a and b. Let a := {a} and let b := {a,b}. Prove or disprove a and b are the same set 
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other members of the University of Göttingen.
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member...
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:
∀
x
(
x
≠
∅
→
∃
y
∈
x
(
y
∩
x
=
∅
)
)
{\displaystyle \forall x\,(x\neq \varnothing \rightarrow \exists y\in x\,(y\cap x=\varnothing ))}
.
The axiom implies...
Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by Zermelo (1908) as a special case of his axiom of elementary sets.
== Formal statement ==
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
∀
A
∀
B
∃
C
∀
D
[
D
∈
C
⟺
(
D
=
A
...
In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Because restricting comprehension solved Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
...
