## Variations on Realizability: Realizing the Propositional Axiom of Choice (2000)

Venue: | Math. Structures Comput. Sci |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Hyland00variationson,

author = {J. M. E. Hyland},

title = {Variations on Realizability: Realizing the Propositional Axiom of Choice},

journal = {Math. Structures Comput. Sci},

year = {2000},

volume = {12}

}

### OpenURL

### Abstract

Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod

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Citation Context ...y toposes provide all sorts of curious (incomplete) Heyting algebras. In particular from the Kleene-Vesley set-up one obtains the opposite of the old lattice of Medvedev degrees of mass problems (see =-=[26]-=-). Hartley Rogers gives Medvedev's result that the opposite of his lattice is a Heyting algebra. Here is the reason. 18 Modified realizability Kreisel [29] introduced typed modified realizability quit... |

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Citation Context ...terest. Work in the categorical tradition then focused in particular on models for impredicative polymorphic calculi such as System F (Girard [12]) and the Calculus of Constructions (Coquand and Huet =-=[7]-=-), and on Synthetic Domain Theory (Hyland [20] and Taylor [50]). The use of realizability in this context has a quite different character from its earlier metamathematical use. For details of realizab... |

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Citation Context ... but) for a variety of exotic non-classical theories of interest. Work in the categorical tradition then focused in particular on models for impredicative polymorphic calculi such as System F (Girard =-=[12]-=-) and the Calculus of Constructions (Coquand and Huet [7]), and on Synthetic Domain Theory (Hyland [20] and Taylor [50]). The use of realizability in this context has a quite different character from ... |

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Citation Context ... Jaap van Oosten studied more subtle interpretations from an abstract point of view. 3 For accounts of Type Theories and the connection with categories and fibrations, consult the recent books Jacobs =-=[25]-=- and Taylor [51]. 2 ffl The propositional axiom of choice (as above) holds at all types of T. ffl T has coequalizers of equivalence relations: so we have a form of quotient types. To get the full forc... |

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Citation Context ...of realizability in this context has a quite different character from its earlier metamathematical use. For details of realizability models of impredicative type theories the reader may consult Crole =-=[8]-=-. Domain theory in a realizability context has been treated in the dissertations of Rosolini [46], Phoa [39], Longley [32]; recent progress along one particular line is described in van Oosten and Sim... |

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Citation Context ...o a continuous functional x, then X `simplies W X 2sand a; b 2simplies asb 2 . 12 This is Scott's term. I originally called them effective objects as generalizing the effective operations (see Hyland =-=[18]-=-). Most published work on realizability interpretations is concerned with such obects. 13 Thomas Streicher has observed that the representation as an internal category needs attention: he has a defini... |

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Citation Context ...alizability models of impredicative type theories the reader may consult Crole [8]. Domain theory in a realizability context has been treated in the dissertations of Rosolini [46], Phoa [39], Longley =-=[32]-=-; recent progress along one particular line is described in van Oosten and Simpson [58]. For formal expositions of Synthetic Domain Theory informed by the realizability experience see Reus [42] and Re... |

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Citation Context ...n focused in particular on models for impredicative polymorphic calculi such as System F (Girard [12]) and the Calculus of Constructions (Coquand and Huet [7]), and on Synthetic Domain Theory (Hyland =-=[20]-=- and Taylor [50]). The use of realizability in this context has a quite different character from its earlier metamathematical use. For details of realizability models of impredicative type theories th... |

49 |
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Citation Context ...at Martin-Lof type theory, 4 at least as explained in [34] makes sense conceptually only with intensional equality: the intensional equality is treated in Nordstrom, Petersson and Smith [38]. Hofmann =-=[16]-=- treats extensional type theory and gives an extensive treatment of quotients in that context. Of course the axiom of choice is provable, and Maietti [33] shows that even in the predicative setting of... |

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Citation Context ...lty for restricted realizability. 13 3.3 Flat sets The motivation for the objects we now consider comes from experience with the continuous functionals (Kreisel [29]) or countable functionals (Kleene =-=[27]-=-). These were first proposed in the 1950s in connection on the one hand with the foundations of analysis 14 , and on the other with generalized recursion theory. Kleene's recursion theoretic interests... |

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Citation Context ...onstruct realizability toposes. These involve rather different starting points. Generally the issue of quotients can be handled in terms of the fundamental notion of the exact completion (see Carboni =-=[6]-=- and Robinson and Rosolini [45]). 3. As observed independently at least by Scott and Prawitz (see Scott [47]) all the basic operations of intuitionistic logic can be defined in second order logic usin... |

