A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
aa r X i v : . [ m a t h . L O ] M a y A REALIZABILITY SEMANTICS FOR INDUCTIVE FORMALTOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE
MARIA EMILIA MAIETTI, SAMUELE MASCHIO, AND MICHAEL RATHJENDipartimento di Matematica “Tullio Levi Civita”, Universit`a di Padova, Italy e-mail address : [email protected] di Matematica “Tullio Levi Civita”, Universit`a di Padova, Italy e-mail address : [email protected] of Mathematics, University of Leeds, UK e-mail address : [email protected]
Abstract.
We present a Kleene realizability semantics for the intensional level of the Min-imalist Foundation, for short mTT , extended with inductively generated formal topologies,Church’s thesis and axiom of choice.This semantics is an extension of the one used to show consistency of the intensionallevel of the Minimalist Foundation with the axiom of choice and formal Church’s thesis inprevious work.A main novelty here is that such a semantics is formalized in a constructive theoryrepresented by Aczel’s constructive set theory
CZF extended with the regular extensionaxiom. Introduction
A main motivation for introducing the Minimalist Foundation, for short MF , in [MS05,Mai09] was the desire to provide a foundation where to formalize constructive point-freetopology in a way compatible with most relevant constructive foundations. In particular, MF was designed with the purpose of formalizing the topological results developed byadopting the approach of Formal Topology by P. Martin-L¨of and G. Sambin introduced in[Sam87]. This approach was further enriched with the introduction of Positive Topology bySambin in [Sam03]. A remarkable novelty of this approach to constructive topology was theadvent of inductive topological methods (see [CSSV03, CMS13]) to represent the point-freetopologies of the real number line, of Baire space and of Cantor space.However, while the basic notions of Formal Topology can be formalized in the Mini-malist Foundation in [Mai09], the construction of inductively generated topologies cannot. Key words and phrases:
Realizability, Axiom of Choice, Church’s thesis, Point-free topology.Projects EU-MSCA-RISE project 731143 ”Computing with Infinite Data” (CID), MIUR-PRIN 2010-2011 and Correctness by Construction (EU 7th framework programme, grant no. PIRSES-GA-2013-612638)provided support for the research presented in the paper.
LOGICAL METHODSIN COMPUTER SCIENCE DOI:10.2168/LMCS-??? c (cid:13)
M. E. Maietti, S. Maschio, and M. RathjenCreative Commons
This is indeed done on purpose since the Minimalist Foundation, for short MF , wasintroduced to be a minimalist foundation compatible with (or interpretable in) the most rel-evant constructive and classical foundations for mathematics in the literature (see [Mai09]).Observe indeed that the intensional level of MF is quite weak in proof-theoretic strength be-ing interpretable in the fragment of Martin-L¨of’s type theory with one universe, or directlyin Feferman’s theory of non-iterative fixpoints c ID as first shown in [MM16].Moreover, MF is presented in [Mai09] as a two level system in accordance with thenotion of constructive foundation in [MS05]. Indeed MF consists of an intensional levelbased on an intensional type theory `a la Martin-L¨of, aimed at exhibiting the computationalcontents of mathematical proofs, and an extensional level formulated in a language as closeas possible to that of present day mathematics which is interpreted in the intensional levelby means of a quotient model (see [Mai09]).Here we present an extension MF ind of MF with the inductive definitions sufficientto define inductively generated formal topologies and necessary to define inductive suplat-tices. This is due to the fact that in [CSSV03] the problem of generating formal topologiesinductively is reduced to that of generating inductive suplattices. The rules added to theintensional level mTT of MF to form the intensional level of MF ind , called mTT ind ,are driven by those of well-founded sets in Martin-L¨of’s type theory in [NPS90] withoutassuming generic well-founded sets as in the representations given in [CSSV03, Val07].The main purpose of our paper is then to show that the intensional level mTT ind of MF ind is consistent with the axiom of choice ( AC ) and the formal Church’s thesis ( CT ).More in detail AC states that from any total relation we can extract a type-theoreticfunction as follows:( AC ) ( ∀ x ∈ A ) ( ∃ y ∈ B ) R ( x, y ) → ( ∃ f ∈ (Π x ∈ A ) B ) ( ∀ x ∈ A ) R ( x, Ap ( f, x ))with A and B generic collections and R ( x, y ) any relation, while CT (see also [Tv88]) statesthat from any total relation on natural numbers we can extract a (code of a) recursivefunction by using the Kleene predicate T and the extracting function U ( CT ) ( ∀ x ∈ N ) ( ∃ y ∈ N ) R ( x, y ) → ( ∃ e ∈ N ) ( ∀ x ∈ N ) ( ∃ z ∈ N ) ( T ( e, x, z ) ∧ R ( x, U ( z ))) . Such a consistency property is essential to fulfill the requirement of the intensional levelof a constructive foundation proposed in [MS05].In order to meet our purpose, we produce a realizability semantics for mTT ind by ex-tending the one used to show the consistency of the intensional level of MF with AC + CT in [IMMS18], which in turn extends Kleene realizability interpretation of intuitionistic arith-metic.A main novelty of our semantics is that it is formalized in a constructive theory asthe (generalized) predicative set theory CZF+REA , namely Aczel’s constructive Zermelo-Fraenkel set theory extended with the regular extension axiom
REA .To this purpose it is crucial to modify the realizability interpretation in [IMMS18]in the line of the realizability interpretations of Martin-L¨of type theories in extensions ofKripke-Platek set theory introduced in [Rat93] (published as [GR94]).Therefore, contrary to the semantics in [IMMS18], which was formalized in a classicaltheory as Feferman’s theory of non-iterative fixpoints c ID , here we produce a proof that mTT ind , and hence mTT , is constructively consistent with AC + CT . REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE3
As in [IMMS18], we actually build a realizability model for a fragment of Martin-L¨of’stype theory [NPS90], called
MLtt ind , where mTT ind extended with the axiom of choicecan be easily interpreted.As it turns out,
CZF + REA and
MLtt ind possess the same proof-theoretic strength.In the future we intend to further extend our realizability to model mTT ind enrichedwith coinductive definitions to represent Sambin’s generated Positive Topologies. Anotherpossible line of investigation would be to employ our realizability semantics to establishthe consistency strength of mTT ind or the extension of mTT with particular inductivelygenerated topologies, like that of the real line.2.
