## A minimalist two-level foundation for constructive mathematics (811)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Maietti811aminimalist,

author = {Maria Emilia Maietti},

title = {A minimalist two-level foundation for constructive mathematics},

year = {811}

}

### OpenURL

### Abstract

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs ” paradigm and acts as a programming language.

### Citations

449 | andDouglasBridges,Constructive Analysis
- Bishop
- 1985
(Show Context)
Citation Context ...f’s type theory in [Mar84]. Quotient model. We will interpret our extensional theory emTT in a quotient model built over mTT. This model is based on the well-known notion of total setoid à la Bishop =-=[Bis67]-=- and the interpretation shows that the design of emTT over mTT satisfies Sambin’s forget-restore principle in [SV98]. Indeed, the interpretation represents the process of restoring all the irrelevant ... |

243 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...e the categorical structure of Q(mTT), as well as that of Q(mTTdp), it is also useful to know that these models are closed under finite products and equalizers (for their definition see, for example, =-=[Mac71]-=-), namely that they are lex categories: Lemma 4.14 The category Q(mTT), as well as Q(mTTdp), is lex (i.e. with terminal object, binary products, equalizers). Proof. In the following we indicate with c... |

179 |
Sketches of an elephant: a topos theory compendium
- Johnstone
- 2002
(Show Context)
Citation Context ... explains why we introduced the concept of Pq-equivalence relation!). Finally note that effective quotients in Q(mTT) and Q(mTT)set, and also in Q(mTTdp), are enough to make these models regular (see =-=[Joh02a]-=- for the categorical definition): indeed one can define the image of f : (A,=A) → (B,=B) as the quotient of (A,=A) over its kernel, namely as (A, f(x) =B f(y) ), after noticing that monic arrows are i... |

133 | On the meanings of the logical constants and the justifications of the logical laws - Martin-Löf - 1985 |

114 |
Computation and reasoning. A type theory for computer science., volume 11
- Luo
- 1994
(Show Context)
Citation Context ...ons towards all collections or of small propositions towards all sets, because the existential quantification would be then equivalent to the corresponding strong indexed sum on the same constituents =-=[Luo94]-=-. The reason to reject the general validity of the axiom of choice in mTT is to get a minimalist foundation compatible with the existing ones, including the internal theory of a generic topos where th... |

101 | Programming in MartinLof’s Type Theory
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...omatic set theory, such as Aczel-Myhill’s CZF theory [AR], or in category theory, such as the internal theory of a topos (for example in [Mai05]), or in type theory, such as Martin-Löf’s type theory =-=[NPS90]-=- and Coquand’s Calculus of Inductive Constructions [Coq90]. There we also argued what it means for a foundation to be constructive. The idea is that a foundation to develop mathematics is constructive... |

93 | Categorical logic - Pitts - 2000 |

87 |
The effective topos
- Hyland
- 1981
(Show Context)
Citation Context ...ent useful constructions on subsets. Two-level theories, where one level is related to the other via a quotient completion, already appeared in the literature. One of this is Hyland’s effective topos =-=[Hyl82]-=-. There the underlying theory is given by a tripos [HJP80], namely a realizability model of many-sorted intuitionistic logic indexed on classical set theory. Then the topos is obtained by freely addin... |

72 |
Foundations of constructive mathematics. Metamathematical studies., volume 6 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Beeson
- 1985
(Show Context)
Citation Context ...ograms requirement. This is due to the well known result by Troelstra [Tro77] that in intuitionistic arithmetics on finite types extensionality of functions is inconsistent with CT+AC (see [TvD88] or =-=[Bee85]-=-). At present, only theories presented in terms of an intensional type theory, such as Martin-Löf’s one in [NPS90], seem to fit into our paradigm. This led us to conclude that in a proofs-as-programs... |

65 |
Metamathematical investigations on a calculus of constructions
- Coquand
- 1990
(Show Context)
Citation Context ... or in category theory, such as the internal theory of a topos (for example in [Mai05]), or in type theory, such as Martin-Löf’s type theory [NPS90] and Coquand’s Calculus of Inductive Constructions =-=[Coq90]-=-. There we also argued what it means for a foundation to be constructive. The idea is that a foundation to develop mathematics is constructive if it satisfies the “proofs-as-programs” paradigm, namely... |

