Editing New Foundations (section)

=== Typed Set Theory ===
New Foundations is closely related to '''Russellian unramified typed set theory''' ('''TST'''), a streamlined version of the theory of types of ''Principia Mathematica'' with a linear hierarchy of types.  In this [[Structure (mathematical logic)#Many-sorted structures|many-sorted]] theory, each variable and set is assigned a type. It is customary to write the ''type indices'' as superscripts: <math>x^n</math> denotes a variable of type ''n''. Type 0 consists of individuals otherwise undescribed. For each (meta-) [[natural number]] ''n'', type ''n''+1 objects are sets of type ''n'' objects; objects connected by identity have equal types and sets of type ''n'' have members of type ''n''-1. The axioms of TST are extensionality, on sets of the same (positive) type, and comprehension, namely that if <math>\phi(x^n)</math>{{Hair space}}is a formula, then the set <math>\{x^n \mid \phi(x^n)\}^{n+1}\!</math> exists. In other words, given any formula <math>\phi(x^n)\!</math>, the formula <math>\exists A^{n+1} \forall x^n [ x^n \in A^{n+1} \leftrightarrow \phi(x^n) ]</math> is an axiom where <math>A^{n+1}\!</math> represents the set <math>\{x^n \mid \phi(x^n)\}^{n+1}\!</math> and is not [[Free variables and bound variables|free]] in <math>\phi(x^n)</math>. This type theory is much less complicated than the one first set out in the ''Principia Mathematica'', which included types for [[Relation (mathematics)|relations]] whose arguments were not necessarily all of the same types.

There is a correspondence between New Foundations and TST in terms of adding or erasing type annotations. In NF's comprehension schema, a formula is stratified exactly when the formula can be assigned types according to the rules of TST. This can be extended to map every NF formula to a set of corresponding TST formulas with various type index annotations. The mapping is one-to-many because TST has many similar formulas. For example, raising every type index in a TST formula by 1 results in a new, valid TST formula.