HISTORICAL NOTES AND GUIDE TO THE BIBLIOGRAPHY CHAPTER 1 Set theory was invented by Georg Cantor. The first attempt to consider infinite sets is at­ tributed to Bolzano (who also introduced the term Menge). It was however Cantor who realized the significance of one-to-one functions between sets and introduced the notion of cardinality of a set. Cantor originated the theory of cardinal and ordinal numbers as well as the investigations of the topology of the real line. Much of the development in the first four sections follows Cantor's work. The main reference to Cantor's work is his collected works, Cantor [1932]. Another source of references to the early research in set theory is HausdorlT's book [1914]' Cantor started his investigations in [1874], where he proved that the set of all real numbers is uncountable, while the set of all algebraic reals is countable. In [1878] he gave the first formulation of the celebrated continuum hypothe·sis. Section 1 The axioms for set theory (except Replacement and Regularity) are due to Zermelo [1908]. Prior to Zermelo, Frege attempted to axiomatize set theory, using the false comprehension schema; Zermelo formulated his axioms following the discovery of several paradoxes by Burali-Forti, Cantor, Russell, and Zermelo. The Replacement schema is due to Fraenkel [1922a] and Skolem (see [1970, pp. 137-152]). Exercises 1.12 and 1.13: Tarsk i [1925a]. Section 2 The theory of well-ordered sets was developed by Cantor, who also introduced transfinite induction. The idea of identifying an ordinal number with the set of smaller ordinals is due to Zermelo and von Neumann. Section 3 Cardinal numbers and alephs were introduced by Cantor. The proof of the Cantor-Bernstein theorem is Bernstein's; see Borel [1898, p. 103]. The first proof of t-:, . t-:, = t-:, appeared in Hessen­ berg [1906, p. 593]. Regularity of cardinals was investigated by Hausdorff, who also raised the question of existence of regular limit cardinals. D-finiteness was formulated by Dedekind. 579 580 HISTORICAL NOTES AND GUIDE TO THE BIBLIOGRAPHY Section 4 Theorems 10 and II are due to Cantor. The construction of real numbers by completion of the rationals is due to Dedekind [1872). Theorem 13: Baire (1899). Theorem 14: Cantor, Bendixson [1883). Exercises 4.3, 4.4, 4.5a: Cantor. Suslin's problem: Suslin (1920). Section 5 The axiom of choice was formulated by Zermelo, who used it to prove the well-ordering theorem in [1904)' The present form of the Kuratowski-Zorn lemma is as in Zorn [1935]; for a related principle, see Kuratowski [1922)' (Hausdorff in [1914, pp. 140-141] proved that every partially ordered set has a maximal linearly ordered subset.) The principle of dependent choices was formulated by Bernays in [1937-1954, III). Exercise 5.1: Felix Bernstein. Section 6 Konig's theorem appeared in J. Konig [1905]. Theorem 18 was found independently by Bukovsky [1965] and Hechler. The discovery that cardinal exponentiation is determined by the gimel function (Theorem 20) was made by Bukovsky; cf. [1965). The inductive computation oft'{~' is as in Jech [l973b]; that paper also investigates properties of the gimel function. Exercise 6.6: L. Patai. Hausdorff's formula (6.18): Hausdorff [1904). Exercise 6.8: Tarski [l925b). Exercises 6.15 and 6.19: Jech [l973b]' Inaccessible cardinals were introduced in the paper by Sierpinski and Tarski [1930]; see Tarski [1938] for more details. Section 7 The notion of filter is. according to Kuratowski's book [1966], due to H. Cartan. The ultrafilter theorem was first proved by Tarski in [1930]' Fodor's theorem appeared in Fodor [1956]. Reduced products were first investigated by -tos in [1955]; see Section 28. Lemma 7.6 is a special case of Theorem 85 of Solovay proved in Section 35. Exercise 7.12: Friedman [1974]. (For related problems, see Prikry and Solovay [1975]). Mahlo cardinals are named after P. Mahlo, who in 1911-1913 investigated what we call weakly Mahlo cardinals. Section 8 Theorem 23 is due to Silver. Silver's proof [1974] combines forcing with the ultrapower method; see Section 35. An elementary proof was subsequently found by Baumgartner and Prikry [1976. 