## Intuitionistic Sets and Ordinals (1996)

Venue: | Journal of symbolic Logic |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Taylor96intuitionisticsets,

author = {Paul Taylor},

title = {Intuitionistic Sets and Ordinals},

journal = {Journal of symbolic Logic},

year = {1996},

volume = {61},

pages = {705--744}

}

### Years of Citing Articles

### OpenURL

### Abstract

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifes the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius’ transitive set objects. This presents Mostowski’s theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski’s theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.

### Citations

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(Show Context)
Citation Context ...d as 0, 1, 3, 15, 65535, ..., the Zermelo numerals as 0, 1, 2, 4, 16, 65536, ... and the von Neumann ordinals as 0, 1, 3, 11, 2059. (This example is due to Wilhelm Ackermann.) (f) John Conway’s games =-=[5]-=- generalise ensembles, with two element relations; the premise of the extensionality axiom is that the children of both kinds agree. Richard Dedekind’s construction of the real numbers as cuts of the ... |

183 |
Topos Theory
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(Show Context)
Citation Context ...volves quantification over (typed) predicates, ordinals are no different from groups. A set in this sense might be a type in a model of simple type theory [17, 30] or an object of an elementary topos =-=[13, 2]-=-. We shall use the word “carrier” for something which is intended to be an arbitrary such object and not necessarily have a set-theoretic structure. (b) In distinction to a “proper class”; this sense ... |

42 |
The theory of constructions: categorical semantics and topostheoretic models
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(Show Context)
Citation Context ...theory, beginning with the general recursion theorem. Corollary 2.6, and so the Russell, Burali-Forti and Hartogs arguments, will fail, cf . the possibility of having a type of types in domain theory =-=[12, 28]-=- versus [24]. These generalisations lie within the realm of synthetic domain theory [29]. Partial correctness — the correspondence between the connectives of logic and category theory — is now very we... |

32 |
Über unendliche, lineare Punktmannigfaltigkeiten
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(Show Context)
Citation Context ...roof theory and infinitary algebra to extend the iterative constructions used in finitary cases, so it would be of great benefit to include them in an intuitionistic categorical account. Georg Cantor =-=[4]-=- defined ordinals as well founded relations which satisfy the trichotomy law, (x ≺ y) ∨ (x = y) ∨ (y ≺ x), but to show that unions of ordinals have this property depends on excluded middle. However tr... |

23 |
Non-Well-Founded Sets, CSLI Lecture Notes 14
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(Show Context)
Citation Context ...to a family of relations labelled by “actions”; extensionality is known as bisimulation and well-foundedness corresponds to termination. Nonterminating processes are also of interest, and Peter Aczel =-=[1]-=- has generalised set theory accordingly. Examples 3.7, 4.2 and 6.3 show the effect of intuitionistic logic, plumpness and directedness on these examples, and on the numerals 2 and 3 in particular; 0 =... |

20 |
Toposes and local set theories: An introduction
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(Show Context)
Citation Context ...volves quantification over (typed) predicates, ordinals are no different from groups. A set in this sense might be a type in a model of simple type theory [17, 30] or an object of an elementary topos =-=[13, 2]-=-. We shall use the word “carrier” for something which is intended to be an arbitrary such object and not necessarily have a set-theoretic structure. (b) In distinction to a “proper class”; this sense ... |

13 | Preframe presentations present
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(Show Context)
Citation Context ...editary semilattice X one can then show that f : X → A preserves binary meets as well as successor and arbitrary joins. The advantage over the situation in [15] is that preframe presentations present =-=[14]-=-. Although the Hartogs method is constructive, its application is not. By considering a case where we know where the fixed point is, we see that H(A) is in general much too small. Proposition 9.8 For ... |

6 |
Über das Problem der Wohlordnung
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(Show Context)
Citation Context ...ntuitionistically, as well as classically, we can define ordinals and use them to iterate functions as often as we like, but when do we stop? Using the Burali-Forti idea, Lemma 9.3 (Freidrich Hartogs =-=[11]-=-) For any carrier A, there is an ordinal α such that there is no injective function α ↩→ A. Proof Let I ⊂ P(A) × P(A × A) be the collection of all subsets U ⊂ A with ordinal structures (≺) ⊂ U ×U. The... |

3 |
The consistency of the axiom of choice and the continuum hypothesis
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Citation Context ...ricted to some subclass. Considering only the decidable ones gives a Boolean model. By admitting those predicates which are definable in terms of X we can build Kurt Gödel’s constructible hierarchy L =-=[9]-=- instead of the full von Neumann hierarchy V . (b) Put p(x0) = {u ∈ X : ∀z ∈ X. z ɛ u ⇒ z ɛ x0} for the collection of subsets which are already representable; then the powerset axiom says that p(x0) i... |

2 |
Π 1 2 logic: Part I, dilators
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(Show Context)
Citation Context ...ons of initial segments. Finally WOrdP ⊂ WOrdT ⊂ Wfr are the categories composed of strictly monotone functions. All of these structures arise in common usage, for example Jean-Yves Girard’s dilators =-=[8]-=- are functors WOrd → WOrd which preserve pullbacks and filtered colimits. 18sProposition 5.2 The set-theoretic union (Corollary 2.12) of a family of transitive or plump ordinals, and the intersection ... |

1 |
questione sui numeri transfiniti. Rendiconti del Circolo matematico di Palermo, 11:154–164, 1897. English translation: A question on tranfinite numbers and on wellordered classes in [32
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(Show Context)
Citation Context ... the use at several points in the argument of equality as an induction predicate. It is interesting to note that Cesare Burali-Forti got the definition of well-foundedness wrong in his original paper =-=[3]-=-, but that the argument remained valid when this was corrected, and was subsequently used in other logical systems. So the idea has some claim to be a real part of the mathematical world. However it i... |

1 |
Über die Grundlagen der Cantor-Zermeloschen Mengenlehre
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(Show Context)
Citation Context ... carrier could be anything we please — but John von Neumann [34] and set theorists following him said that the order had to be membership. For them, ω2 does not exist without the axiom of replacement =-=[6, 19, 26]-=-, whereas for us it is easy to make it “by hand” as the even numbers followed by the odd ones. In categorical terms, replacement says that the topos of sets is (externally) complete and cocomplete wit... |

1 |
The lack of definable witnesses and provably total functions in intuitionistic set theories
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(Show Context)
Citation Context ...ve that this also gives the fixed point. However collection destroys the existence property, which is the outstanding feature of intuitionistic logic, and alters the class of provably total functions =-=[7]-=-. Personally, I can see no justification of this axiom by examples in parts of mathematics other than set theory, and I also feel that constructive mathematicians ought to emphasise the fact that infi... |

1 |
Heyting valued models for intuitionistic set theory
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(Show Context)
Citation Context ...es credit. 1sof transitive sets,” i.e. transitive extensional well founded relations. William Powell [25] showed that this definition allows intuitionistic transfinite recursion, but as Robin Grayson =-=[10]-=- pointed out, the successor operation, α ↦→ α ∪ {α}, is poorly behaved. A new kind of successor, more intuitionistic in character, is defined in Proposition 5.5. (For the sake of orientation, you may ... |