The following system is Mendelson's (1997, 289–293) ST
type theory
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 fou ...
.
ST is equivalent with Russell's ramified theory plus the
Axiom of reducibility
The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis ...
.
The
domain of quantification is partitioned into an ascending hierarchy of types, with all
individuals assigned a type. Quantified variables range over only one type; hence the underlying logic is
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. ST is "simple" (relative to the type theory 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. ...
'') primarily because all members of the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
and
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
of any
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
must be of the same type.
There is a lowest type, whose individuals have no members and are members of the second lowest type. Individuals of the lowest type correspond to the
urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual.
Theory
There ...
s of certain set theories. Each type has a next higher type, analogous to the notion of
successor
Successor may refer to:
* An entity that comes after another (see Succession (disambiguation))
Film and TV
* ''The Successor'' (film), a 1996 film including Laura Girling
* ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
in
Peano arithmetic. While ST is silent as to whether there is a maximal type, a
transfinite number
In mathematics, transfinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to q ...
of types poses no difficulty. These facts, reminiscent of the Peano axioms, make it convenient and conventional to assign a
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 ...
to each type, starting with 0 for the lowest type. But type theory does not require a prior definition of the naturals.
The symbols peculiar to ST are primed variables and
infix operator
Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in .
Usage
Binary relations a ...
. In any given formula, unprimed variables all have the same type, while primed variables (
) range over the next higher type. The
atomic formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
s of ST are of two forms,
(
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
) and
. The infix-operator symbol
suggests the intended
interpretation, set membership.
All variables appearing in the definition of identity and in the axioms ''Extensionality'' and ''Comprehension'', range over individuals of one of two consecutive types. Only unprimed variables (ranging over the "lower" type) can appear to the left of '
', whereas to its right, only primed variables (ranging over the "higher" type) can appear. The first-order formulation of ST rules out quantifying over types. Hence each pair of consecutive types requires its own axiom of Extensionality and of Comprehension, which is possible if ''Extensionality'' and ''Comprehension'' below are taken as
axiom schemata "ranging over" types.
* Identity, defined by