in which a set is defined as containing all sets that do not contain themselves

One of the most famous and absurd recursive Set Theory examples is that of Russell's Paradox, in which a set is defined as containing all sets that do not contain themselves.

The above argument , known as Russell's Paradox

The above argument, known as Russell's Paradox, was discovered by Bertrand Russell in 1901.

set theory Paradox

If this is the case, the writer should inform himself so as to understand this logical, set theory Paradox (set forth by Russell in 1901) and of the existence and nature of several response arguments of logic to Russell's Paradox.

for naive set theory

My position, at any rate, is that (a) the Stone Paradox is just as much of a logical problem for the claim that an omnipotent being exists as Russell's Paradox is for naive set theory, (b) we have powerful empirical evidence against the existence of God in the form of the Problem of Evil, and that, (c) given the total absence of anything resembling evidence for the existence of God, even if there were no evidence against the existence of God, and theism were a logically coherent position, and we were in an 'all else being equal' situation, atheism would still win hands down as a matter of sheer ontological simplicity.

the paradox

In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox.

either of two interrelated logical antinomies

Russell’s paradox represents either of two interrelated logical antinomies.

that every set theory ... that contai

Russell's paradox shows that every set theory that contai ...

a barber

Specifically, Russell's paradox describes a barber who is defined such that Bertrand Russell both shaves himself and does not shave himself.

showed that the naive set theory of Frege leads to a contradiction

Part of the foundation of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.

Cantorian set theory led to contradictions

Russell's paradox served to show that Cantorian set theory led to contradictions, meaning not only that set theory had to be rethought, but most of mathematics (due to resting on set theory) was technically in doubt.

that don't belong to themselves

Russell's paradox arises once we contemplate these units that don't belong to themselves.

which illustrates in a simple fashion the problems inherent in naive set theory

Russell discovers 'Russell's paradox' which illustrates in a simple fashion the problems inherent in naive set theory.

a famous example of an impredicative construction— namely the set of all sets that do not contain themselves

Russell's paradox is a famous example of an impredicative construction—namely the set of all sets that do not contain themselves.

as a result of naive set theory's so-called unrestricted comprehension (or abstraction) axiom

Russell's paradox arises as a result of naive set theory's so-called unrestricted comprehension (or abstraction) axiom.

a contradiction——at the very heart of the system of logic upon which Russell had hoped to build the whole of mathematics now known as Russell’s paradox

In 1901 Russell discovered a contradiction—now known as Russell’s paradox—at the very heart of the system of logic upon which Russell had hoped to build the whole of mathematics.

a road marker suggesting a boundary of reason, or at least precision in language

Russell’s paradox was a road marker suggesting a boundary of reason, or at least precision in language.

in the set theory developed by Frege, in which Bertrand Russell discovered a contradiction known as Russell's Paradox

Russell also famously highlighted a flaw in the set theory developed by Frege, in which Bertrand Russell discovered a contradiction known as Russell's Paradox.

a set-theoretic echo of an earlier paradox, one that was known to the ancient Greeks and is called the Epimenides paradox by some philosophers

By the way, the Russell paradox is a set-theoretic echo of an earlier paradox, one that was known to the ancient Greeks and is called the Epimenides paradox by some philosophers.

that every set theory ... Berry's paradox that contains a

Russell's paradox shows that every set theory that contains a ... Berry's paradox

a self-contradiction in one of a self-contradiction's basic laws

The goal of this work was to show that mathematics was just a part of logic, but Russell's Paradox revealed a self-contradiction in one of a self-contradiction's basic laws.