The extension MF ind with inductively generated formal topologies Here we describe the extension MF ind of MF capable of formalizing most relevant examplesof formal topologies defined by inductive methods introduced in [CSSV03].In that paper, the problem of generating the minimal formal topology which satisfiessome given axioms is reduced to show how to generate a complete suplattice in terms of aninfinitary relation called basic cover relation a ✁ V between elements a of a set A , thought of as basic opens , and subsets V of A , meaning that the basic open a is covered by the union of basic opens in the subset V .Then the elements of the generated suplattice would be fixpoints of the associated closure operator ✁ ( − ) : P ( A ) −→ P ( A )defined by putting ✁ ( V ) ≡ { x ∈ A | x ✁ V } which are complete with respect to families of subsets indexed over a set .Furthermore, a formal topology is defined as a basic cover relation satisfying a con-vergence property and a positivity predicate (see [CSSV03, MV04, CMS13]). Indeed inthis case the resulting complete suplattice of ✁ -fixpoints actually forms a predicative locale which is overt (or open in the original terminology by Joyal and Tierney) for the presenceof the positivity predicate.The tool of basic covers appears to be the only one available in the literature to representcomplete suplattices in most-relevant predicative constructive foundations including Aczel’s CZF , Martin-L¨of’s type theory and also MF .The reason is that there exist no non-trivial examples of complete suplattices that forma set in such predicative foundations (see [Cur10]). As a consequence, there exist no non-trivial examples of locales which form a set and the approach of formal topology based ona cover relation seems to be compulsory (see also [MS13a]) when developing topology in aconstructive predicative foundation, especially in MF .In [CSSV03] it was introduced a method for generating basic covers inductively startingfrom an indexed set of axioms , called axiom set . Such a method allows to generate a formaltopology inductively when the basic cover relation ✁ is defined on a preordered set ( A, ≤ )and it is generated by an axiom set satisfying a so called localization condition which refersto the preorder defined on A . An algebraic study of the relation between basic covers andformal covers including their inductive generation is given in [CMS13]. M. E. MAIETTI, S. MASCHIO, AND M. RATHJEN
In the following we describe a suitable extension of MF capable of representing induc-tively generated basic covers, and hence also formal topologies.We start by describing how to enrich the extensional level emTT of MF in [Mai09]with such inductive basic covers. The reason is that the language of emTT is more apt torepresent the topological axioms given that it is very close to that of everyday mathematicalpractice (with proof-irrelevance of propositions and an encoding of the usual language offirst order arithmetic and of subsets of a set, see [Mai09]).We recall that in emTT we have four kinds of types, namely collections , sets , propo-sitions and small propositions according to the following subtyping relations: small propositions (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / sets (cid:127) _ (cid:15) (cid:15) propositions (cid:31) (cid:127) / / collections where collections include the power-collection P ( A ) (which is not a set!) of any set A andsmall propositions are defined as those propositions closed under intuitionistic connectivesand quantifiers restricted to sets.We first extend emTT with new primitive small propositions a ⊳ I,C
V prop s expressing that the basic open a is covered by the union of basic opens in V for any a elementof a set A , V subset of A , assuming that the basic cover is generated by a family of (open)subsets of A indexed on a family of sets I ( x ) set [ x ∈ A ] and representing by C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] . The precise rules extending emTT to form a new type system emTT ind are the fol-lowing:
Rules of inductively generated basic covers in emTT ind F- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] V ∈ P ( A ) a ∈ A a ⊳
I,C
V prop s rf- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] V ∈ P ( A ) a ǫ V truea ⊳ I,C
V true tr- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] a ∈ A i ∈ I ( a ) V ∈ P ( A ) ∀ yǫC ( a,i ) y ⊳ I,C
V true a ⊳
I,C
V true ind- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] P ( x ) prop [ x ∈ A ] V ∈ P ( A ) cont ( V, P ) truea ∈ A a ✁ I,C
V true P ( a ) true REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE5 where cont ( V, P ) ≡ ∀ x ∈ A ( x ǫ V → P ( x ) )& ∀ x ∈ A ( ∀ j ∈ I ( x ) ∀ y ∈ A ( y ǫ C ( x, j ) → P ( y ) ) → P ( x ) )where above we adopted the convention of writing φ true for a proposition φ instead of true ∈ φ as in [Mai09].A main example of formal topology that can be formalized in emTT ind with the rulesabove is that of real line , represented by Joyal’s inductive formal cover ✁ r of Dedekind realnumbers defined on the set Q × Q which acts as A in the rules above and where Q is the setof rational numbers . This formal cover is generated by a family of open subsets C ( h p, q i , j )indexed on j ∈ I ( h p, q i ) for h p, q i ∈ Q × Q which is defined as an encoding of the followingrules: q ≤ p h p, q i ✁ r U h p, q i ǫ U h p, q i ✁ r U p ′ ≤ p < q ≤ q ′ h p ′ , q ′ i ✁ r U h p, q i ✁ r Up ≤ r < s ≤ q h p, s i ✁ r U h r, q i ✁ r U h p, q i ✁ r U wc wc ( h p, q i ) ✁ r U h p, q i ✁ r U where in the last axiom we have used the abbreviation wc ( h p, q i ) ≡ { h p ′ , q ′ i ∈ Q × Q | p < p ′ < q ′ < q } ( wc stands for ‘well-covered’). For relevant applications see for instance [Pal05, MS13b] andloc.cit.It is worth noting that different presentations of basic covers may yield to the samecomplete suplattice. For example, any complete suplattice presented by (the collectionof fixpoints associated to) a basic cover ✁ I,C on a quotient set
B/R , can be equivalentlypresented by a cover on the set B itself which behaves like ✁ I,C but in addition it considersequal opens those elements which are related by R .In order to properly show this fact, which it will be useful in the next, we define acorrespondence between subsets of B/R and subsets of B as follows: Definition 2.1. In emTT ind , given a quotient set B/R , for any subset W ∈ P ( B/R ) wedefine es ( W ) ≡ { b ∈ B | [ b ] ǫ W } and given any V ∈ P ( B ) we define es − ( V ) ≡ { z ∈ B/R | ∃ b ∈ B ( b ǫ V ∧ z = B/R [ b ] ) } . Definition 2.2.
Given an axiom set represented by a set A ≡ B/R with I ( x ) set [ x ∈ A ] and C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] , we define a new axiom set as follows: A R ≡ B I R ( x ) ≡ I ([ x ]) + (Σ y ∈ B ) R ( x, y ) for x ∈ B where C R ( b, j ) is the formalization of C R ( b, j ) ≡ ( es − ( C ([ b ] , j ) ) if j ∈ I ([ b ]) { π ( j ) } if j ∈ (Σ y ∈ B ) R ( b, y ) for b ∈ B and j ∈ I R ( x ) .We then call ✁ RI,C the inductive basic cover generated from this axiom set.