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Citation Context ... Proposition 3.5 Suppose that X 2 S[R] is separable and Y 2 S[R] is flat. Then 8x 2 X:9y 2 Y:OE(x; y) ! 9f 2 Y X :8x 2 X:OE(x; f(x)) holds in the logic of S[P ]. Proof. The terminology we use is from =-=[19]-=-. First we consider the general issue. Typical realizability models fail to satisfy axioms of choice for all but the simplest modest sets. Suppose that X and Y are modest (or X seprable and Y modest) ... |

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Type theory via exact categories
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Citation Context ...ility and sheaf models. Recently the subject has come back into play in connection with Equilogical Spaces (Scott [48]) and related categories (see in particular Birkedal, Carboni, Rosolini and Scott =-=[4]-=-). For us the main point will be the old but unpublished fact that choice principles hold for finite types in appropriate realizability models based on domains with continuous functions. Here we will ... |

25 | The discrete objects in the effective topos
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(Show Context)
Citation Context ...s are assemblies where distinct elements have disjoint realizing sets. Abstractly they are separated objects orthogonal to the codiscrete object \Delta2. For details see Hyland, Robinson and Rosolini =-=[24]-=- where there is an account of the internal and indexed category of such orthogonality classes of objects, the `discrete objects' in any standard realizability topos. Again the definition of modest set... |

24 | Local realizability toposes and a modal logic for computability
- Awodey, Birkedal, et al.
(Show Context)
Citation Context ...and an exponential ideal in 6 This goes back to Higgs [15]: the tripos to topos construction mimics the H-valued sets approach to sheaves. 7 The proof is given in detail in Awodey, Birkedal and Scott =-=[1]-=-. 8 If (f ` fs) : R ! P is an inclusion then f commutes with reindexing u , so taking right adjoints fscommutes with 8u . The exponential ideal property familiar from locale theory is a special case o... |

22 |
Interpretation of analysis by means of functionals of finite type
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Citation Context ...nal category carry over without difficulty for restricted realizability. 13 3.3 Flat sets The motivation for the objects we now consider comes from experience with the continuous functionals (Kreisel =-=[29]-=-) or countable functionals (Kleene [27]). These were first proposed in the 1950s in connection on the one hand with the foundations of analysis 14 , and on the other with generalized recursion theory.... |

22 |
Qualitative Distinctions between some Toposes of Generalized Graphs
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(Show Context)
Citation Context ...localic extension The idea of extensional realizability is old, but should be well known as it provides about the simplest case of Lawvere's `unity and identity of opposites' (see for example Lawvere =-=[30]-=-) in the context of realizability toposes. Suppose that (A; \Delta) is a partial applicative structure. Then we have the standard tripos P of ordinary realizability: P(I) = (P (A) I ; `) : But we can ... |

17 |
Matching typed and untyped realizability
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(Show Context)
Citation Context ...as developed by Hyland and Ong [23]. And recently Thomas Streicher and Jaap van Oosten have made calculations in this area (see van Oosten [55] for example). Both van Oosten [56] and [57] and Longley =-=[31]-=- at the Workshop addressed aspects of the interplay between typed and untyped realizability. One result which is significant in this contex is a theorem of Bezem [3] which we briefly explain. In any s... |

14 |
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Citation Context ...d V are the inf and sup in the lattice of subsets containing 0. Finally the generic proposition is the natural indexed family of all truth values. The beautiful and mathematically natural Diller-Nahm =-=[10]-=- variant of the Dialectica interpretation and its variants (for which see Diller [9] and Stein [49]) can also be treated as above, and thus give rise to toposes. Generally the investigations of the Mu... |

14 | Modified realizability toposes and strong normalization proofs
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(Show Context)
Citation Context ...choice; but that does not hold for the extensional hierarchy. 20 ) An extremely general setting for modified realizability employing a weakening of the notion of a PCA was developed by Hyland and Ong =-=[23]-=-. And recently Thomas Streicher and Jaap van Oosten have made calculations in this area (see van Oosten [55] for example). Both van Oosten [56] and [57] and Longley [31] at the Workshop addressed aspe... |

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Citation Context ...ese concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley =-=[28]-=-; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the met... |

10 |
About effective quotients in Constructive Type Theory
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Citation Context ... A more specific motivation for this paper is that I have been stimulated by a question put to me by Maria-Emilia Maietti. Maietti's question arose naturally from her work ( see in particular Maietti =-=[33]-=-) in type theory; but in categorical form 3 her question is essentially this. How close to the structure of a topos can one get with the propositional axiom of choice holding, but not the law of the e... |

