Page 155 - Textos de Matemática Vol. 40

A note on the history of functional self-application 143
In fact, there are many presentations of Russell’s paradox which are not based on sets (or classes). For example, Hilbert and Wilhelm Ackermann (1896–1962) [HA28, Chapter 4, §4] use a form based on predicates to discuss it. With reference to Heinrich Behmann (1891–1970), a former collaborator of Hilbert, Walter Dubislav (1895–1937) [Dub81, p. 94f.] presents such a predicate version in a “functionalized” way, very close to Hilbert’s paradoxical function, but using the negation function for predicates. Such a version was also discussed by Curry, [Cur30, p. 511]36 and [CF58, Section 0D],37 as well as Church, [Chu32, p. 347]. Since von Neumann’s set theory is based on the notion of function it also links Russell’s paradox to functions.38 In fact, his presentation of the paradox resembles Hilbert’s argumentation, [vN28, footnote 5, p. 346]:
Diese [Russell’s] Antinomie kann z.B. so hergeleitet werden: Wir w¨ahlen f nach III.1, dann ist fu¨r jedes I II-Ding (d.h. jede Argument- Funktion) [f, ⟨[x, x], A⟩] ̸= A oder = A, je nachdem [x, x] = A oder ̸= A ist. Also ist stets [f, ⟨[x, x], A⟩] ̸= [x, x]. Wir k¨onnen nun [f, ⟨[x, x], A⟩] mit Hilfe der Axiome II. 1.–7. auf die Form [g, x] bringen [...]; damit haben wir ein II-Ding (Funktion) g gewonnen, bei dem fu¨r jedes I II-Ding (jede Argument-Funktion) [g, x] ̸= [x, x] ist.
W¨are nun jedes II-Ding gleichzeitig auch I-Ding (jede Funktion auch Argument), so wu¨rde hieraus die Unm¨oglichkeit [g, g] ̸= [g, g] folgen.
Unsere Funktion g entspricht der Russellschen ”Menge aller Men- gen, die sich selbst nicht enthalten.“39
a consistent framework of set theory, as it is exemplified in the work on non-well-founded sets by Mirimanoff and Finsler mentioned above. Both are motivated by the paradoxes and Finsler argues that the non-well-founded sets provide the correct solution to them. For a modern account to non-well-founded sets see [Acz88]; this reference also contains a short appendix Notes Towards a History of non-well-founded sets.
36English translation in [Sel80b, p. 17f].
37In this context, the following passage from a letter of Curry to Dana Scott from May 1st, 1979 might be of interest [Sco80, p. 261]: “At a seminar at Harvard about 1926 Whitehead cited Suzanne Langer for the functional (or predicate) form of the Russell paradox; but I [Curry] think I saw it in Russell’s Principles of Mathematics.”
38Based on the fact that the functional base of the set theory of Von Neumann involves some axioms related to combinatory logic (cf. [CF58, p. 10f] and [CH0x, footnote 4]) one can speculate whether von Neumann was influenced by Scho¨nfinkel, cf. [CH0x, §2].
39“This [Russell’s] antinomy can, e.g., be derived in the following way: We choose f ac- cording to III.1, then for every I II-thing (i.e., every argument-function) [f,⟨[x,x],A⟩] ̸= A or = A, depending on [x,x] = A or ̸= A. So we always have [f,⟨[x,x],A⟩] ̸= [x,x]. We can now transform [f, ⟨[x, x], A⟩] with the help of the axioms II. 1.–7. into [g, x] [...]; with it we have won a II-thing (function) g, for which for every I II-thing (every argument-function) [g, x] ̸= [x, x].
If every II-thing now would be at the same time also I-thing (every function also argument), then from this would follow the impossibility [g, g] ̸= [g, g].
Our function g corresponds to the Russellian “set of all sets which do not contain itself.”