It is then easy to check that
M. E. MAIETTI, S. MASCHIO, AND M. RATHJEN
Lemma 2.3.
For any axiom set in emTT ind represented by a set A ≡ B/R with I ( x ) set [ x ∈ A ] and C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )], the suplattice defined by ✁ I,C is isomorphic tothat defined by ✁ RI,C by means of an isomorphism of suplattices.
Proof.
It is immediate to check that for any subset W of B/R which is a fixpoint for ✁ I,C the subset es ( W ) is a fixpoint for ✁ RI,C and that, conversely, for any subset V of B which isa fixpoint for ✁ RI,C the subset es − ( V ) is a fixpoint for ✁ I,C . Moreover, this correspondencepreserves also the suprema defined as in [CMS13]. Alternatively, one could check that therelation z F b ≡ Id ( B/R , z , [ b ] ) defines a basic cover isomorphism in the sense of [CMS13]between the basic cover ✁ I,C and ✁ RI,C .3.
The intensional level mTT ind
Here we describe the extension mTT ind of the intensional level mTT of MF capable ofinterpreting the extension emTT ind .We recall that in mTT as well as in emTT we have the same four kinds of typeswith the difference that in mTT power-collections of sets are replaced by the existenceof a collection of small propositions prop s and function collections A → prop s for any set A . Such collections are enough to interpret power-collections of sets in emTT within aquotient model of dependent extensional types built over mTT , as explained in [Mai09].Therefore, in order to define mTT ind we cannot simply add the rules of inductivelygenerated basic covers of emTT ind but we need to add an intensional version of them. Tothis purpose in mTT ind in addition to the new small proposition a ⊳ I,C
V prop s we need to add new proof-term constructors associated to it in such a way that judgementsasserting that some proposition is true in emTT ind are turned into judgements of mTT ind producing a proof-term of the corresponding proposition.It is worth noting that the equality rules of the inductive basic covers are driven bythose of well-founded sets in Martin-L¨of’s type theory in [NPS90] without assuming genericwell-founded sets as in the representations given in [CSSV03, Val07]. However, in accor-dance with the idea that proof-terms of propositions of mTT represent just a constructiverendering of the proofs of propositions in emTT , we do restrict the elimination rules ofinductive basic covers to act toward propositions non depending on their proof-terms, sincethese proof-terms do not appear at the extension level of emTT .When expressing the rules of inductive basic covers we use the abbreviation a ǫ V to mean Ap ( V , a )for any set A , any small propositional function V ∈ A → prop s and any element a ∈ A .The precise rules of inductive basic covers extending mTT to form a new type system mTT ind are the following: REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE7
Rules of inductively generated basic covers in mTT ind F- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )] V ∈ A → prop s a ∈ Aa ⊳
I,C
V prop s rf- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )] V ∈ A → prop s a ∈ A q ∈ a ǫ V rf ( a, q ) ∈ a ⊳ I,C V tr- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )] V ∈ A → prop s a ∈ A i ∈ I ( a ) q ∈ ∀ x ∈ A ( x ǫ C ( a, i ) → x ⊳ I,C V ) tr ( a, i, q ) ∈ a ⊳ I,C V ind- ⊳ A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ P ( A ) [ x ∈ A, j ∈ I ( x )] P ( x ) prop [ x ∈ A ] V ∈ A → prop s a ∈ A m ∈ a ✁ I,C Vq ( x, z ) ∈ P ( x ) [ x ∈ A, z ∈ x ǫ V ] q ( y, j, f ) ∈ P ( y ) [ y ∈ A, j ∈ I ( y ) , f ∈ ∀ z ∈ A ( z ǫ C ( y, j ) → P ( z ) )] ind ( m, q , q ) ∈ P ( a )C -ind A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )] P ( x ) prop [ x ∈ A ] V ∈ A → prop s a ∈ A q ∈ a ǫ Vq ( x, z ) ∈ P ( x ) [ x ∈ A, z ∈ x ǫ V ] q ( y, j, f ) ∈ P ( y ) [ y ∈ A, j ∈ I ( y ) , f ∈ ∀ z ∈ A ( z ǫ C ( y, j ) → P ( z ) )] ind ( rf ( a, q ) , q , q ) = q ( a, q ) ∈ P ( a )C -ind A set I ( x ) set [ x ∈ A ] C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )] P ( x ) prop [ x ∈ A ] V ∈ A → prop s a ∈ A i ∈ I ( a ) q ∈ ∀ x ∈ A ( x ǫ C ( a, i ) → x ⊳ I,C V ) q ( x, z ) ∈ P ( x ) [ x ∈ A, z ∈ x ǫ V ] q ( y, j, f ) ∈ P ( y ) [ y ∈ A, j ∈ I ( y ) , f ∈ ∀ z ∈ A ( z ǫ C ( y, j ) → P ( z ) )] ind ( tr ( a, i, q ) , q , q ) = q ( a , i , λz.λu. ind ( Ap ( Ap ( q, z ) , u ) , q , q ) ) ∈ P ( a )Note that the cover relation preserves extensional equality of subsets represented assmall propositional functions thanks to the induction principle: Lemma 3.1.
For any axiom set in mTT ind represented by a set A with I ( x ) set [ x ∈ A ]and C ( x, j ) ∈ A → prop s [ x ∈ A, j ∈ I ( x )], for any propositional functions V ∈ A → prop s M. E. MAIETTI, S. MASCHIO, AND M. RATHJEN and V ∈ A → prop s , there exists a proof-term q ∈ V = ext V → a ⊳ I,C V = ext a ⊳ I,C V where for any small propositional functions W and W on a set A we abbreviate W = ext W ≡ ∀ x ∈ A ( W ( x ) ↔ W ( x ) )Recalling that the interpretation of emTT in mTT in [Mai09] interprets a set A as anextensional quotient defined in mTT as a set A J of mTT equipped with an equivalencerelation = A J over A J , as well as families of sets are interpreted as families of extensionalsets preserving the equivalence relations in their telescopic contexts, it is crucial to uselemma 2.3 to interpret basic covers of emTT ind on a base A as basic covers ✁ = AJ I J ,C J of mTT ind defined on the support A J : Proposition 3.1.
