New Foundations - The Type Theory TST

The Type Theory TST

The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality and membership . TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number 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 (notation to be improved).

The axioms of TST are:

• Extensionality: 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 is an axiom where represents the set .

This type theory is much less complicated than the one first set out in the Principia Mathematica, which included types for relations whose arguments were not necessarily all of the same type. In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchy of sets described here.