Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name. We will prove this by reductio ad absurdum. Let Ω be a set consisting of all ordinal numbers. Ω is transitive because for every element x of Ω (which is an ordinal number and can be any ordinal number) and every element y of x (i.e. under the definition of Von Neumann ordinals, for every ordinal number < ), we have that y is an element of Ω because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction. Ω is well ordered by the membership relation because all its elements are also well ordered by this relation. So, by steps 2 and 3, we have that Ω is an ordinal class and also, by step 1, an ordinal number, because all ordinal classes that are sets are also ordinal numbers. This implies that Ω is an element of Ω. Under the definition of Von Neumann ordinals, < is the same as Ω being an element of Ω. This latter statement is proven by step 5. But no ordinal class is less than itself, including Ω because of step 4 (Ω is an ordinal class), i.e. ≮. We have deduced two contradictory propositions ( < and ≮ ) from the sethood of Ω and, therefore, disproved that Ω is a set. The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (16)

Quotient method for controlling the acrobot

Dominique Bonvin, Philippe Müllhaupt

This paper describes a two-sweep control design method to stabilize the acrobot, an input-afﬁne under-actuated system, at the upper equilibrium point. In the forward sweep, the system is successively reduced, one dimension at a time, until a two-dimensiona ...

2009

Axiom schema of specification

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.

New Foundations

In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NF with urelements (NFU), an important variant of NF due to Jensen and clarified by Holmes. In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets.

Zermelo–Fraenkel set theory

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.