The interpretation of emTT in mTT in [Mai09] extends to an inter-pretation of emTT ind in mTT ind by interpreting a ✁ I,C V for a ∈ A and V ∈ P ( A ) in thecorresponding basic cover in mTT ind induced over the support of A J which is an extensionalproposition in the sense of [Mai09] . The fragment
MLtt ind of intensional Martin-L¨of’s type theory withinductive basic covers
We here briefly describe the theory
MLtt ind obtained by adding the rules of inductive basiccovers to the first order fragment of intensional Martin-L¨of’s type theory in [NPS90] withone universe.This is essentially a fragment of intensional Martin-L¨of’s type theory which interprets mTT ind as soon as propositions are identified as sets following the Curry-Howard corre-spondence in [NPS90] but with the warning that we strengthen the elimination rule ofinductive basic covers to act towards sets depending on their proof-terms according toinductive generation of types in Martin-L¨of’s type theory.Actually the interpretation of mTT ind into
MLtt ind validates also the axiom of choice AC as formulated in the introduction.Therefore in order to show the consistency of mTT ind with AC + CT (with CT formu-lated as in the introduction) is enough to show the consistency of MLtt ind extended with(the translation of) CT .Here we adopt the notation of types and terms within the first order fragment MLtt of intensional Martin-L¨of’s type theory with one universe U `a la Tarsky in [IMMS18] andwe just describe the rule of inductive basic covers added to it.To this purpose we add to MLtt the code a b ⊳ s,i, v ∈ U for a ∈ T ( s ) and v ∈ T ( s ) → U meaning that the element a of a small set T ( s ) represented by the code s ∈ U is coveredby the subset v represented by a small propositional function from T ( s ) to the (large) set ofsmall propositions identified with U by the propositions-as-sets correspondence.Moreover, we use the abbreviations a ⊳ s,i,c v ≡ T ( a b ⊳ s,i,c v ) x ǫ y ≡ T ( Ap ( y, x ))and the notation axcov ( s, i, c ) REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE9 to abbreviate the following judgements s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))]Then, the precise rules of inductive basic covers extending MLtt to form a new typesystem MLtt ind are the following:
Rules of inductively generated basic covers in MLtt ind F- ⊳ s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))] a ∈ T ( s ) v ∈ T ( s ) → U a b ⊳ s,i,c v ∈ U rf- ⊳ s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))] a ∈ T ( s ) v ∈ T ( s ) → U r ∈ a ǫ v rf ( a, r ) ∈ a ⊳ s,i,c v tr- ⊳ s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))] a ∈ T ( s ) j ∈ T ( i ( a )) v ∈ T ( s ) → U r ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v ) tr ( a, j, r ) ∈ a ⊳ s,i,c v ind- ⊳ axcov ( s, i, c ) v ∈ T ( s ) → U P ( x, u ) type [ x ∈ T ( s ) , u ∈ x ⊳ s,i,c v ] a ∈ T ( s ) m ∈ a ⊳ s,i,c vq ( x, z ) ∈ P ( x, rf ( x, z )) [ x ∈ T ( s ) , z ∈ x ǫ v ] q ( x, j, f, k ) ∈ P ( x, tr ( x, j, k ))[ x ∈ T ( s ) , j ∈ T ( i ( x )) ,f ∈ (Π z ∈ T ( s ))( z ǫ c ( x, j ) → P ( z )) , k ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v )] ind ( m, q , q ) ∈ P ( a, m )C -ind- ⊳ axcov ( s, i, c ) v ∈ T ( s ) → U P ( x, u ) type [ x ∈ T ( s ) , u ∈ x ⊳ s,i,c v ] a ∈ T ( s ) r ∈ a ǫ vq ( x, z ) ∈ P ( x, rf ( x, z )) [ x ∈ T ( s ) , z ∈ x ǫ v ] q ( x, j, f, k ) ∈ P ( x, tr ( x, j, k ))[ x ∈ T ( s ) , j ∈ T ( i ( x )) ,f ∈ (Π z ∈ T ( s ))( z ǫ c ( x, j ) → P ( z )) , k ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v )] ind ( rf ( a, r ) , q , q ) = q ( a, r ) ∈ P ( a, rf ( a, r )) C -ind- ⊳ axcov ( s, i, c ) v ∈ T ( s ) → U P ( x, u ) type [ x ∈ T ( s ) , u ∈ x ⊳ s,i,c v ] a ∈ T ( s ) j ∈ T ( i ( a )) r ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v ) q ( x, z ) ∈ P ( x, rf ( x, z )) [ x ∈ T ( s ) , z ∈ x ǫ v ] q ( x, j, f, k ) ∈ P ( x, tr ( x, j, k ))[ x ∈ T ( s ) , j ∈ T ( i ( x )) ,f ∈ (Π z ∈ T ( s ))( z ǫ c ( x, j ) → P ( z )) , k ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v )] ind ( tr ( a, j, r ) , q , q ) = q ( a, j, λz.λu. ind ( Ap ( Ap ( r, z ) , u ) , q , q ) , r ) ∈ P ( a, tr ( a, j, r )))A crucial difference from the ordinary versions of Martin-L¨of’s type theory is that for MLtt ind we postulate just the replacement rule repl)repl) c ( x , . . . , x n ) ∈ C ( x , . . . , x n ) [ x ∈ A , . . . , x n ∈ A n ( x , . . . , x n − ) ] a = b ∈ A . . . a n = b n ∈ A n ( a , . . . , a n − ) c ( a , . . . , a n ) = c ( b , . . . , b n ) ∈ C ( a , . . . , a n ) in place of the usual congruence rules which would include the ξ -rule in accordance with therules of mTT in [Mai09], and hence of mTT ind .The motivation for this restriction in mTT ind and in MLtt ind is due to the fact that therealizability semantics we present in the next sections, based on that in [IMMS18] and henceon the original Kleene realizability in [Tv88], does not validate the ξ -rule of lambda-terms ξ c = c ′ ∈ C [ x ∈ B ] λx B .c = λx B .c ′ ∈ (Π x ∈ B ) C which is instead valid in [NPS90].It is indeed an open problem whether the original intensional version of Martin-L¨of’stype theory in [NPS90], including the ξ -rule of lambda terms, is consistent with CT .It worth noting that the lack of the ξ -rule does not affect the possibility of adopting mTT as the intensional level of a two-level constructive foundation as intended in [MS05],since its term equality rules suffice to interpret an extensional level including extensionalityof functions, as that represented by emTT , by means of the quotient model as introducedin [Mai09] and studied abstractly in [MR12, MR13, MR15].Furthermore our realizability semantics interprets terms as applicative terms in the firstKleene algebra and their equality as numerical equality turning into an extensional equalityin the context-dependent case. Hence we need a suitable encoding of lambda-terms whichvalidates the replacement rule under the interpretation. As observed in [IMMS18] not eachtranslation of pure lambda calculus in the first Kleene algebras satisfies this requirement(see pp.881-882 in [IMMS18]). Theorem 4.1.
