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Dialectica Categories and Cardinalities of the Continuum (March2014)

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Slides not used in talk at EBL2014, given by Samuel Gomes da Silva.

Slides not used in talk at EBL2014, given by Samuel Gomes da Silva.

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  • 1. Dialectica Categories and Cardinalities of the Continuum Valeria de Paiva and Samuel Gomes da Silva EBL2014 March, 2014 Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 1 / 31
  • 2. Outline Outline 1 Introduction 2 Dialectica Categories 3 Cardinal characteristics 4 Computational Complexity Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 2 / 31
  • 3. Introduction Introduction [...]It often happens that there are similarities between the solutions to problems, or between the structures that are thrown up as part of the solutions. Sometimes, these similarities point to more general phenomena that simultaneously explain several different pieces of mathematics. These more general phenomena can be very difficult to discover, but when they are discovered, they have a very important simplifying and organizing role, and can lead to the solutions of further problems, or raise new and fascinating questions. – T. Gowers, The Importance of Mathematics, 2000 Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 3 / 31
  • 4. Dialectica Categories Hilbert’s Program Main goal of Hilbert’s program: provide secure foundations for all mathematics. How? Formalization all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules. Sounds good, doesn’t it? Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 4 / 31
  • 5. Dialectica Categories Hilbert’s Program: there is no ignorabimus in mathematics... Wir müssen wissen, wir werden wissen. Consistent: no contradiction can be obtained in the formalism of mathematics. Complete: all true mathematical statements can be proven in the formalism. Consistency proof use only "finitistic" reasoning about finite mathematical objects. Conservative: any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without ideal objects. Decidable: an algorithm for deciding the truth or falsity of any mathematical statement. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 5 / 31
  • 6. Dialectica Categories Gödel’s Incompleteness Gödel’s incompleteness theorems (1931) showed that Hilbert’s program was impossible to achieve, at least if interpreted in the most obvious way. BUT: The development of proof theory itself is an outgrowth of Hilbert’s program. Gentzen’s development of natural deduction and the sequent calculus [too]. Gödel obtained his incompleteness theorems while trying to prove the consistency of analysis. The tradition of reductive proof theory of the Gentzen-Schütte school is itself a direct continuation of Hilbert’s program. R. Zach 2005 Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 6 / 31
  • 7. Dialectica Categories Gödel’s Dialectica Interpretation If we cannot do Hilbert’s program with finitistic means, can we do it some other way? Can we, at least, prove consistency of arithmetic? Try: liberalized version of Hilbert’s programme – justify classical systems in terms of notions as intuitively clear as possible. G’s approach: computable (or primitive recursive) functionals of finite type, using the Dialectica Interpretation (named after the Swiss journal Dialectica, special volume dedicated to Paul Bernays 70th birthday) in 1958. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 7 / 31
  • 8. Dialectica Categories Dialectica Categories Hyland suggested that to provide a categorical model of the Dialectica Interpretation, one should look at the functionals corresponding to the interpretation of logical implication. The categories de Paiva came up with proved to be a model of Linear Logic... Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 8 / 31
  • 9. Dialectica Categories Dialectica categories are models of Linear Logic Linear Logic was created by Girard (1987) as a proof-theoretic tool: the dualities of classical logic plus the constructive content of proofs of intuitionistic logic. Linear Logic: a tool for semantics of Computing. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 9 / 31
  • 10. Dialectica Categories Dialectica Interpretation: motivations... For Gödel (in 1958) the interpretation was a way of proving consistency of arithmetic. For me (in 1988) an internal way of modelling Dialectica that turned out to produce models of Linear Logic instead of models of Intuitionistic Logic, which were expected... For Blass (in 1995) a way of connecting work of Votjás in Set Theory with mine and also his own work on Linear Logic and cardinalities of the continuum. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 10 / 31
  • 11. Dialectica Categories Dialectica categories: useful for proving what...? Blass (1995) Dialectica categories as a tool for proving inequalities between cardinalities of the continuum. Questions and Answers – A Category Arising in Linear Logic, Complexity Theory, and Set Theory in Advances in Linear Logic (ed. J.-Y. Girard, Y. Lafont, and L. Regnier) London Math. Soc. Lecture Notes 222 (1995). Also Propositional connectives and the set theory of the continuum (1995) and the survey Nearly Countable Cardinals. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 11 / 31