61 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...ed functors (we remind from [Mai05] that this overcomes the problem, first solved in [Hof94], of interpreting substitution correctly when following the informal interpretation first given by Seely in =-=[See84]-=- and recalled in [Joh02b]). But this interpretation requires a choice of Q(mTT)-structure in order to interpret the various constructors. Unfortunately, we are not able to fix such a choice, if we tak... |

58 | Equality in hyperdoctrines and comprehension schema as an adjoint functor - Lawvere - 1968 |

54 |
D.: Constructivism in mathematics. An introduction., volume 1
- Troelstra, Dalen
- 1988
(Show Context)
Citation Context ...ive if it satisfies the “proofs-as-programs” paradigm, namely if it enjoys a realizability model where to extract programs from proofs. If such a semantics is defined in terms of Kleene realizability =-=[Tv88]-=-, then the foundation turns out to be consistent with the formal Church thesis (for short CT) and the axiom of choice (for short AC). In [MS05] we took such a consistency property as our formal notion... |

44 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1994
(Show Context)
Citation Context ...emTTdp in Q(mTTdp), at a first glance it seems that we could simply use the interpretation in [Mai05] given by fibred functors (we remind from [Mai05] that this overcomes the problem, first solved in =-=[Hof94]-=-, of interpreting substitution correctly when following the informal interpretation first given by Seely in [See84] and recalled in [Joh02b]). But this interpretation requires a choice of Q(mTT)-struc... |

41 |
Categorical Logic and Type Theory, volume 141
- Jacobs
- 1999
(Show Context)
Citation Context ...re exists a proofterm pr2(x) ∈ f(x) =B g(x) [x ∈ A ] The category Q(mTT) comes naturally equipped with an indexed category (or split fibration) satisfying the universal property of comprehension (see =-=[Jac99]-=- for its definition) thanks to the closure of mTT collections under strong indexed sums: Def. 4.2 The indexed category: Pq : Q(mTT) OP → Cat 10 is defined as follows. For each object (A,=A) in Q(mTT) ... |

40 | Regular and exact completions - Carboni, Vitale - 1998 |

40 | Internal type theory
- Dybjer
- 1996
(Show Context)
Citation Context ... certain dependent products, it will be useful to know that Q(mTT)- morphisms correspond to extensional dependent collections defined in an analogous way to dependent sets in [Bis67, Pal05] (see also =-=[Dyb96]-=-) as follows: Def. 4.9 (extensional dependent collection) Given an object (A,=A) of Q(mTT), abbreviated with A=, we define an extensional dependent collection on (A,=A) written B=(x) [x ∈ A= ] as a de... |

39 |
Some free constructions in realizability and proof theory
- Carboni
- 1995
(Show Context)
Citation Context ... namely a realizability model of many-sorted intuitionistic logic indexed on classical set theory. Then the topos is obtained by freely adding quotients to a regular category associated to the tripos =-=[Car95]-=-. However, the effective topos can be seen as obtained by a quotient completion on a lex category, too [Car95]. This latter completion is closer to our quotient completion. The precise correspondence ... |

37 | Type theories, toposes and constructive set theory: predicative aspects of AST
- Moerdijk, Palmgren
- 2002
(Show Context)
Citation Context ...tness is derivable anyway 2. 2The proof of disjointness in emTT is similar to that for mTT mentioned in the proof of theorem 4.20. 9 Our emTT is also compatible with the notion of a predicative topos =-=[MP02]-=-, being its set-theoretic part a fragment of the internal theory of a locally cartesian closed pretopos. In particular, if we perform over Martin-Löf’s type theory with universes the quotient model w... |

32 | Setoids in type theory - Barthe, Capretta, et al. |

30 | Semantics of type theory - Streicher - 1991 |

27 | Type theory via exact categories - Birkedal, Carboni, et al. - 1998 |

27 | Some points in formal topology - Sambin |

26 | Observational equality, now - Altenkirch, McBride, et al. - 2007 |

26 | Adjointness in foundations - Lawvere - 1969 |

24 | Locally cartesian closed exact completions - Carboni, Rosolini - 2000 |

24 | Extensional equality in intensional type theory - Altenkirch - 1999 |

23 |
About models for intuitionistic type theories and the notion of definitional equality
- Martin-Lof
- 1975
(Show Context)
Citation Context ...rt of the mTT version presented here, called mTTset. We say “essentially” because here we adopt a version of type theory with explicit substitution rules and without the ξ-rule for lambda-terms as in =-=[Mar75]-=-. The reason to adopt such a modified version is due to the fact that, as suggested to us by P. Martin-Löf and T. Streicher, its set-theoretic fragment mTTset directly enjoys Kleene’s realizability i... |