1977] and Jensen. The combinatorial argument of Baumgartner and Prikry was extended by Galvin and Hajnal in [1975] to obtain Theorem 24. The proof of Lemma 8.7 is a modification of Galvin-Hajnal's method. Lemma 8.5: Erdos. Hajnal, and Milner [1968]. Models where the singular cardinals hypothesis fails have been constructed by Magidor, cf. [1977b] and [1977c]. Section 9 The axiom of regularity was introduced by von Neumann in [1925]. although a similar prin­ ciple had been considered previously by Skolem (see [1970. pp. 137-152]). The transitive collapse is HISTORICAL NOTES AND GUIDE TO THE BIBLIOGRAPHY 581 defined in Mostowski [1949]; see Theorem 28 in Section II. Induction on well-founded relations (Theorem 25') was formulated by Montague in [1955]. Appendix The axiomatic system BG was introduced by Bernays in [1937-1954, I] and used by Godel in his consistency proof [1940]. Shoenfield's result was published in [1954]. F or more references on the history of axioms of set theory consult Fraenkel et al. [1973]. CHAPTER 2 The main results, namely consistency of the axiom of choice and ofthe generalized continuum hypothesis, are due to Kurt Godel, as is the concept of constructible sets. The results were an­ nounced in [1938], and an outline of proof appeared in [1939]. Godel's monograph [1940] contains a detailed construction of L and the proof that L satisfies AC and GCH. Section 10 For concepts of model theory, the history of the subject and for model-theoretical terminology, I refer the reader to Chang and Keisler's book [1973]. The impossibility of a consistency proof of existence of inaccessible cardinals follows from Godel's theorem [1931]' An argument that more or less establishes the consistency of the axiom of regularity appeared in Skolem's work in 1923 (see [1970, pp. 137-152]). Exercise 10.6: Montague and Vaught [1959]. Section 11 The investigation of transitive models of set theory was of course motivated by GOdel's construction of the model L. The first systematic study of transitive models was done by Shepherd­ son in [1951-1953]. Bernays in [1937-1954, I], employed a finite number of operations on classes to give a finite axiomatization of BG (see Exercise 11.9). Theorem 31 is explicitly stated by Hajnal in [ 1956]. The collapsing theorem 28 appeared in Mostowski [1949]. The reflection principle was in­ troduced by Montague; see [1961] and Uvy [1960a]. Exercise 11.2: Galvin. Section 12 In [1939] Godel defined constructible sets using LH 1 = {all subsets of L. definable over L.}; in [1940] he used finitely many operations (and worked in the system BG). Exercise 12.5: The J. hierarchy was used by Jensen in [1972]. Section 13 Both Theorems 33 and 34 are due to GOdel. Section 14 The 1:. hierarchy was introduced by Uvy in [1965b]. Uvy proved in [1965b] Theorem 36 (the absoluteness lemma), using the principle of dependent choices and Shoenfield's absoluteness theorem (Theorem 98 in Section 41 )-see Exercise 41.5; the present proof is due to Barwise and Fisher [1970]' Another result of Levy [1965] is that the truth predicate ~.+ 1 is 1:.+ 1 (Exercise 14.20); for Tarski's theorem, see, e.g., Tarski [1939]. 582 HISTORICAL NOTES AND GUIDE TO THE BIBLIOGRAPHY Karp's paper [1967] investigates LI relations and gives a detailed computation verifying that constructibility is L I . Primitive recursive functions appear in Jensen and Karp [1971]. The charac­ terization of the sets L. as transitive models of a single sentence (J is a result of Boolos [1970]. Exercises 14.23 and 14.24: Jensen [1972]. Section 15 Relative constructibility was investigated by Hajnal [1956], Shoen field [1959] and most gen- erally by Levy [1957] and [1960b]. Lemma 15.1: Vopenka and Balcar [1967]. Lemma 15.2 and Exercise 15.10: Levy [1957] and [1960b]. Exercise 15.16: see Theorem 98. The concept of ordinal definability was suggested by Godel in his talk [1946]; the theory was developed independently by Myhill and Scott in [1971] and by Vopenka, Balcar, and Hajek in [1968]. The former authors use the definition from Exercise 15.18, the latter use (15.16). CHAPTER 3 The method of forcing was invented by Paul Cohen who used it to prove the independence of the continuum hypothesis and of the axiom of choice (see [1963-1964] and the book [1966]). The present formulation of Cohen's method using Boolean-valued models has been developed by Scott, Solovay, and Vopenka. Section 16 The two fundamental theorems of the method of forcing, the forcing theorem and the generic model theorem, are due to Cohen. (To be accurate, Cohen's original method was formulated for particular examples of a notion of forcing, and under the assumption that 9)1 is a countable transitive model of ZFC. The definition (16.3) of an 9)I-generic filter on a partially ordered set is due to Solovay.) The forcing conditions defined in (16.4) are the ones used by Cohen in his proof of independence of the continuum hypothesis; the" typical example," a collapsing mapping of w onto WI is also due to Cohen. Section 17 Most of the facts about Boolean algebras can be found in Sikorski's book [1964], which also contains an extensive bibliography. The representation theorem for Boolean algebras (Theorem 41) is due to Stone [1936]. The important lemma 17.6 on the chain condition was proved by Erdos and Tarski in [1943]. For a more detailed discussion of the role of the prime ideal theorem, see my book on the axiom of choice [1973c]; for the model-theoretical concepts involved, consult Chang and Keisler [1973].