Some facts and definitions concerning choice and description in intuitionistic logic.

References:

[Bell] John L. Bell, Choice principles in intuitionistic set theory, unpublished. [Bell93] John L. Bell, Hilbert's Epsilon Operator in Intuitionistic Type Theories, Mathematical Logic Quarterly, volume 39, 1993. [Carlström04] Jesper Carlström, EM + Ext + AC_int is equivalent to AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004. [Carlström05] Jesper Carlström, Interpreting descriptions in intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. [Werner97] Benjamin Werner, Sets in Types, Types in Sets, TACS, 1997.

Definitions

Choice, reification and description schemes We make them all polymorphic. Most of them have existentials as conclusion so they require polymorphism otherwise their first application (e.g. to an existential in Set) will fix the level of A.

Constructive choice and description

AC_rel = relational form of the (non extensional) axiom of choice (a "set-theoretic" axiom of choice)

AC_fun = functional form of the (non extensional) axiom of choice (a "type-theoretic" axiom of choice)

AC_fun_dep = functional form of the (non extensional) axiom of choice, with dependent functions

AC_trunc = axiom of choice for propositional truncations (truncation and quantification commute)

DC_fun = functional form of the dependent axiom of choice

ACw_fun = functional form of the countable axiom of choice

AC! = functional relation reification (known as axiom of unique choice in topos theory, sometimes called principle of definite description in the context of constructive type theory, sometimes called axiom of no choice)

AC_dep! = functional relation reification, with dependent functions see AC!

AC_fun_repr = functional choice of a representative in an equivalence class

AC_fun_setoid = functional form of the (so-called extensional) axiom of choice from setoids

AC_fun_setoid_gen = functional form of the general form of the (so-called extensional) axiom of choice over setoids

AC_fun_setoid_simple = functional form of the (so-called extensional) axiom of choice from setoids on locally compatible relations

ID_epsilon = constructive version of indefinite description; combined with proof-irrelevance, it may be connected to Carlström's type theory with a constructive indefinite description operator

ID_iota = constructive version of definite description; combined with proof-irrelevance, it may be connected to Carlström's and Stenlund's type theory with a constructive definite description operator)

Weakly classical choice and description

GAC_rel = guarded relational form of the (non extensional) axiom of choice

GAC_fun = guarded functional form of the (non extensional) axiom of choice

GAC! = guarded functional relation reification

OAC_rel = "omniscient" relational form of the (non extensional) axiom of choice

OAC_fun = "omniscient" functional form of the (non extensional) axiom of choice (called AC* in Bell [Bell])

D_epsilon = (weakly classical) indefinite description principle

D_iota = (weakly classical) definite description principle

Generalized schemes

Subclassical schemes PI = proof irrelevance

IGP = independence of general premises (an unconstrained generalisation of the constructive principle of independence of premises)

Drinker = drinker's paradox (small form) (called Ex in Bell [Bell])

EM = excluded-middle

Extensional schemes Ext_prop_repr = choice of a representative among extensional propositions

Ext_pred_repr = choice of a representative among extensional predicates

Ext_fun_repr = choice of a representative among extensional functions

We let also

• IPL_2 = 2nd-order impredicative minimal predicate logic (with ex. quant.)
• IPL^2 = 2nd-order functional minimal predicate logic (with ex. quant.)
• IPL_2^2 = 2nd-order impredicative, 2nd-order functional minimal pred. logic (with ex. quant.)

with no prerequisite on the non-emptiness of domains

Table of contents

1. Definitions 2. IPL_2^2 |- AC_rel + AC! = AC_fun 3.1. typed IPL_2 + Sigma-types + PI |- AC_rel = GAC_rel and IPL_2 |- AC_rel + IGP -> GAC_rel and IPL_2 |- GAC_rel = OAC_rel 3.2. IPL^2 |- AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker 3.3. D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker 4. Derivability of choice for decidable relations with well-ordered codomain 5. AC_fun = AC_fun_dep = AC_trunc 6. Non contradiction of constructive descriptions wrt functional choices 7. Definite description transports classical logic to the computational world 8. Choice -> Dependent choice -> Countable choice 9.1. AC_fun_setoid = AC_fun + Ext_fun_repr + EM 9.2. AC_fun_setoid = AC_fun + Ext_pred_repr + PI

