i)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

If by an increase in the content of a notion we understand the adjoining of constituents without the existing connection between constituents being destroyed, then, as the author establishes in conjunction with a remark of Bolzano, extension need not be at all diminished by an increase in content. For example, in this sense “blue or red” has greater content than “blue”. The author then investigates the different kinds of increase in content that are possible from this point of view (conjunctive and disjunctive joining, negation, substitution of a name, adjoining of quantifiers) with regard to their effect on the extension of notions.
Review of Kalmar 1932: A contribution to the decision problem (1932l)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

The decision problem for the restricted functional calculus is reduced to the case in which the expression under consideration possesses, in normal form, a prefix of the form That is, it is shown that for every expression A of the restricted functional calculus one can specify another expression B, with a prefix of the form displayed above, so that A is satisfiable (or satisfiable in a finite domain of individuals) if and only if the same is true of B. In particular, if B is satisfiable in a domain I, then A is satisfiable in that same domain I. If A is satisfiable in a domain I, then B is satisfiable in a more extensive domain I’, whose cardinality may easily be computed from that of J. B will contain a few more functional variables than A, but, as is shown, one can always ensure that B contains only two-place functional variables whenever A does.
[ Über die Parryschen Axiome ] (1933)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

“p impliziert q analytisch” kann man vielleicht so interpretieren: “q ist aus p und den logischen Axiomen ableitbar und enthiilt keine anderen Begriffe als p” und es ware, nachdem man diese Definition genauer priizisiert hat, ein Vollstiindigkeitsbeweis fiir die Parryschen Axiome [Parry 1933] zu erstreben, in dem Sinn, daB alle Siitze, welche fur die obige Interpretation von–gelten, ableitbar sind.
[ On Parry’s axioms] (1933)
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

One can perhaps interpret “p analytically implies q” as follows: “q is derivable from p and the axioms of logic, and q contains no notions other than those in p”; and, after this definition has been made more precise, it would be appropriate to strive for a proof of the completeness of Parry’s axioms [Parry 1933], in the sense that all statements that hold when is interpreted as suggested above are derivable.
Introductory note to 1999a
Authors: Feferman Solomon , Dawson John W , Kleene Stephen C , Moore Gregory H , Solovay Robert M , Heijenoort Jean van
Published: November 2023

Hahn had asked whether every independence proof for formulas of the propositional calculus could be carried out with a finite model. Gödel here presents an independence proof using an infinite model and shows that no finite model would suffice. It was less than three months after this colloquium that Gödel answered a previous query of Hahn’s about the intuitionistic propositional calculus by again showing the inadequacy of finite models; see Gödel 1992.
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

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

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

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

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

Jede eineindeutige Abbildung </) 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

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

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

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

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

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 > .
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

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

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

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

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  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 ( 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  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

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 > ¢(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

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

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

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

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

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

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

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

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

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.
Truth and Sacrifice: Sati Immolations in India
Authors: Harlan Lindsey
Published: November 2023

To associate the sati, a term referring to a woman who immolates herself on her husband’s funeral pyre, with the martyr, a word with indisputably positive valuation in Christianity and other traditions originating in European and Middle Eastern antiquity, is to invite charges of insensitivity from many people engaged in moral discourse outside India and from those opposed to sati immolation and worship within India. As Ashis Nandy has shown, educated urban elites as well as feminists in India have been adamant in their condemnation of sati immolation, which they view as a horrifying practice and an embarrassing legacy of the Hindu past.2 It was therefore with trepidation that I accepted an invitation to contemplate the issue of sati immolation within the context of a colloquium on martyrdom.
Stephen’s Diary in Joyce’s Portrait—The Shape of Life
Authors: Levenson Michael
Published: November 2023

Despite the Unflagging Controversy over the character of Stephen Dedalus, remarkably little attention has been paid to the ending of A Portrait of the Artist as a Young Man, where one might have supposed that the problem would become acute. The diary that concludes the novel has had the rare dis tinction of being a virtually unannotated specimen within the Joycean oeuvre, and while it is tempting to keep it free from the endless profusion of inky scholia, it raises issues too important to leave unremarked. One of the most intricate mechanisms in an intricate novel, its workings bear directly on the problem of interpreting Stephen’s development. Questions of characterization join with questions of genre, and the relationship between individual agency and impersonal form becomes vivid. In elaborate, though submerged, patterns the diary reinterprets the narrative that it will soon conclude and offers some trenchant answers to a question that governs the novel as it governs this essay: What is the shape of a life?
Introduction
Authors: Lindow John
Published: November 2023

When most of us use the word “myth” in conversation, we refer to something that is not true. When historians of religion use it, they generally refer to a representation of the sacred in words. When anthropologists use it, they often refer to narratives that tell about the formation of some social institution or behavior. None of the definitions, however, will hold directly for the characters and stories this book treats.