basic constructions:
strong axioms
further
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Russell’s paradox is a famous paradox of set theory^{1} that was observed around 1902 by Ernst Zermelo^{2} and, independently, by the logician Bertrand Russell. The paradox received instantly wide attention as it lead to a contradiction in Frege’s monumental “Foundations of Arithmetic” (1893/1903) whose second volume was just about to go to print when Frege was informed about the inconsistency by Russell.
The paradox entangles a concept with its own extension in a vicious circle. The attempt to overcome this circularity in set formation had a huge impact on subsequent forms of axiomatic set theory and in the aftermath mathematical logic became heavily focussed on consistency proofs for fully specified formal theories: the paradoxes triggered a shift in the foundation of mathematics away from the mathematics to the foundation itself.
Doch zur Sache selbst! Herr Russell hat einen Widerspruch aufgefunden, der nun dargelegt werden mag.^{3}
If one assumes a naive, full axiom of comprehension, one can form the set
One then asks: is $R\in R$? If so, then $R\notin R$ by definition, whereas if not, then $R\in R$ by definition. Thus we have both $R\in R$ and $R\notin R$, a contradiction.
Russell’s paradox is closely related to the classical liar paradox (“this sentence is false”), to Gödel’s incompleteness theorem, and to the halting problem — all use a diagonalization argument to produce an object which talks about itself in a contradictory or close-to-contradictory way.
On the other hand, Cantor's paradox can be said to “beta-reduce” to Russell’s paradox when we apply Cantor's theorem to the supposed set of all sets. See Cantor's paradox for explanation.
Also related:
There are a number of possible resolutions of Russell’s paradox.
Russell himself (1903,1908,1910) proposed the introduction of type theory as a solution e.g. in Principia Mathematica (1910) an intricate system of ramified types tracks the variables of propositional functions in order to prevent circular propositions. This is inspired by Poincaré‘s ideas on impredicativity and can be viewed as a radical generalisation of Frege’s ontological distinction between an argument as a satured object and a function or concept as an unsaturated object.
The “classical” solution, adopted in ZFC and thus by the mathematical “mainstream”, is to restrict the axiom of comprehension so as to disallow the formation of the set $R$: one requires that the set being constructed be a subset of some already existing set. The restricted axiom is usually given a different name such as the axiom of separation.
Another solution is to distinguish between sets and proper classes (= collections that are “too big” to be sets) as e.g. in NBG “set” theory.^{4} Here we may write down the definition of $R$, but from $R \notin R$ we may conclude $R \in R$ only if we already know that $R$ is a set; the $x$ in the definition must be a set. So we have no contradiction, but only a proof that $R$ is a proper class.
In the set theory called New Foundations, the axiom of comprehension is restricted in a rather different way, by requiring the set-defining formula to be “stratifiable”. Since the formula $x\notin x$ is not stratifiable, the set $R$ cannot be formed. This related to Russell’s ideas on ramified types.
In most structural set theories, the featurelessness of the elements of the structural sets secures the consistency of set formation. If sets cannot be elements of other sets, then the “definition” of $R$ is just a type error. The same is true in other structural foundational systems such as (modern, non-Russellian) type theory. However, Russell’s paradox can be recreated in structural foundations with inconsistent universes by constructing pure sets within them.
Alternatively, one can change the underlying logic. Passing to constructive logic does not help: although there is a seeming appeal to excluded middle (either $R\in R$ or $R\notin R$), without using excluded middle we can obtain that $R$ is both not in $R$ and not not in $R$, which is also a contradiction. However, passing to linear logic (or even affine logic) does help: there is an unavoidable use of contraction in the paradox. There exist consistent linear set theories? with the full comprehension axiom, in which $R\in R$ implies $R\notin R$ and vice versa, but we can never get both $R\in R$ and $R\notin R$ at the same time to produce a paradox. (See the Ph.D. thesis Linear Set Theory of Masaru Shirahata, completed under Grigori Mints supervision, Stanford University, 1994.)
Finally, and perhaps most radically, one can decide to allow contradictions, choosing to use a paraconsistent logic. There exist nontrivial paraconsistent set theories with full comprehension in which the set $R$ exists, being both a member of itself and not a member of itself.
Zermelo’s observation which is mentioned in a footnote of his 1908 paper on the well-ordering theorem is analyzed in
Russell indicated the contradiction leading to the inconsistency of G. Frege’s system of “Grundgesetze der Arithmetik” in a famous letter to the latter on June 16th 1902 which together with Frege’s reply is reprinted pp.124-128 in
For an account of Russell’s encounter with the paradox:
The first published account is presumably in the appendix of
Russell discusses the paradox extensively in chapter X of
The influence of Poincaré‘s views shows in
See also
Discussion of a paradox similar to Russell’s in type theory with W-types is in
naive material set theory that is! ↩
Zermelo was apparently led to the paradox by considering a purported proof of Ernst Schröder in his Algebra der Logik (1890) that Boole’s concept of a universal class 1 was contradictory. Interestingly, the solution suggested by Schröder is reminiscent of type theory or even nested universes where a universal class 1 recurs at each level. ↩
This solution proposed by J. von Neumann in the 1920s can be viewed as related to ideas of G. Cantor. The latter knew about similar phenomena concerning “the set of all sets” (in fact, Russell hit upon the paradox in a reflection on Cantor’s proof of the inexistence of a largest cardinal number), and had already pointed out in 1885 in a review of Frege’s “Grundlagen der Arithmetik” that not every concept has an extension. Therefore Cantor proposed in letters to Dedekind and Jourdain to differentiate between ‘consistent multiplicities’ (“compossible” multiplicities one could say) where things can coexist or compose to a consistent whole - ‘ensemble’ (fr.) which correspond to sets in the usual sense and, as completed collections, can in turn be elements in other sets, from ‘inconsistent multiplicities’ whose elements cannot consistently completed to a whole and cannot be member of other collections due to this lack of ‘unity’. ↩