AC_rel + AC! = AC_fun

We show that the functional formulation of the axiom of Choice (usual formulation in type theory) is equivalent to its relational formulation (only formulation of set theory) + functional relation reification (aka axiom of unique choice, or, principle of (parametric) definite descriptions) This shows that the axiom of choice can be assumed (under its relational formulation) without known inconsistency with classical logic, though functional relation reification conflicts with classical logic

Connection between the guarded, non guarded and omniscient choices

We show that the guarded formulations of the axiom of choice are equivalent to their "omniscient" variant and comes from the non guarded formulation in presence either of the independence of general premises or subset types (themselves derivable from subtypes thanks to proof- irrelevance)

AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel

OAC_rel = GAC_rel

AC_fun + IGP = GAC_fun = OAC_fun = AC_fun + Drinker

AC_fun + IGP = GAC_fun

AC_fun + Drinker = OAC_fun This was already observed by Bell [Bell]

OAC_fun = GAC_fun This is derivable from the intuitionistic equivalence between IGP and Drinker but we give a direct proof

D_iota -> ID_iota and D_epsilon <-> ID_epsilon + Drinker

D_iota -> ID_iota

ID_epsilon + Drinker <-> D_epsilon

Derivability of choice for decidable relations with well-ordered codomain

Countable codomains, such as nat, can be equipped with a well-order, which implies the existence of a least element on inhabited decidable subsets. As a consequence, the relational form of the axiom of choice is derivable on nat for decidable relations. We show instead that functional relation reification and the functional form of the axiom of choice are equivalent on decidable relation with nat as codomain

AC_fun = AC_fun_dep = AC_trunc

Choice on dependent and non dependent function types are equivalent

The easy part

Deriving choice on product types requires some computation on singleton propositional types, so we need computational conjunction projections and dependent elimination of conjunction and equality

Functional choice and truncation choice are equivalent

Reification of dependent and non dependent functional relation are equivalent

The easy part

Deriving choice on product types requires some computation on singleton propositional types, so we need computational conjunction projections and dependent elimination of conjunction and equality

Non contradiction of constructive descriptions wrt functional axioms of choice

Non contradiction of indefinite description

Non contradiction of definite description

Remark, the following corollaries morally hold: Definition In_propositional_context (A:Type) := forall C:Prop, (A -> C) -> C. Corollary constructive_definite_descr_in_prop_context_iff_fun_reification : In_propositional_context ConstructiveIndefiniteDescription <-> FunctionalChoice. Corollary constructive_definite_descr_in_prop_context_iff_fun_reification : In_propositional_context ConstructiveDefiniteDescription <-> FunctionalRelReification. but expecting FunctionalChoice (resp. FunctionalRelReification) to be applied on the same Type universes on both sides of the first (resp. second) equivalence breaks the stratification of universes.

Excluded-middle + definite description => computational excluded-middle

The idea for the following proof comes from [ChicliPottierSimpson02] Classical logic and axiom of unique choice (i.e. functional relation reification), as shown in [ChicliPottierSimpson02], implies the double-negation of excluded-middle in Set (which is incompatible with the impredicativity of Set). We adapt the proof to show that constructive definite description transports excluded-middle from Prop to Set. [ChicliPottierSimpson02] Laurent Chicli, Loïc Pottier, Carlos Simpson, Mathematical Quotients and Quotient Types in Coq, Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646, Springer Verlag.

Choice => Dependent choice => Countable choice

About the axiom of choice over setoids

Consequences of the choice of a representative in an equivalence class

This is a variant of Diaconescu and Goodman-Myhill theorems

AC_fun_setoid = AC_fun_setoid_gen = AC_fun_setoid_simple

AC_fun_setoid = AC! + AC_fun_repr

Note: What characterization to give of RepresentativeFunctionalChoice? A formulation of it as a functional relation would certainly be equivalent to the formulation of SetoidFunctionalChoice as a functional relation, but in their functional forms, SetoidFunctionalChoice seems strictly stronger

AC_fun_setoid = AC_fun + Ext_fun_repr + EM

This is the main theorem in [Carlström04]

Note: all ingredients have a computational meaning when taken in separation. However, to compute with the functional choice, existential quantification has to be thought as a strong existential, which is incompatible with the computational content of excluded-middle

AC_fun_setoid = AC_fun + Ext_pred_repr + PI

Note: all ingredients have a computational meaning when taken in separation. However, to compute with the functional choice, existential quantification has to be thought as a strong existential, which is incompatible with proof-irrelevance which requires existential quantification to be truncated

Compatibility notations