The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)

Zermelos Zeit in Göttingen (1897?1910) kann als wissenschaftlich fruchtbarste Periode in seiner Karriere angesehen werden. Gleichwohl stehen bisher Untersuchungen aus. die eine Einbettung von Zermelos Werk in den biographischen und sozialen Kontext ermöglichen Die vorliegende Studie will diese Lücke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfüllen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten über die Grundlagen der Mengenlehre; (2) die Vorgeschichte und näheren Umstände des 1907 an Zermelo verliehenen Lehrauftrages für mathematische Logik und verwandte Gegenstände. mit dem ein erster (...) Schritt zur Institutionalisierung dieses Faches als mathematischer Teildisziplin gemacht wurde. Beides wird in enger Verbindung zur ersten Phase des Hilbertschen Programms zur Grundlegung der Mathematik gesehen. Es wird aber auch gezeigt, daß für die Erteilung des Lehrauftrages neben diesen systematischen und forschungspolitischen Motiven auch persönliche und institutspolitische Rücksichten ausschlaggebend waren. Im Anhang werden Teile eines Briefwechsels zwischen Ernst Zermelo und dem Göttinger Philosophen Leonard Nelson ediert sowie die Anträge der Direktoren des Göttinger mathematisch-physikalischen Seminars auf Erteilung des Lehrauftrags und auf Ernennung Zermelos zum außerordentlichen Professor. Zermelo's Göttingen period (1897?1910) can be regarded as the most fruitful period in his scientific career. Nevertheless, up till now there have been no investigations which enable us to embed his work in its biographical and social contexts. This study tries to partially fill the gap, concentrating on two major themes: (I) the historical context of the development of Zermelo's early work on the foundations of set theory; (2) the prehistory and particulars of his lectureship for mathematical logic and related topics, this lectureship, which he was awarded in 1907, being the first step taken in Germany towards the establishment of this subject as a separate mathematical discipline. Both will be seen in close connection with the first phase of Hilbert's programme of founding mathematics. It is shown, however, that this lectureship was not only created to further the aims of Hilbert's research programme but that personal and institutional motives also played a role. In the appendices, parts of a correspondence between Ernst Zermelo and the Göttingen Philosopher Leonard Nelson are edited, as well as the applications of the directors of the Göttingen Mathematisch-physikalisches Seminar for awarding the lectureship to Zermelo in 1907, and for appointing Zermelo to an extraordinary professorship in 1910. (shrink)

Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

A key idea in both Frege's development of arithmetic in theGrundlagen[7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains betweenconceptandnumber, in Zermelo's (through the axiom of choice), betweensetandmember. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type reducing (...) correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondencecbetween second-level entities (here calledconcepts) and first-level ones (here calledobjects) induces a well-ordering relationW(c) in a canonical manner. We shall see that, whencis the “Fregean” correspondence between concepts and cardinal numbers,W(c) is (the well-ordering of) the ordinalω+ 1, and whencis a “Zermelian” choice function on concepts,W(c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion ofextensionof a concept. To each conceptXwe assume there is assigned an objecte(X) in such a way that, for any conceptsX, Ysatisfying a certain predicateE, we havee(X) =e(Y) iff the same objects fall underXandY. (shrink)

It is claimed that cantor had the technical apparatus available to derive russell's paradox some ten years before russell's discovery.