The interpretation of mTT into
MLtt given in [Mai09] extends to thatof mTT ind in MLtt ind by interpreting each basic cover ✁ I,C of mTT ind associated to anaxiom set I ( − ) and C ( − , − ) in the corresponding basic cover of MLtt ind associated to theinterpreted axiom set.Proof.
Note that small propositions are encoded in the universe U as well as axiom setsgenerating a basic cover inductively in mTT ind . Notice that a trivial instance of the ξ -rule is derivable from repl) when c and c ′ don’t depend on x B . REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE11
Remark 4.2.
It is worth recalling that for any axiom set represented by a set A with I ( x ) set [ x ∈ A ] and C ( x, j ) ∈ A → U [ x ∈ A, j ∈ I ( x )] and any propositional function V ∈ A → U representing a subset of A , the propositional function representing the subset ✁ I,C ( V ) ≡ { x ∈ A | x ✁ I,C V } is definable in the extension with well-founded sets as shown in [Val07]. A direct represen-tation of ✁ I,C ( V ) ∈ A → U is obtained as the well founded set ( W x ∈ D ) B ( x ) where D ≡ (Σ x ∈ A ) ( a ǫ V + I ( x ) )and B ( x ) [ x ∈ D ] is the inductive type defined by recursion on D toward the first universe U satisfying the following conditions: B ( x ) ≡ ( N if x = h a, inl ( z ) i for z ∈ a ǫ VC ( a , j ) if x = h a, inr ( j ) i for j ∈ I ( a )where we recall that inl and inr are the injections in the sum and π and π are the projectionsof the indexed sum. Then the terms of the introduction and elimination rules for basic coverscan be represented by means of those of well founded sets. For example we can put rf ( a, r ) ≡ sup ( h a, inl ( q ) i , λx.r o ( x ) ) tr ( a, i, q ) ≡ sup ( h a, inr ( j ) i , λy.q ( a, j, y ) )where r ( x ) is the eliminator of the empty set N .5. A realizability interpretation of
MLtt ind with Formal Church’s Thesis
Here we are going to describe a realizability model of
MLtt ind with CT extending that of MLtt in [IMMS18].A main novelty here is that we formalize such a model in the (generalized) predicativeand constructive theory CZF + REA where
CZF stands for Constructive Zermelo-FraenkelSet Theory and
REA stands for the regular extension axiom (for details see [AR01, AR10]).Since the interpretation in [IMMS18] is performed in d ID which is a classical theoryof fixed points, we cannot follow the proof technique in [IMMS18] to fulfill our purpose.Moreover d ID is a too weak theory to accommodate inductively defined topologies as it canbe gleaned from [CR12]. The solution is to adopt the proof-technique in [Rat93, GR94] tofulfill our goal.As usual in set theory we identify the natural numbers with the finite ordinals, i.e. N := ω . To simplify the treatment we will assume that CZF has names for all (meta) naturalnumbers. Let n be the constant designating the n th natural number. We also assume that CZF has function symbols for addition and multiplication on N as well as for a primitiverecursive bijective pairing function p : N × N → N and its primitive recursive inverses p and p , that satisfy p ( p ( n, m )) = n and p ( p ( n, m )) = m . We also assume that CZF isendowed with symbols for a primitive recursive length function ℓ : N → N and a primitiverecursive component function ( − ) − : N × N → N determining a bijective encoding of finitelists of natural numbers by means of natural numbers. CZF should also have a symbol T forKleene’s T -predicate and the result extracting function U . Let P ( { e } ( n )) be a shorthandfor ∃ m ( T ( e, n, m ) ∧ P ( U ( m ))). Further, let p ( n, m, k ) := p ( p ( n, m ) , k ), p ( n, m, k, h ) := p ( p ( n, m, k ) , h ), etc. . . . We use a, b, c, d, e, d, f, n, m, l, k, s, t, j, i as metavariables for naturalnumbers. We first need to introduce some abbreviations:(1) n is p (0 , n is p (0 ,
1) and n is p (0 , σ ( a, b ) is p (1 , p ( a, b )), π ( a, b ) is p (2 , p ( a, b )) and +( a, b ) is p (3 , p ( a, b ))(3) list ( a ) is p (4 , a ) and id ( a, b, c ) is p (5 , p ( a, b, c )))(4) a e ⊳ c,d,e b is p (6 , p ( a, b, c, d, e ))(5) ρ ( a, r ) is p (7 , p ( a, r ))(6) τ ( a, j, r ) is p (8 , p ( a, j, r ))Recall that, in intuitionistic set theories, ordinals are defined as transitive sets all ofwhose members are transitive sets, too. Unlike in the classical case, one cannot prove thatthey are linearly ordered but they are perfectly good as a scale along which one can iteratevarious processes. The trichotomy of 0, successor, and limit ordinal, of course, has to bejettisoned. Definition 5.1.