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Continuity in Spatial Toposes
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Citation Context ...tinuous functions. Here we will give one considerable generalization. Recall that the continuous functionals can be represented as quotients of subspaces of domains as described for example in Hyland =-=[17]-=-. 15 In that treament the domains vary with the types; however using just one universal domain gives exactly the representation as a modest set in the realizability topos. Now the equivalence classes ... |

9 | The S-replete construction
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(Show Context)
Citation Context ...o be flat just when X is a modest set stably orthogonal to \Sigma. (We intend this in an internal or indexed sense. A discussion of orthogonality from the indexed point of view is in Hyland and Moggi =-=[22]-=-.) The crucial property of flat objects is as the following. Proposition 3.3 Let X be flat. Suppose that a realizing x 2 X and a 0 realizing x 0 2 X are such that asa 0 . Then x = x 0 Proof. To repres... |

5 |
Injectivity in the Topos of Complete Heyting Algebra valued Sets
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Citation Context ...those of P . It is an elementary but important fact (important especially for realizability triposes) that if R is a functional substructure of P and an exponential ideal in 6 This goes back to Higgs =-=[15]-=-: the tripos to topos construction mimics the H-valued sets approach to sheaves. 7 The proof is given in detail in Awodey, Birkedal and Scott [1]. 8 If (f ` fs) : R ! P is an inclusion then f commutes... |

4 |
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Citation Context ... 14 Remarkably Kreisel's paper also gives the first account of his modified realizability. 15 The basic phenomenon was first observed by Ershov. For a succinct account of his original perspective see =-=[11]-=-. 11 Furthermore the equivalence classes are incomparable: ffl if ; j ` D are the equivalence classes for x 6= y, then a 2sand b 2 j imply a and b are incomparable. There are two sides to this: the eq... |

4 |
1978], Choice implies excluded middle, Zeitschrift fiir Mathematische Logik und Grundlagen der
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Citation Context ...oice implies the law of excluded middle. (This was observed by Diaconescu. For arguments in the internal logic of toposes see Scott [47] or in the constructive set theory tradition Goodman and Myhill =-=[14]-=-.) Secondly the specific formulation given above may not be fair to Maietti: she has certainly considered variations, some of which she may prefer from the point of view of type theory. Note in partic... |

3 |
Goodman's theorem and beyond
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(Show Context)
Citation Context ...ng use of iteration is in the proof of Goodman's Theorem (see [13]). Goodman shows that the propositional axiom of choice is conservative over Heyting Arithmetic with intensional higher types. Beeson =-=[2]-=- gives an analysis of a proof using a realizability extension of a sheaf model: the sheaf model is used to create a generic function N ! N, and then we use Kleene application for functions recursive i... |

3 |
Isomorphisms between HEO and HRO
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(Show Context)
Citation Context ... Oosten [56] and [57] and Longley [31] at the Workshop addressed aspects of the interplay between typed and untyped realizability. One result which is significant in this contex is a theorem of Bezem =-=[3]-=- which we briefly explain. In any standard realizability topos there is a weakly cartesian closed category whose objects are closed subobjects of the natural modest set of realizers. Take either the e... |

1 |
Concepts and Aims of Functional Interpretations: towards a Functional Interpretation of Constructive Set Theory
- Burr
- 1999
(Show Context)
Citation Context ...es. Generally the investigations of the Munster school deserve to be better known and understood: for example there has been some very interesting work on interpreting constructive set theory by Burr =-=[5]-=- using an extension of the ideas of Diller-Nahm. We do not have an explanation of this interpretation in terms of categorical proof theory and so it presents an interesting challenge to the abstract p... |

1 |
Functional interpretation of Heyting's Arithmetic in all finite types. Nieuw Archief voor Wiskunde
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- 1979
(Show Context)
Citation Context ...roposition is the natural indexed family of all truth values. The beautiful and mathematically natural Diller-Nahm [10] variant of the Dialectica interpretation and its variants (for which see Diller =-=[9]-=- and Stein [49]) can also be treated as above, and thus give rise to toposes. Generally the investigations of the Munster school deserve to be better known and understood: for example there has been s... |

1 |
Relativized realisability in intuitionistic arithmetic of all finite types
- Goodman
- 1978
(Show Context)
Citation Context ...s q-realizability and mq-realizability (see Troelstra [52]). The resulting toposes can also be obtained by glueing. Perhaps the most telling use of iteration is in the proof of Goodman's Theorem (see =-=[13]-=-). Goodman shows that the propositional axiom of choice is conservative over Heyting Arithmetic with intensional higher types. Beeson [2] gives an analysis of a proof using a realizability extension o... |