Simple theory of types

The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal systems, some of which can serve as alternatives to naive set theory as a foundation for all mathematics.

It has been tied to formal mathematics since Principia Mathematica to today's proof assistants.

In a letter to Gottlob Frege (1902) Russell announced his discovery of the paradox in Frege's Begriffsschrift. Frege promptly responded, acknowledging the problem and proposing a solution in a technical discussion of "levels". To quote Frege:

Incidentally, it seems to me that the expression "a predicate is predicated of itself" is not exact. A predicate is as a rule a first-level function, and this function requires an object as argument and cannot have itself as argument (subject). Therefore I would prefer to say "a concept is predicated of its own extension".

He goes about showing how this might work but seems to pull back from it. As a consequence of what has become known as Russell's paradox both Frege and Russell had to quickly amend works that they had at the printers. In an Appendix B that Russell tacked onto his The Principles of Mathematics (1903) one finds his "tentative" theory of types. The matter plagued Russell for about five years.

Willard Quine presents a historical synopsis of the origin of the theory of types and the "ramified" theory of types: after considering abandoning the theory of types (1905), Russell proposed in turn three theories:

Quine observes that Russell's introduction of the notion of "apparent variable" had the following result:

the distinction between 'all' and 'any': 'all' is expressed by the bound ('apparent') variable of universal quantification, which ranges over a type, and 'any' is expressed by the free ('real') variable which refers schematically to any unspecified thing irrespective of type.

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