20 | Foundations of Constructive Analysis (McGraw-Hill - Bishop - 1967 |

20 | The free exact category on a left exact one - Carboni, Magno - 1982 |

20 |
Building up a toolbox for Martin-Löf ’s type theory: subset theory, in Twenty-five years of constructive type theory
- Sambin, Valentini
- 1995
(Show Context)
Citation Context ... the whole foundation relies on the fact that the extensional level must be implemented over the intensional level, but not only this. Indeed, in [MS05] following Sambin’s forget-restore principle in =-=[SV98]-=- we required that extensional concepts must be abstractions of intensional ones as result of forgetting irrelevant computational information. Such information is then restored when we implement extens... |

19 |
Intuitionistic Type Theory, notes by G. Sambin of a series of lectures given
- Martin-Löf
- 1980
(Show Context)
Citation Context ...n of a set with ε-relation and comprehension used in everyday mathematical practice. The set-theoretic part of emTT includes the fragment without universes of extensional Martin-Löf’s type theory in =-=[Mar84]-=-. Quotient model. We will interpret our extensional theory emTT in a quotient model built over mTT. This model is based on the well-known notion of total setoid à la Bishop [Bis67] and the interpreta... |

15 |
Hyperdoctrines, natural deduction and the beck condition
- Seely
- 1983
(Show Context)
Citation Context ...nd these exponentials are stable under pullback. Lastly, the indexed category Pq validates first-order intuitionistic logic with equality, namely it is an intuitionistic hyperdoctrine in the sense of =-=[See83]-=- (see also [Law70, Law69, Pit00]). - The category Q(mTTdp) enjoys the same properties as Q(mTT), but in addition is also locally cartesian closed (i.e. with dependent products). Proof. Thanks to lemma... |

14 |
Extensional Constructs in Intensional Type Theory. Distinguished Dissertations
- Hofmann
- 1997
(Show Context)
Citation Context ...s type theory in [Mar84] and the intensional one in [NPS90]. It consists in the fact that while type judgements in the intensional version are decidable, those in the extensional one are no longer so =-=[Hof97]-=-. Another difference is that in emTT propositions are mono as in [Mai05], that is they 7 are inhabited by at most one proof by introducing in emTT the following rule: prop-mono) A prop [Γ] p ∈ A [Γ] q... |

13 | Telescopic mapping in typed lambda calculus
- Bruijn
- 1991
(Show Context)
Citation Context ...dgement (expressing the definitional equality between terms of the same type), respectively, all under a context Γ. The contexts Γ of these judgements are formed as in [NPS90] and they are telescopic =-=[dB91]-=- since types are dependent, namely they are allowed to depend on variables ranging over other types. The precise rules of mTT are given in the appendix 6. Types include collections, sets, propositions... |

12 |
Choice implies excluded middle
- Goodman, Myhill
- 1978
(Show Context)
Citation Context ...emTT. Indeed, we can prove that the validity of the axiom of choice in emTT yields that all propositions are decidable as shown in [ML06, Car04, MV99], whose proof goes back to GoodmanMyhill’s one in =-=[GM78]-=-. To prove this, we use a choice property valid for effective quotients thanks to the fact that propositions are mono: Lemma 4.44 In emTT for any quotient set A/R set [Γ] we can derive a proof of ∀z∈A... |

12 |
Toward a minimalist foundation for constructive mathematics
- Maietti, Sambin
- 2005
(Show Context)
Citation Context ...-as-programs” paradigm and acts as a programming language. MSC 2000: 03G30 03B15 18C50 03F55 Keywords: intuitionistic logic, set theory, type theory. 1 Introduction In a previous paper with G. Sambin =-=[MS05]-=- we argued about the necessity of building a foundation for constructive mathematics to be taken as a common core among relevant existing foundations in axiomatic set theory, such as Aczel-Myhill’s CZ... |

11 |
About effective quotients in constructive type theory
- Maietti
(Show Context)
Citation Context ...as well as that small propositions are mono sets. Indeed, if we identify small propositions with sets simply, or propositions with collections, quotient effectiveness may lead to classical logic (see =-=[Mai99]-=-), because it yields to a sort of choice operator, and hence it is no longer a constructive rule. Moreover, observe that the set-theoretic part of our emTT, called emTTset, is a variation of the inter... |