By transfinite recursion on ordinals (cf. [AR10] , Proposition 9.3.3) wedefine simultaneously two relations
Set α ( n ) and n ε α m on N in CZF + REA .In the following definition we use the shorthand
Fam α ( e, k ) to convey that Set α ( k ) and ∀ j ( j ε α k → Set α ( { e } ( j ))) and we shall write Set ∈ α ( n ) for ∃ β ∈ α ( Set β ( n )) , n ε ∈ α m for ∃ β ∈ α ( n ε β m ) and Fam ∈ α ( e, k ) for ∃ β ∈ α ( Fam β ( e, k )) . (1) Set α ( n j ) iff j = 0 or j = 1 , and m ε α n j iff m < j ; (2) Set α ( n ) holds, and m ε α n iff m ∈ N . (3) If Fam ∈ α ( e, k ) , then Set α ( π ( k, e )) and Set α ( σ ( k, e )) ;if Fam ∈ α ( e, k ) , then (a) n ε α π ( k, e ) iff there exists β ∈ α such that Fam β ( e, k ) and ∀ i ( i ε β k → { n } ( i ) ε β { e } ( i ))(b) n ε α σ ( k, e ) iff there exists β ∈ α such that Fam β ( e, k ) , p ( n ) ε β k ∧ p ( n ) ε β { e } ( p ( n ))(4) If there exists β ∈ α such that Set β ( n ) and Set β ( m ) , then Set α (+( n, m )) , and i ε α + ( n, m ) iff there exists β ∈ α such that Set β ( n ) , Set β ( m ) and ( p ( i ) = 0 ∧ p ( i ) ε β n ) ∨ ( p ( i ) = 1 ∧ p ( i ) ε β m )(5) If there exists β ∈ α such that Set β ( n ) , then Set α ( list ( n )) , and i ε α list ( n ) iff there exists β ∈ α such that Set β ( n ) and ∀ j ( j < ℓ ( i ) → ( i ) j ε β n ) . (6) If Set ∈ α ( n ) , then Set α ( id ( n, m, k )) , and s ε α id ( n, m, k ) iff there exists β ∈ α such that Set β ( n ) , m ε β n and s = m = k . (7) Let β ∈ α . Suppose that the following conditions (collectively called ∗ β ) are satisfied: (a) Set β ( s ) , (b) a ε β s , (c) Fam β ( v, s ) , (d) Fam β ( i, s ) and (e) ∀ x ∀ y ( x ε β s ∧ y ε β { i } ( x ) → Fam β ( {{ c } ( x ) } ( y ) , s )) ,then Set α ( a e ⊳ s,i,c v ) ; REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE13 assuming ∗ β , let C β ( a e ⊳ s,i,c v ) be the smallest subsets of N such that whenever r ε β { v } ( a ) then ρ ( a, r ) ∈ C β ( a e ⊳ s,i,c v ) and whenever j ε β { i } ( a ) and ∀ z ∀ s ( z ε β a ∧ s ε β {{{ c } ( a ) } ( j ) } ( z ) → { r } ( z, s ) ∈ C β ( z e ⊳ s,i,c v ))) then τ ( a, j, r ) ∈ C β ( a e ⊳ s,i,c v ) .The existence of the set C β ( a e ⊳ s,i,c v ) is guaranteed by the axiom REA .Finally we define q ε α a e ⊳ s,i,c v iff ∃ β ∈ α ( ∗ β ∧ q ∈ C β ( a e ⊳ s,i,c v )) . Remark 5.2.
It is worth noting that in the above definition the interpretation of thePropositional Identity b Id ( s, a, b ) ∈ U for s ∈ U and a ∈ T ( s ) and b ∈ T ( s ) agrees with thatin [IMMS18] which validates the rules of the extensional Propositional Identity in [NPS90].Then also our realizability semantics actually validates the extensional version of MLtt ind .Hence the elimination rule of inductive basic covers can be equivalently weakened to acttowards types non dipendenting on proof-terms of basic covers, as soon as we add a suitable η -rule in a similarly way to what happens to the rules of first-order types (like disjoint sumsor natural numbers or list types) in the extensional type theories in [Mai05].Here we have a crucial lemma. Lemma 5.1. In CZF + REA , for all m ∈ N , if Set α ( m ) , then for all ρ such that Set ρ ( m ) , ∀ i ∈ N ( i ε α m ↔ i ε ρ m ) . Proof.
We proceed by induction on α . Suppose Set α ( m ) and Set ρ ( m ). We look at the forms m can have.If m is n , n or n , then the claim is immediate in view of clauses (1) and (2) in theprevious definition.If m is of the form π ( k, e ), then there exists β ∈ α such that Fam β ( e, k ). The inductionhypothesis applied to β yelds that whenever Fam ξ ( e, k ), then ∀ j ∈ N ( i ε β m ↔ i ε ξ m ) ∀ i ∈ N ∀ j ∈ N ( i ε β m → ( j ε β { e } ( i ) ↔ j ε ξ { e } ( i )))The thesis follows from these. If m is either σ ( k, e ), +( a, b ), list ( a ) or id ( a, b, c ) the argumentproceeds as in the previous case.If m is of the form a e ⊳ s,i,c v , the proof is similar, although more involved. Definition 5.3.
We define in
CZF + REA the formula
Set ( n ) as ∃ α ( Set α ( n )) and x ε y as ∃ α ( x ε α y ) . Theorem 5.4.
The theory
MLtt ind is consistent with the formal Church thesis CT .Proof. We outline a realizability semantics in
CZF + REA . Every preterm is interpretedas a K -applicative term (that is, a term built with numerals and Kleene application) as itis done in [IMMS18]. We only need to interpret the new preterms of MLtt ind that is:(1) ( a b ⊳ s,i,c v ) I is defined as { p } (6 , { p } ( a I , v I , s I , Λ x.i I , Λ x. Λ y.c I )), where p and p arenumeral representing the encoding of pairs of natural numbers and of 5-tuples ofnatural numbers, respectively ;(2) ( rf ( a, r )) I := { p } (7 , { p } ( a I , r I ));(3) ( tr ( a, j, r )) I := { p } (8 , { p } ( a I , j I , r I )), where p is a numeral representing the en-coding of triples of natural numbers; when we write { b } ( a , ..., an ), we mean { ... { b } ( a ) } ( a ) ... } ( a n ) (4) ( ind ( m, q , q )) I is { ind q ,q } ( m I ) where ind q ,q is the code of a recursive functionsuch that(a) ind q ,q ( ρ ( a, r )) ≃ {{ Λ x. Λ z.q I } ( a I ) } ( r I )(b) ind q ,q ( τ ( a, j, r )) ≃ { Λ x. Λ k. Λ f. Λ k.q I } ( a, j, Λ y. Λ s. ind q ,q ( {{ r } ( y ) } ( s )) , r )If τ is an K -applicative term, we will define τ ε A as an abbreviation for φ [ τ /x ].We will interpret pretypes as definable subclasses of N in CZF + REA as follows:(1) N I := { x ∈ N | ⊥} (2) N I := { x ∈ N | x = 0 } (3) ((Σ y ∈ A ) B ) I := { x ∈ N | p ( x ) ∈ A I ∧ p ( x ) ∈ B I [ p ( x ) /y ] } (4) ((Π y ∈ A ) B ) I := { x ∈ N | ∀ y ∈ N ( y ∈ A I → { x } ( y ) ∈ B I ) } (5) ( A + B ) I := { x ∈ N | ( p ( x ) = 0 ∧ p ( x ) ∈ A I ) ∨ ( p ( x ) = 1 ∧ p ( x ) ∈ B I ) } (6) ( List ( A )) I := { x ∈ N | ∀ i ∈ N ( i < ℓ ( x ) → ( x ) i ∈ A I ) } (7) ( Id ( A, a, b )) I := { x ∈ N | x = a I ∧ a I = b I ∧ a I ∈ A I } (8) U I := { x | Set ( x ) } (9) T ( a ) I := { x | x ε a I } Precontexts are interpreted as conjunctions of formulas of
