Abstract
After 1918, most important contributions to the foundations of abstract set theory relied on modem axiom systems. But until the 1920s few authors adopted Zermelo’s axiom system explicitly.2 As we saw in the preceding chapter, many favored the theory of types because it seemed to offer a safer framework, and at the same time it was sufficient for the limited amount of set theory that is necessary in so-called classical mathematics. As late as 1939 Alonzo Church was writing that the simplified theory of types and Zermelo’s set theory were essentially similar, and the “safest cities of refuge” for classicist mathematicians at the time.3 But the Zermelo system had to compete with another alternative, the system of von Neumann, presented in 1925 and developed later by Bernays and Gödel (see §3).
We shall carefully investigate those ways of forming notions and those modes of inference that are fruitful; we shall nurse them, support them, and make them usable, whenever there is the slightest promise of success. No one shall be able to drive us from the paradise that Cantor created for us.1
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References
[Hilbert 1926] as translated in [van Heijenoort 1967, 375–76].
Among the few examples are Hessenberg [1909] and Hartogs [1915]. The latter proved that, in Zermelo’s system without AC, the comparability of cardinals implies Well-Ordering.
See [Church 1937, 95; 1939, 69–70].
Sierpinski was interested in open problems, like those related to AC and CH, their consequences and equivalences. See [Moore 1982; 1989].
The first to argue in earnest for Replacement, and to work out its consequences, was von Neumann in 1923. The history of Replacement has been carefully studied by Hallett [1984].
Fraenkel, who was a zionist, moved to Palestina and became a professor at the Hebrew University in Jerusalem in 1929. See his autobiography [Fraenkel 1967].
The most important instance was Separation, apart from which only the axiom of Elementary Sets was reduced to postulating pairs {a,b} (whence the name in use today, Axiom of Pairs).
But, unlike von Neumann, he could not see whether this detoured way of dealing with ordinals “presents real advantages,” although it throws new light on Cantor’s theory [1917a, 217]. Mirimanoff [1917, 45] considered also a criterion of set existence that would be characteristic of von Neumann’s axiomatization of set theory.
[Mirimanoff 1917, 38]: “Quelles sont les conditions nécessaires et suffisantes pour qu’un ensemble d’individus existe?”
See [Mirimanoff 1919, 35]. In this later paper his lack of understanding for axiomatic issues comes out clearly.
This point was emphasized by Hallett [1984].
This had become clear to him in correspondence with Fraenkel himself [1928, 323].
[von Neumann 1928, 334]. I have introduced a slight change to simplify the original notation.
Zermelo did not spell out the details of his logical standpoint, particularly the effect of second-order logic for the Skolem paradox. But see [Zermelo 1931] and [Grattan-Guinness 1979; Moore 1980; Dawson 1985a]. A clear discussion of the logical aspects can be found in [Shapiro 1991] or [Lavine 1994].
[Zermelo 1930, 31]: “Axiom der Fundierung: Jede (rückschreitende) Kette von Elementen, in welcher jedes Glied Element des vorangehenden ist, bricht mit endlichem Index ab bei einem Urelement. Oder, was gleichbedeutend ist: Jeder Teilbereich T enthält wenigstens ein Element t o , das kein Element t in That.“
Today one would require (with first-order quantification) that every non-empty set s in the domain have at least one element t, such that no element oft is also an element of s.
The `layers’ Q a of Zermelo are not cumulative, in contrast to his `sections’ P a [1930, 36]. Therefore it is the second which correspond to the ranks of the usual cumulative hierarchy; instead of P a we shall write V a for the ranks, as has become customary.
This term is taken from [Hellman 1989] and [Lavine 1994].
The same applies to different models with a single, common `basis’ of urelements (that differ in their `characteristic’). Likewise, of two different models with the same `characteristic,’ one is always isomorphic to a subdomain of the other [1930, 42].
For a careful analysis of this paper, see Hallett’s introduction to his translation of the paper in [Ewald 1996, vol.2].
Indeed, a double sequence of models, since we also have to take into account the many possible `bases’ of urelements [Zermelo 1930, 42, 47].
Gödel had already presented this conception in conferences during the 1930s; see volume 3 of his Collected Works.
