In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a field of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician, after whom the Burali-Forti paradox is named.
Biography
Burali-Forti was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 18 ...
, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor.
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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.
Stated in terms of von Neumann ordinals
We will prove this by a deliberational deconstruction.
# Let be a set consisting of all ordinal numbers.
# is
transitive because for every element of (which is an ordinal number and can be any ordinal number) and every element of (i.e. under the definition of
Von Neumann ordinal
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
s, for every ordinal number ), we have that is an element of because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.
# 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 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 is an element of .
# Under the definition of Von Neumann ordinals, is the same as being an element of . This latter statement is proven by step 5.
# But no ordinal class is less than itself, including because of step 4 ( is an ordinal class), i.e. .
We have deduced two contradictory propositions ( and ) from the sethood of and, therefore, disproved that is a set.
Stated more generally
The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
, 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.
Here is an account with fewer presuppositions: suppose that we associate with each
well-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
an object called its
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
in an unspecified way (the order types are the ordinal numbers). The order types (ordinal numbers) themselves are well-ordered in a natural way,
and this well-ordering must have an order type
. It is easily shown in
naïve set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It de ...
(and remains true in
ZFC but not in
New Foundations) that the order
type of all ordinal numbers less than a fixed
is
itself.
So the order
type of all ordinal numbers less than
is
itself. But
this means that
, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals,
but the latter is
itself by definition. This is a contradiction.
If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed
is
itself must be true. The collection of von Neumann ordinals, like the collection in the
Russell paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
, cannot be a set in any set theory with classical logic. But the collection of order types in New Foundations (defined as equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of the ordinals less than
turns out not to be
.
Resolutions of the paradox
Modern
axioms for formal set theory such as ZF and ZFC circumvent this antinomy by not allowing the construction of sets using
terms like "all sets with the property ", as is possible in
naive set theory and as is possible with
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
's axiomsspecifically Basic Law Vin the "Grundgesetze der Arithmetik." Quine's system
New Foundations (NF) uses a
different solution. showed that in the original version of Quine's system "Mathematical Logic" (ML), an extension of New Foundations, it is possible to derive the Burali-Forti paradox, showing that this system was contradictory. Quine's revision of ML following Rosser's discovery does not suffer from this defect, and indeed was subsequently proved equiconsistent with NF by
Hao Wang.
See also
*
Absolute Infinite
The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor.
It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfin ...
References
*
*
Irving Copi (1958) "The Burali-Forti Paradox",
Philosophy of Science
Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
25(4): 281–286,
*
*
External links
*
Stanford Encyclopedia of Philosophy:
Paradoxes and Contemporary Logic—by Andrea Cantini.
{{Set theory
Ordinal numbers
Paradoxes of naive set theory