CZF + REA as follows.(1) [ ] I is the formula ⊤ ;(2) [Γ , x ∈ A ] I is the formula Γ I ∧ x I ∈ A I .Validity of judgements J in the model is defined as follows:(1) A type [Γ] holds if Γ I ⊢ CZF + REA ∀ x ( x ∈ A I → x ∈ N )(2) A = B type [Γ] holds if Γ I ⊢ CZF + REA ∀ x ( x ∈ A I ↔ x ∈ B I )(3) a ∈ A [Γ] holds if Γ I ⊢ CZF + REA a I ∈ A I (4) a = b ∈ A [Γ] holds if Γ I ⊢ CZF + REA a I ∈ A I ∧ a I = b I where x is a fresh variable.The encoding of lambda-abstraction in terms of K -applicative terms can be chosen(see [IMMS18]) in such a way that if a and b are terms and x is a variable which is notbounded in a , then the terms ( a [ b/x ] ) I and a I [ b I /x I ] coincide.The proof that for every judgement if MLtt ind ⊢ J , then J holds in the realizabilitymodel is a long, but straightforward verification.We just prove for the sake of example that the rules for the inductively generated covers(rf- ⊳ ) and (tr- ⊳ ) preserve the validity of judgments in the model in the empty-context case.(rf- ⊳ ) Suppose the premisses of the following rule are valid in the model.rf- ⊳ s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))] a ∈ T ( s ) v ∈ T ( s ) → U r ∈ a ǫ v rf ( a, r ) ∈ a ⊳ s,i,c v Then, in particular a I ε s I and r I ε { v I } ( a I ) hold in CZF + REA . As a consequenceof definition 5.1, we hence have that rf ( a, r ) I = ρ ( a I , r I ) ε a I e ⊳ s I ,i I ,c I v I holds in CZF + REA , but this is equivalent to the validity of the judgement rf ( a, r ) ∈ a⊳ s,i,c v in the model.(tr- ⊳ ) Suppose the premisses of the following rule are valid in the model.tr- ⊳ s ∈ U i ( x ) ∈ U [ x ∈ T ( s )] c ( x, y ) ∈ T ( s ) → U [ x ∈ T ( s ) , y ∈ T ( i ( x ))] a ∈ T ( s ) j ∈ T ( i ( a )) v ∈ T ( s ) → U r ∈ (Π x ∈ T ( s ))( x ǫ c ( a, j ) → x ⊳ s,i,c v ) tr ( a, j, r ) ∈ a ⊳ s,i,c v REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE15
Then, in
CZF + REA , a I ε s I , j I ε { i I } ( a I ), ∀ x ∈ N ( x ε s I → Set ( { v I } ( x ))) and ∀ x ∈ N ∀ y ∈ N ( x ε s I ∧ y ε { c I ( a I , j I ) } ( x ) → { r I } ( x, y ) ε ⊳ ( s I , i I , c I ; x, v I ))Thus in particular, by definition 5.1, ( tr ( a, j, r )) I = τ ( a I , j I , r I ) ε a I e ⊳ s I ,i I ,c I v I , whichmeans that tr ( a, j, r ) ∈ a ⊳ s,i,c v is valid in the model. Corollary 5.2.
The theory mTT ind is consistent with the axiom of choice, AC , and FormalChurch thesis, CT .Proof. This follows from theorems 4.1 and 5.4.
Corollary 5.3.
The theory mTT ind + AC + CT has an interpretation in the intensionalversion of the type theory ML V in Definition 5.1 of [Rat93] (or [GR94] ).Proof. This is a consequence of the proof of the above Theorem 5.4 and Proposition 5.3 in[Rat93], namely the interpretability of
CZF + REA in ML V . Remark 5.5.
In a certain sense there is nothing special about inductively generated basiccovers in that the interpretation of
MLtt ind in CZF + REA would also work if one addedfurther inductive types such as generic well founded sets to
MLtt ind . In the same veinone could add more universes or even superuniverses (see [Pal98, Rat01]) after beefing upthe interpreting set theory by adding large set axioms. As a consequence one can concludethat intensional Martin-L¨of type theory with some or all these type constructors added, butcrucially missing the ξ -rule, is compatible with Church’s thesis. Theorem 5.6. MLtt ind and
CZF + REA have the same proof-theoretic strength.Proof.
It follows from [Rat93], Theorem 5.13, Theorem 6.9, Theorem 6.13 (or the sametheorems in [GR94]) together with the observation that the theory
IARI of [Rat93] inDefinition 6.2 can already be interpreted in
MLtt ind using the interpretation of [Rat93] inDefinition 6.5.We just recall that
IARI is a subsystem of second order intuitionistic number theory.It has a replacement schema and an axiom of inductive generation asserting that for everybinary set relation R on the naturals the well-founded part of this relation is a set. Theinterpretation for the second order variables are the propositions on the naturals with truthconditions in U .The crucial step is to interpret the axiom of inductive generation of IARI in MLtt ind .To this purpose one has to show that if s ∈ U and R ∈ T ( s ) × T ( s ) → U then thewell-founded part of R , WP ( R ), can be given as a predicate WP ( R ) ∈ T ( s ) → U . Tothis end define i ∈ T ( s ) → U by i ( x ) := s , v ∈ T ( s ) → U by v ( p ) := n , c ( x, y ) ∈ T ( s ) → U by c ( x, y )( z ) := R ( z, x ) (so y is dummy) for x ∈ T ( s ) and y ∈ T ( s ). Now let WP ( R )( a ) := a ⊳ s,i,c v for a ∈ T ( s ). Then it follows that a is in the well-founded partexactly when WP ( R )( a ) is inhabited. To see this, suppose we have a truth maker r for(Π x ∈ T ( s ))( R ( x, a ) → WP ( R )( x )). Then r ∈ (Π x ∈ T ( s ))( x ǫ c ( a, a ) → x ⊳ s,i,c v ), hence tr ( a, a, r ) ∈ a ⊳ s,i,c v by (tr- ⊳ ), whence tr ( a, a, r ) ∈ WP ( R )( a ). Thus WP ( R ) satisfies theappropriate closure properties characterizing the well-founded part of R . The pertaininginduction principle is then a consequence of (ind- ⊳ ). Remark 5.7.
