41. In contrast in intensional type theory type checking is integer numbers, rational numbers, and real numbers. 42. The simplest way I can imagine having an expanding multiverse would be with a bubble universe type theory . 43. The type theory has a number of " type formers " and 4 kinds rules for each one. 44. They may also use formal logics that are stronger than first-order logic, such as type theory . 45. The common usage of " type theory " is when those types are used with a term rewrite system. 46. Multiple type theories are supported by Mizar is an example of a proof system that only supports set theory. 47. Induction-Recursion came out of investigations to the rules of Martin-L�f's intuitionistic type theory . 48. There is an element of U for every type in the type theory ( except U itself ! ). 49. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like Epigram. 50. Type theory extends this identification to predicate logic by introducing dependent types, that is, types that contain values.