Strictly speaking, it is wrong to say that Gödel offered the iterative conception as justification of the ZFC axioms. This was done by later authors. See [Klaua 1964], [Kreisel 1965], [Shoenfield 1967], [Boolos 1971], [Wang 1974], [Parsons 1977].
Comprehension obtains sets by dividing the universal class into two categories — objects that comply or do not comply with a given condition.
Von Neumann to Zermelo, August 1923, partly reproduced in [Meschkowski 1967, 28991].
The latter had already been stressed by Fraenkel, whose work seems to have guided many of von Neumann’s reflections.
[Meschkowski 1967, 290]: “Eine Menge ist dann und nur dann `zu gross’, wenn sie der Menge aller Dinge aequivalent ist.” In this letter, von Neumann employed naive terminology, as he would keep doing later, but it is clear that he was aware of the dangers.
In his opinion the development of the system became much simpler this way [1928a, 346; 1929, 494].
This is the “class theorem” proved in the first installment of Bernays’ series of papers, nowadays called theorem of predicative existence of classes.
In the 60s, Kripke, Cohen and Solovay, working independently, established that NBG with Global Choice is also a conservative extension of ZFC. The result was published later by Feigner [1971].
We shall not enter into more details regarding NBG here. Readers interested in a detailed analysis may turn, e.g., to [Fraenkel, Bar-Hillel & Levy 1973]. I would also like to emphasize that the historical interplay between ZFC and NBG ought to be the subject of more detailed research.
He refrained from publishing immediately, since he hoped to get further results regarding the independence of AC and CH (see Moore in [Gödel 1990, 158]).
Shepherdson [1951/53]. It was a very special inner model: the minimal one that contains all the ordinals and is transitive, i.e., such that, whenever x is in the model, so are elements of X.
Letter to Wang, 1968, in [Wang 1974, 10].
See the 1939 abstract in [Gödel 1990, 27]. Gödel writes Ma instead of the usual L a , and I have modernized his logical symbolism slightly.
As Solovay writes in [Gödel 1990, 8].
In ZF one writes: Vx3 a (xe L a ). Gödel’s axiom has been extensively studied by logicians, specially since the 60s; see Solovay in [Gödel 1990, 14–25].
Theorem 2 in [1938, 29]. This is the version of the axiom of reducibility that Gödel mentioned (see above). For further details, see Solovay in [Gödel 1990, 8–12] and [Gödel 1938].
In [1940], working within NBG, Gödel was able to give detailed proofs of his results without having to deal with metatheoretical notions within axiomatic set theory. The result, however, was a much less intuitive proof than the one offered for ZF in [1938]. Gödel himself admitted that the first exposition exhibited more clearly the basic idea of the proof in a note added in 1965 [Gödel 1940, 97] (see also Solovay in [Gödel 1990, 12–13]).
See [Moore 1988a], [Kanamori 1996] and Moore in [Godel 1990, 158–59]. Forcing is a powerful method, based on first-order logic, for defining models with prescribed properties.
As second-order logic can be seen to do, when the system is interpreted extensionally (§X.5).
I skip here over the issue of choosing between ZFC and NBG (see §3).
Another relevant but slightly earlier contribution is [Bemays 1935], which deals with Platonism in mathematics.
By the long sought proof of consistency of the theory of real numbers. As regards axiomatic set theory, von Neumann thought that a consistency proof was beyond reach [1929, 495].
See [Lakatos 1967], who calls this `quasi-empiricism,’ while I would prefer to speak of a hypothetical conception of (large parts of) mathematics. The most important popularizer of this view among philosophers has been Quine; see his [1953].
For Bourbaki’s hypothetical conception, see [1949, 3]. Perhaps I should mention that Bourbaki was no real person, but a fictitious character invented by a group of French mathematicians, who published their joint work under that name.
When the first volume of his treatise was published, Bourbaki [1954] had changed his axiom system slightly. Some indication of the difficulties can be found in [Corry 1996].
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Ferreirós, J. (1999). Consolidation of Axiomatic Set Theory. In: Labyrinth of Thought. Science Networks. Historical Studies, vol 23. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5049-0_11
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