Algebra pp 2-42

Foreword on Set Theory

• Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

Abstract

The building blocks of mathematics are sets, yet no one should say what a set is. An axiomatic treatment of set theory postulates the existence of certain undefined or primitive objects, called sets, together with symbols and axioms governing their use. A graphic analogy can be made with geometry where there are given undefined objects called points, lines, and planes, together with a collection of axioms relating these objects to each other. An axiom of projective geometry is P: If L 1 and L 2 are distinct lines, there is one and only one point P on both L 1 and L 2. Here the relation “P is on L 1” is undefined. Another axiom is P*: If P 1 and P 2 are distinct points, there is one and only one line L on both P 1 and P 2. Any collection of objects which, when properly named as “points” and “lines”, satisfy the axioms of geometry, serve as points and lines, equally as well as any other such collection. It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug (David Hilbert, quoted by Weyl [44, p. 635]). Nevertheless, from the axioms we soon discover that some objects in a geometry are not “points” and some objects are not “lines”. Analogously in the theory of sets, after we have named our candidates for sets, and listed the undefined symbols and the axioms governing them, we find that not everything in sight is a set. For example, as Cantor observed, there is no such set as “the set of all sets”.

Keywords

Continuum Hypothesis Duality Principle Order Isomorphism Transfinite Induction Generalize Continuum Hypothesis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

General References

1. Abian, A.: The Theory of Sets and Transfinite Arithmetic. Philadelphia: Saunders 1965.
2. Birkhoff, G.: Lattice Theory, Colloquium Publication. Vol. 25 (revised). Amer. Math. Soc., Providence 1948, 1967.Google Scholar
3. Bourbaki, N.: Theorie des Ensembles. Actualités Scientifiques et Industrielles, Nos. 1212, 1243. Paris: Hermann 1960, 1963, Chapters 1–3.Google Scholar
4. Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publ. 1942.Google Scholar
5. Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: Benjamin 1966.Google Scholar
6. Cohen, P. J.: The independence of the continuum hypothesis, I, II. Proc. Nat. Acad. Sci. U.S.A. 50, 1143–1148 (1963); 51, 105–110 (1964).
7. Cohn, P. M.: Universal Algebra. New York: Harper & Row 1965.
8. Fraenkel, A.: Über den Begriff „definit“ und die Unabhängigkeit des Auswahl axioms. Sitzungsber. Preuss. Akad. wiss. Phil.-math. 1922, 253–257.Google Scholar
9. Bar-Hillel, Y.: Set Theory. Amsterdam North-Holland Publ. 1958.Google Scholar
10. Gödel, K.: The Consistency of the Continuum Hypothesis. Princeton: Princeton University Press 1940; 1958.
11. Halmos, P.: Naive Set Theory. New York: Van Nostrand 1960.
12. Hausdorff, F.: Set Theory. New York: Chelsea 1957.
13. Hausdorff, F.: Mengenlehre. Leipzig 1914.Google Scholar
14. Kamke, E.: Theory of Sets. New York: Dover Publ. 1950.
15. Kelley, J.: General Topology. New York: Van Nostrand 1955.
16. Kurosch, A.: Theory of Groups, Vols. 1 and 2. New York: Chelsea 1955, 1956.Google Scholar
17. Kurosch, A.: Lectures on General Algebra. New York: Chelsea 1963.Google Scholar
18. Lawvere, F.: Elementary theory of the category of sets. Proc. Nat. Acad. Sci. U.S.A. 52, 1506–1511 (1964).
19. Lawvere, F.: The category of categories as a foundation for mathematics. Proc. Conf. Categorical Algebra. Amsterdam: North-Holland Publ. 1965, pp. 1–20.Google Scholar
20. Sierpinski, W.: L’Hypothèse généralisée du continu et l’axiome du choix. Fund. Math. 24, 1–5 (1947).
21. Sierpinski, W.: Cardinal and Ordinal Numbers. Monographs of the Polish Academy, Warsaw, 1958.
22. Solovay, R.: Independence results in the theory of cardinals (Preliminary Report), Notices of the Amer. Math. Soc. 10, 595, (1963).Google Scholar
23. Zermölo, E.: Beweis, daß jede Menge wohlgeordnet werden kann. Math. Ann. 59, 514–516 (1904).