(in category theory/type theory/computer science)
of all homotopy types
of (-1)-truncated types/h-propositions
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
In a constructive weakly predicative structural set theory, such as doing mathematics in a well-pointed cartesian closed pretopos with a natural numbers object and an inaccessible object, it is still possible to define a category inside of the set theory, and in particular, it is still possible to define a well-pointed cartesian closed pretopos object with a natural numbers object inside of any well-pointed cartesian closed pretopos with a natural numbers object. These objects are called universes or Grothendieck universes in structural set theory. From here, one could form the sub-(0,1)-category of subterminal objects in , the set of -small subsingletons, and subsingletons model propositions in the universe, so could be called a set of -small propositions as well.
If a set theory has multiple such universes and , then there are multiple such sets of propositions, one of which is -small, and another one of which is -small. In general, the set of -small propositions and the set of -small propositions cannot be proven to be equivalent to each other. However, propositional resizing is the axiom that for universe and , the set of -small propositions is in bijection with the set of -small propositions, .
In homotopy type theory, the type of h-propositions of a Tarski universe (or Russell universe) is not in general essentially -small. Propositional resizing is then the statement that the type of h-propositions is essentially -small.
There is a separate axiom of propositional resizing for a hierarchy of universes, where the type of h-propositions of each universe, where one universe embeds into the other universe, are equivalent to each other.
Propositional resizing is a form of impredicativity for h-propositions, and by avoiding its use, the universe or hierarchy is said to remain predicative.
However, when using Tarski universes, while universes and universe hierarchies may be impredicative, the overarching type theory is still predicative if it has a judgment ‘’, since it is impossible to talk about all types.
We work in a structural set theory which externally forms a well-pointed cartesian closed lextensive coherent category with a natural numbers object, namely a structural set theory with
Any well-pointed cartesian closed coherent category with a natural numbers object inside of the set theory is thus an internal model of the set theory, and thus could be considered to be a universe inside of the set theory.
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There are many different notions of propositional resizing in type theory. These include
propositional resizing for individual universes
propositional resizing for universe hierarchies
propositional resizing for the entire dependent type theory, if the dependent type theory is defined via universes and universe levels.
A universe hierarchy satisfies propositional resizing if it satisfies both local propositional resizing and global propositional resizing. The universe hierarchy is then said to be impredicative.
Similarly, if the dependent type theory is defined via universes and universe levels, the dependent type theory satisfies propositional resizing if it satisfies both local propositional resizing and global propositional resizing, and then the dependent type theory is then said to be impredicative.
Let be a weakly Tarski universe and let
be the type of all propositions in . The weakly Tarski universe is impredicative or satisfies propositional resizing if it has a term and an equivalence
A weakly Tarski universe is strictly impredicative or satisfies strict propositional resizing if the above equivalence becomes a definitional equality:
In particular, every impredicative strictly Tarski universe is strictly impredicative.
Any universe of propositions satisfying propositional resizing is a contractible type.
Given a universe of propositions , the type of all propositions in is itself. Then propositional resizing says that has an element such that its type reflection is itself. This implies that itself a mere proposition by definition of universe of proposiions, which implies that is a contractible type by the element and the fact that for all other elements , .
Any universe of propositions closed under the empty type does not satisfy propositional resizing.
Suppose that is closed under the empty type represented in by the element , and satisfies propositional resizing. Then one has that since is a mere proposition, and by transport one has , but since is contractible by propositional resizing, the empty type is also contractible, which is false.
The universe of all propositions does not satisfy propositional resizing.
Last revised on December 7, 2024 at 10:46:38. See the history of this page for a list of all contributions to it.