The anecdotes Berger recounted in the thirty-minute monologue are adaptations of well-known paradoxes: Zeno's paradoxes of motion and Bertrand
Russell's paradox of set theory, which concerns the impossibility of a set's containing itself--hence the exhibition's title, "Not to Belong to Themselves." Berger's monologue, scripted by Garcia Torres together with the comedian and producer Eduardo Donjuan, transplants these intellectual games into everyday life, where they produce absurd situations.
But a similar point arises with certain predicates, and indeed even the proper concept of a predicate has been lost, producing the most notorious of the standard puzzles:
Russell's Paradox. The paper inevitably makes reference to contemporary logical symbolism in places, but its purpose is to show that the only way forward is to use a symbolism that is abbreviatory only, that is, one that maps directly onto natural speech
Soon I began thmking of this as a form of
Russell's Paradox: "Borges" only shaves those Borgeses who do not shave themselves!
The antonym of autological is heterological--and don't ask whether this word is autological or heterological because either way you'll get a contradiction (this is known to logicians as the Grelling-Nelson Paradox, and it's closely related to
Russell's Paradox the discovery of which was one of the milestones leading to the development of modern logic).
Frege Arithmetic is the core second-order logic of GgA minus the incriminated Basic Law V, which leads to
Russell's Paradox, but supplemented with HP, which Frege derives with the help of Basic Law V.
Just four years later,
Russell's paradox would put a closely related result in vivid form, as the demonstration of the contradiction that follows necessarily from the supposition that there exists a set of all sets that are not members of themselves.
Some paradoxes require the revision of their intuitive conception (
Russell's paradox, Cantor's paradox), others depend on the inadmissibility of their description (Grelling's paradox), others show counterintuitive features of formal theories (Material implication paradox, Skolem Paradox), others are self-contradictory [Smarandache Paradox: "All is <A> the <Non-A> too!", where <A> is an attribute and <Non-A> its opposite; for example "All is possible the impossible too!" (Weisstein, 1998)].
Russell's paradox is still topical: it will not go away (Moorcroft 1993).
SUMMARY: This paper is concerned with locating the specific assumption that led Frege into
Russell's Paradox. His understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking one could invariably replace a two-place relation with a one-place property.
Some paradoxes require the revision of their intuitive conception (
Russell's paradox, Cantor's paradox), others depend on the inadmissibility of their description (Grelling's paradox), others show counter-intuitive features of formal theories (Material implication paradox, Skolem Paradox), others are self-contradictory--Smarandache Paradox: "All is <A> the <Non-A> too!", where <A> is an attribute and <Non-A> its opposite; for example "All is possible the impossible too!" (Weisstein, 1998 [2]).
The contradictions in
Russell's paradox of the barber who only
In order to illustrate the theoretical merits of this conception, I use the process-oriented epistemology in order to deal with
Russell's paradox, which has amazed me for many years.