The Definitional Side of the Forcing

International audienceThis paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper ; ii) one call-by-name, whose intuitions already appear in Kriv-ine and Miquel's work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions , which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom

Reflection and potentialism

It was widely thought that the paradoxes of Russell, Cantor, and Burali-Forti had been solved by the iterative conception of set. According to this conception, the sets occur in a well-ordered transfinite series of stages. On standard articulations – for example, those in Boolos (1971, 1989) – the sets are implicitly taken to constitute a plurality. Although sets may fail to exist at certain stages, they all exist simpliciter. But if they do constitute a plurality, what could stop them from forming a set? Without a satisfactory answer to this question, the paradoxes threaten to reemerge. In response, it has been argued that we should think of the sets as an inherently potential totality: whatever things there are, there could have been a set of them. In other words, any plurality could have formed a set. Call this potentialism. Actualism, in contrast, is the view that there could not have been more sets than there are: whatever sets there could have been, there are. This thesis explores a particular consideration in favour of actualism; namely, that certain desirable second-order resources are available to the acutalist but not the potentialist. In the first part of chapter 1 I introduce the debate between potentialism and actualism and argue that some prominent considerations in favour of potentialism are inconclusive. In the second part I argue that potentialism is incompatible with the potentialist version of the second-order comprehension schema and point out that this schema appears to be required by strong set-theoretic reflection principles. In chapters 2 and 3 I explore the possibilities for reflection principles which are compatible with potentialism. In particular, in chapter 2 I consider a recent suggestion by Geoffrey Hellman for a modal structural reflection principle, and in chapter 3 I consider some influential proposals by William Reinhardt for modal reflection principles

From the beginning of set theory to Lebesgue’s measure problem

Descriptive set theory has its origins in Cantor’s work on pointsets in the 1870s. Cantor’s construction of real numbers and proof of the non-denumerability of real numbers were the first results towards a new theory. Later, in 1878, Cantor formulated his continuum hypothesis for the first time, which lead to the first result in descriptive set theory: the Cantor-Bendixson theorem. Two decades later, Cantor’s theory awoke interest in French analysts Borel, Baire and Lebesgue. Their work on measure theory and classification of functions rested heavily on Cantorian ideas. The most important problem for the development of descriptive set theory was Lebesgue’s measure problem: which subsets of the real line are Lebesgue measurable? After Vitali’s impossibility result in 1904 and Zermelo’s axiomatization of set theory ZFC in 1908 Lebesgue’s measure problem gained, in addition to its mathematical framework, a philosophical one as well. This allowed for a better understanding of the underlying situation and also proof-theoretic considerations. The limit to what could be proved to be measurable in ZFC was soon achieved. However, new ideas arose through the works of Banach and Ulam. Their more general measure problem was identified as being dependant on strong axioms of infinity, the large cardinals, which are still linked to real numbers. Cantor’s theory of cardinal numbers had thus found applications even at the level of real numbers.Cantor konstruoi reaaliluvut ensimmäistä kertaa 1870-luvun alussa ja osoitti, että ne muodostavat ylinumeroituvan joukon. Myöhemmin vuonna 1878 Cantor esitti kontinuumihypoteesin, joka johti kuvailevan joukko-opin syntyyn sekä ensimmäiseen tulokseen – Cantor–Bendixsonin lauseeseen. Kaksi vuosikymmentä myöhemmin Cantorin työ sai huomiota Ranskassa. Ranskalaisten matematiikkojen Borelin, Bairen ja Lebesguen ideoissa mittateoriassa ja funktioiden luokittelussa hyödynnettiin laajasti Cantorin ajatuksia. Tärkeimmäksi yksittäiseksi ongelmaksi kuvailevassa joukko-opissa muodostui Lebeguen mittaongelma: mitkä reaalilukujen osajoukot ovat Lebesgue mitallisia? Vital näytti vuonna 1904 vedoten valinta-aksioomaan, että on olemassa reaalilukujen osajoukko, joka ei ole mitallinen. Tämän tuloksen ja Zermelon vuonna 1908 esittämien joukko-opin aksioomien myötä Lebesguen mittaongelma sai matemaattisen näkökulman lisäksi myös filosofisen. Seuraavien vuosikymmenten aikana osoitettiin, että Zermelon aksioomien avulla ei voida vastata moniin keskeisiin joukko-opin kysymyksiin. Banach ja Ulam onnistuivat kehittämään uusia aksioomia näiden ongelmien ratkaisemiseksi. Osoittautui, että heidän esittämä yleisempi mittaongelma riippuu voimakkaista suurten kardinaliteettien aksioomista. Täten Cantorin kardinaalilukujen teorialle löytyi sovelluksia jopa reaalilukujen osajoukoille

