ST type theory

The following system is Mendelson's (1997, 289–293) ST type theory. ST is equivalent with Russell's ramified theory plus the Axiom of reducibility. 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. ST is "simple" (relative to the type theory of Principia Mathematica) primarily because all members of the domain and codomain of any relation 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

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

${\displaystyle \in }$. In any given formula, unprimed variables all have the same type, while primed variables (${\displaystyle x'}$) range over the next higher type. The

${\displaystyle x=y}$

${\displaystyle y\in x'}$

${\displaystyle \in }$

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 '${\displaystyle \in }$', 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

• Identity, defined by ${\displaystyle x=y\leftrightarrow \forall z'[x\in z'\leftrightarrow y\in z']}$.
• Extensionality. An axiom schema. ${\displaystyle \forall x[x\in y'\leftrightarrow x\in z']\rightarrow [y'=z']}$.

Let ${\displaystyle \Phi (x)}$ denote any

${\displaystyle x}$

• Comprehension. An axiom schema. ${\displaystyle \exists z'\forall x[x\in z'\leftrightarrow \Phi (x)]}$.

Remark. Any collection of elements of the same type may form an object of the next higher type. Comprehension is schematic with respect to ${\displaystyle \Phi (x)}$ as well as to types.

• Infinity. There exists a nonempty binary relation ${\displaystyle R}$ over the individuals of the lowest type, that is irreflexive, transitive, and strongly connected: ${\displaystyle \forall x,y[x\neq y\rightarrow [xRy\vee yRx]]}$ and with codomain contained in domain.

Remark. Infinity is the only true axiom of ST and is entirely mathematical in nature. It asserts that ${\displaystyle R}$ is a strict total order, with a codomain contained in its domain. If 0 is assigned to the lowest type, the type of ${\displaystyle R}$ is 3. Infinity can be satisfied only if the (co)domain of ${\displaystyle R}$ is

ZFC

of other set theories could not be married to ST.

ST reveals how type theory can be made very similar to

.

Formulations based on equality

This section needs expansion. You can help by adding to it. (September 2009)

Church's type theory has been extensively studied by two of Church's students,

• Type theory

References