As an evidence of the validity of the previous theorem, one can notice thatwell founded sets of small sets in
MLtt ind can be represented by suitable inductive basiccovers. Hence the claim essentially follows thanks to theorem 6.13 in [Rat93].Indeed, given a small set s ∈ U and a family of small sets b ( x ) ∈ U [ x ∈ T ( a )] then thewell founded set ( W x ∈ T ( s )) T ( b ( x )) on this family can be interpreted as the open coveron the empty subset ∅ ≡ λx. b N ∈ T ( s ) → U ( W x ∈ T ( s )) T ( b ( x )) ≡ ✁ s,i,c ∅ of the inductive basic cover generated by i ( x ) = b N c ( x, j ) ≡ b ( x )for x ∈ T ( s ) and j ∈ N .Then the term sup ( a, f ), for a ∈ T ( s ) and f ( x ) ∈ ( W x ∈ T ( s )) T ( b ) [ x ∈ T ( s )] - withthe notation of p.98 in [NPS90] - can be defined to be tr ( a, ⋆, λx.λy.f ( a ) ). Moreover, asone could expect, there is no term of the form rf ( a, j ) since N is the empty set.The elimination constructor of well-founded sets wrec ( e, f ) is defined as the term ind ( e, r , f ) where r is the elimination constructor of the empty set. Conclusions.
In the future we aim to further extend the realizability semantics presentedhere to model MF ind enriched with coinductive definitions capable of representing generatedPositive Topologies in [Sam03].A further goal would be to study the consistency strength of mTT ind or of mTT extended with specific inductive formal topologies such as that of the real line. Acknowledgments.
The first author acknowledges very helpful discussions and sugges-tions with F. Ciraulo, P. Martin-L¨of, G. Sambin and T. Streicher. The third author wassupported by a grant from the John Templeton Foundation (“A new dawn of intuition-ism: mathematical and philosophical advances,” ID 60842). The opinions expressed in thispublication are those of the authors and do not necessarily reflect the views of the JohnTempleton Foundation.
References [AR01] P. Aczel and M. Rathjen. Notes on constructive set theory. Mittag-Leffler Technical Report No.40,2001.[AR10] P. Aczel and M. Rathjen. Notes on constructive set theory. Available at , 2010.[CMS13] F. Ciraulo, M. E. Maietti, and G. Sambin. Convergence in formal topology: a unifying notion.
J.Logic & Analysis , 5, 2013.[CR12] Giovanni Curi and Michael Rathjen. Formal Baire space in constructive set theory. In
Logic, con-struction, computation , volume 3 of
Ontos Math. Log. , pages 123–135. Ontos Verlag, Heusenstamm,2012.[CSSV03] T. Coquand, G. Sambin, J. Smith, and S. Valentini. Inductively generated formal topologies.
Annals of Pure and Applied Logic , 124(1-3):71–106, 2003.[Cur10] G. Curi. On some peculiar aspects of the constructive theory of point-free spaces.
MathematicalLogic Quarterly , 56(4):375–387, (2010).[GR94] Edward Griffor and Michael Rathjen. The strength of some Martin-L¨of type theories.
Arch. Math.Logic , 33(5):347–385, 1994.
REALIZABILITY SEMANTICS FOR INDUCTIVE FORMAL TOPOLOGIES, CHURCH’S THESIS AND AXIOM OF CHOICE17 [IMMS18] H. Ishihara, M.E. Maietti, S. Maschio, and T. Streicher. Consistency of the intensional level of theminimalist foundation with church’s thesis and axiom of choice.
Arch. Math. Log. , 57(7-8):873–888,2018.[Mai05] M.E. Maietti. Modular correspondence between dependent type theories and categories includingpretopoi and topoi.
Mathematical Structures in Computer Science , 15(6):1089–1149, 2005.[Mai09] M. E. Maietti. A minimalist two-level foundation for constructive mathematics.
Annals of Pure andApplied Logic , 160(3):319–354, 2009.[MM16] M.E. Maietti and S. Maschio. A predicative variant of a realizability tripos for the minimalistfoundation.
IfCoLog journal of Logics and their applications , 3(4):595–668, 2016.[MR12] M. E. Maietti and G. Rosolini. Elementary quotient completion.
Theory and Applications of Cate-gories , 27, 2012.[MR13] M. E. Maietti and G. Rosolini. Quotient completion for the foundation of constructive mathematics.
Logica Universalis , pages 1–32, 2013. DOI 10.1007/s11787-013-0080-2.[MR15] M. E. Maietti and G. Rosolini. Unifying exact completions.
Applied Categorical Structures , 23(1):43–52, 2015.[MS05] M. E. Maietti and G. Sambin. Toward a minimalist foundation for constructive mathematics. InL. Crosilla and P. Schuster, editor,
From Sets and Types to Topology and Analysis: PracticableFoundations for Constructive Mathematics , number 48 in Oxford Logic Guides, pages 91–114.Oxford University Press, 2005.[MS13a] M. E. Maietti and G. Sambin. Why topology in the Minimalist Foundation must be pointfree.
Logicand Logical Philosophy , 22(2):167–199, 2013.[MS13b] Maria Emilia Maietti and Giovanni Sambin. Why topology in the minimalist foundation must bepointfree.
Logic and Logical Philosophy , 22(2):167–199, 2013.[MV04] M. E. Maietti and S. Valentini. A structural investigation on formal topology: coreflection of formalcovers and exponentiability.
Journal of Symbolic Logic , 69:967–1005, 2004.[NPS90] B. Nordstr¨om, K. Petersson, and J. Smith.
Programming in Martin L¨of ’s Type Theory.
ClarendonPress, Oxford, 1990.[Pal98] Erik Palmgren. On universes in type theory. In
Twenty-five years of constructive type theory(Venice, 1995) , volume 36 of
Oxford Logic Guides , pages 191–204. Oxford Univ. Press, New York,1998.[Pal05] Erik Palmgren. Continuity on the real line and in formal spaces. In
From Sets and Types to Topologyand Analysis: Towards Practicable Foundations of Constructive Mathematics, Oxford Logic Guides .University Press, 2005.[Rat93] Michael Rathjen. The strenght of some Martin-L¨of type theory. Preprint, Department of Mathe-matics, Ohio State University, 1993.[Rat01] Michael Rathjen. The strength of Martin-L¨of type theory with a superuniverse. II.
Arch. Math.Logic , 40(3):207–233, 2001.[Sam87] G. Sambin. Intuitionistic formal spaces - a first communication.
Mathematical logic and its appli-cations , pages 187–204, 1987.[Sam03] Giovanni Sambin. Some points in formal topology.
Theor. Comput. Sci. , 305(1-3):347–408, 2003.[Tv88] A. S. Troelstra and D. van Dalen. Constructivism in mathematics, an introduction, vol. I. In
Studiesin logic and the foundations of mathematics . North-Holland, 1988.[Val07] Silvio Valentini. Constructive characterizations of bar subsets.