In
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, New Foundations (NF) is an
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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, ...
, conceived by
Willard Van Orman Quine
Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...
as a simplification of the
theory of types
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
of ''
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
''. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and clarified by Holmes (1998). 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.
New Foundations has a
universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
, so it is a
non-well-founded set theory
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axio ...
. That is to say, it is an axiomatic set theory that allows infinite descending chains of membership, such as
… x
n ∈ x
n-1 ∈ … ∈ x
2 ∈ x
1. It avoids
Russell's 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 ...
by permitting only
stratifiable formulas to be defined using the
axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but x ∈ x is not.
The Type Theory TST
The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
(
) and
membership
Member may refer to:
* Military jury, referred to as "Members" in military jargon
* Element (mathematics), an object that belongs to a mathematical set
* In object-oriented programming, a member of a class
** Field (computer science), entries in ...
(
). TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-)
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''n'', type ''n''+1 objects are sets of type ''n'' objects; sets of type ''n'' have members of type ''n''-1. Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules:
and
. (Quinean set theory seeks to eliminate the need for such superscripts to denote types.)
The axioms of TST are:
*
Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
: sets of the same (positive) type with the same members are equal;
* An axiom schema of comprehension, namely:
::If
is a formula, then the set
exists.
:In other words, given any formula
, the formula