11 | Predicative topos theory and models for constructive set theory
- Berg
- 2006
(Show Context)
Citation Context ... of implication, of universal quantification and of dependent product set do not seem to be preserved by the functor ξ : Q(mTT)set → C(emTTset) sending an extensional type into its quotient (see also =-=[vdB06]-=-), where C(emTTset) is the syntactic category of emTTset defined as in [Mai05]. However, if we take the set-theoretic coherent fragment cemTT of emTTset, then we expect cemTT to be an internal languag... |

10 |
Modular correspondence between dependent type theories and categories including pretopoi and topoi
- Maietti
(Show Context)
Citation Context ...ken as a common core among relevant existing foundations in axiomatic set theory, such as Aczel-Myhill’s CZF theory [AR], or in category theory, such as the internal theory of a topos (for example in =-=[Mai05]-=-), or in type theory, such as Martin-Löf’s type theory [NPS90] and Coquand’s Calculus of Inductive Constructions [Coq90]. There we also argued what it means for a foundation to be constructive. The i... |

9 | Can you add power-sets to Martin-Löf ’s intuitionistic set theory - Maietti, Valentini - 1999 |

9 | Semantics of Type Theory Birkhäuser - Streicher - 1991 |

9 |
Constructivism in mathematics. Vol
- Troelstra, Dalen
- 1988
(Show Context)
Citation Context ...roofs-as-programs requirement. This is due to the well known result by Troelstra [Tro77] that in intuitionistic arithmetics on finite types extensionality of functions is inconsistent with CT+AC (see =-=[TvD88]-=- or [Bee85]). At present, only theories presented in terms of an intensional type theory, such as Martin-Löf’s one in [NPS90], seem to fit into our paradigm. This led us to conclude that in a proofs-... |

7 |
Notes on constructive set theory. Mittag-Leffler
- Aczel, Rathjen
(Show Context)
Citation Context ...d about the necessity of building a foundation for constructive mathematics to be taken as a common core among relevant existing foundations in axiomatic set theory, such as Aczel-Myhill’s CZF theory =-=[AR]-=-, or in category theory, such as the internal theory of a topos (for example in [Mai05]), or in type theory, such as Martin-Löf’s type theory [NPS90] and Coquand’s Calculus of Inductive Constructions... |

6 | Wellfounded trees in categories. Annals of Pure and Applied Logic - Moerdijk, Palmgren - 2000 |

5 | The generalized type-theoretic interpretation of constructive set theory
- Gambino, Aczel
(Show Context)
Citation Context ...tensional theories, such as the internal calculus of a generic topos (as devised, for example, in [Mai05]) or Aczel-Myhill’s CZF theory, with its extensional level. Also logic enriched type theory in =-=[GA06]-=- can be compared with our mTT. Indeed it appears as a fragment of our mTT except that, being just a many-sorted logic on Martin-Löf’s type theory, its propositions are not inhabited with proofs and t... |

4 |
quotients and partial functions in Martin-Löf’s type theory. In Types for proofs and programs
- Subsets
- 2003
(Show Context)
Citation Context ...y of a topos. Finally, it is worth noting that, thanks to proof irrelevance of propositions, we can implementing functions between subsets as in [SV98] without running into the problem pointed out in =-=[Car03]-=-. The desire of representing power collections of sets together with proof irrelevance of propositions is a key motivation to work in a two-level foundation. Indeed, such constructions can not be dire... |

4 |
The Basic Picture. Structures for constructive topology
- Sambin
(Show Context)
Citation Context ...ule is relevant to turn a small propositional function on a set into a set, and hence to represent functions between subsets as in [SV98] and to represent families indexed on a subset as advocated in =-=[Samar]-=-. The same can be said about subcollections. Moreover, the identification of a proposition with the collection (or set) of its proofs allows also to derive all the induction principles for proposition... |

4 | The forget-restore principle: a paradigmatic example - Valentini - 1995 |

3 |
Ext- + ACint is equivalent to ACext
- EM
(Show Context)
Citation Context ...∈ B R(x, y) −→ ∃f ∈ A→ B ∀x ∈ A R(x,Ap(f, x)) on all emTT sets A and B and small relation R(x, y) props [x ∈ A, y ∈ B] implies that all small propositions are decidable. Proof. We follow the proof in =-=[Car04]-=-. Let us define the following equivalence relation on the boolean set Bool ≡ N1 +N1 whose elements are called true ≡ inl(?) and false ≡ inr(?): given any proposition P we put R(a, b) ≡ a =Bool b ∨ P T... |