Reflection and potentialism

It was widely thought that the paradoxes of Russell, Cantor, and Burali-Forti had been solved by the iterative conception of set. According to this conception, the sets occur in a well-ordered transfinite series of stages. On standard articulations – for example, those in Boolos (1971, 1989) – the sets are implicitly taken to constitute a plurality. Although sets may fail to exist at certain stages, they all exist simpliciter. But if they do constitute a plurality, what could stop them from forming a set? Without a satisfactory answer to this question, the paradoxes threaten to reemerge. In response, it has been argued that we should think of the sets as an inherently potential totality: whatever things there are, there could have been a set of them. In other words, any plurality could have formed a set. Call this potentialism. Actualism, in contrast, is the view that there could not have been more sets than there are: whatever sets there could have been, there are. This thesis explores a particular consideration in favour of actualism; namely, that certain desirable second-order resources are available to the acutalist but not the potentialist. In the first part of chapter 1 I introduce the debate between potentialism and actualism and argue that some prominent considerations in favour of potentialism are inconclusive. In the second part I argue that potentialism is incompatible with the potentialist version of the second-order comprehension schema and point out that this schema appears to be required by strong set-theoretic reflection principles. In chapters 2 and 3 I explore the possibilities for reflection principles which are compatible with potentialism. In particular, in chapter 2 I consider a recent suggestion by Geoffrey Hellman for a modal structural reflection principle, and in chapter 3 I consider some influential proposals by William Reinhardt for modal reflection principles

Logics of formal inconsistency

Orientadores: Walter Alexandre Carnielli, Carlos M. C. L. CaleiroTexto em ingles e portuguesTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasTese (doutorado) - Universidade Tecnica de Lisboa, Instituto Superior TecnicoResumo: Segundo a pressuposição de consistência clássica, as contradições têm um cará[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocínio sensato pode então ter lugar. Uma lógica é paraconsistente se ela rejeita uma tal pressuposição, e aceita ao invés que algumas teorias inconsistentes conquanto não-triviais façam perfeito sentido. A? Lógicas da Inconsistência Formal, LIFs, formam uma classe de lógicas paraconsistentes particularmente expressivas nas quais a noção meta-teónca de consistência pode ser internalizada ao nível da linguagem obje[c]to. Como consequência, as LIFs são capazes de recapturar o raciocínio consistente pelo acréscimo de assunções de consistência apropriadas. Assim, por exemplo, enquanto regras clássicas tais como o silogismo disjuntivo (de A e {não-,4)-ou-13, infira B) estão fadadas a falhar numa lógica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presunção de que estamos raciocinando em um ambiente consistente (neste caso, pelo acréscimo de (consistente-.A) como uma hipótese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustrações destas lógicas e de suas propriedades, mostrando que tais lógicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocínio consistente dentro de tais sistemas inconsistentes são também ilustradas Em cada caso, interpretações em termos de semânticas polivalentes, de traduções possíveis ou modais são fornecidas, e os problemas relacionados à provisão de contrapartidas algébricas para tais lógicas são examinados. Uma abordagem formal abstra[cjta é proposta para todas as definições relacionadas e uma extensa investigação é feita sobre os princípios lógicos e as propriedades positivas e negativas da negação.Abstract: According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and (not-A)-or-B, infer B) are bound to fail in a paraconsistent logic (because A and (not-.4) could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of (consistent-/!) as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation.DoutoradoFilosofiaDoutor em Filosofia e Matemátic

The Universality Problem

The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof

Follow the Flow: sets, relations, and categories as special cases of functions with no domain

We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of doing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science