  • 12. Dialectica Categories Questions and Answers: The category GSets Is this simply a coincidence? The second dialectica category in my thesis GSets (for Girard’s sets) is the dual of GT the Galois-Tukey connections category in Votjás work. Blass calls this category PV. The objects of PV are triples A = (U, X, α) where U, X are sets and α ⊆ U × X is a binary relation, which we usually write as uαx or α(u, x). Blass writes it as (A−, A + , A) but I get confused by the plus and minus signs. Two conditions on objects in PV (not the case in GSets): 1. U, X are sets of cardinality at most |R|. 2. The condition in Moore, Hrusák and Dzamonja (MHD) holds: ∀u ∈ U, ∃x ∈ X such that α(u, x) and ∀x ∈ X∃u ∈ U such that ¬α(u, x) We work with sets 0, 1, 2, N and R and possible relations on those, mostly. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 12 / 31
  • 13. Dialectica Categories Questions and Answers: The category PV In Category Theory morphisms are more important than objects. Given objects A = (U, X, α) and B = (V, Y, β) a map from A to B in GSets is a pair of functions f : U → V and F : Y → X such that α(u, Fy) implies β(fu, y). Usually write maps as U α X ⇓ ∀u ∈ U, ∀y ∈ Y α(u, Fy) implies β(fu, y) V f ? β Y 6 F But a map in PV satisfies the dual condition that is β(fu, y) → α(u, Fy). Trust set-theorists to create morphisms A → B where the relations go in the opposite direction!... Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 13 / 31
  • 14. Dialectica Categories Questions and Answers Objects in PV are only certain objects of GSets, as they have to satisfy the two extra conditions. What happens to the structure of the category when we restrict ourselves to this subcategory? Fact: GSets has products and coproducts, as well as a terminal and an initial object. Given objects of PV, A = (U, X, α) and B = (V, Y, β) the product A × B in GSets is the object (U × V, X + Y, choice) where choice: U × V × (X + Y ) → 2 is the relation sending (u, v, (x, 0)) to α(u, x) and (u, v, (y, 1)) to β(v, y). The terminal object is T = (1, 0, e) where e is the empty relation, e: 1 × 0 → 2 on the empty set. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 14 / 31
  • 15. Dialectica Categories Questions and Answers Similarly the coproduct of A and B in GSets is the object (U + V, X × Y, choice) where choice: U + V × (X × Y ) → 2 is the relation sending ((u, 0), x, y) to α(u, x) and ((v, 1), x, y) to β(v, y) The initial object is 0 = (0, 1, e) where e is the empty relation. Now if the basic sets U, V, X, Y have all cardinality up to |R| then (co)products will do the same. But the MHD condition is a different story. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 15 / 31
  • 16. Dialectica Categories The structure of PV Note that neither T or 0 are objects in PV, as they don’t satisfy the MHD condition. ∀u ∈ U, ∃x ∈ X such that α(u, x) and ∀x ∈ X∃u ∈ U such that ¬α(u, x) To satisfy the MHD condition neither U nor X can be empty. Also the object I = (1, 1, id) is not an object of PV, as it satisfies the first half of the MHD condition, but not the second. And the object ⊥ = (1, 1, ¬id) satisfies neither of the halves. The constants of Linear Logic do not fare too well in PV. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 16 / 31
  • 17. Dialectica Categories The structure of PV Back to morphisms of PV, the use that is made of the category in this application is simply of the pre-order induced by the morphisms. It is somewhat perverse that here, in contrast to usual categorical logic, A ≤ B ⇐⇒ There is a morphism from B to A Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 17 / 31
  • 18. Dialectica Categories Examples of objects in PV 1. The object (N, N, =) where n is related to m iff n = m. To show MHD is satisfied we need to know that ∀n ∈ N∃m ∈ N(n = m), can take m = n. But also that ∀m ∈ N∃k ∈ N such that ¬(m = k). Here we can take k = suc(m). 2. The object (N, N, ≤) where n is related to m iff n ≤ m. 3. The object (R, R, =) where r1 and r2 are related iff r1 = r2, same argument as 1 but equality of reals is logically much more complicated. 4. The objects (2, 2, =) and (2, 2, =), with usual equality. 5.The object (NN , NN , ≤) where ≤ is the eventual majorization ordering in NN , the set of all functions from N to N. The eventual majorization order says that given f, g in NN , g eventually majorizes f iff f(n) ≤ g(n) for all but finitely many ns. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 18 / 31
  • 19. Cardinal characteristics What have Set Theorists done with PV? Cardinal Characteristics of the Continuum Write N for the natural numbers and R for the reals. All the cardinal characteristics considered will be smaller or equal to the cardinality of the reals, or the continuum. They are of little interest if the Continuum Hypothesis holds, as then there are no cardinalities between the cardinality of the integers ω and the cardinality of the reals 2ω . But if the continuum hypothesis does NOT hold there are many interesting connections (in general in the form of inequalities) between various characteristics that Votjás discusses. Votjás work can be characterized as a way of extracting from the inequality proofs information that is of interest even if the Continuum Hypothesis holds. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 19 / 31
  • 20. Cardinal characteristics Cardinals from Analysis - If X and Y are two subsets of N we say that X splits Y if both X ∩ Y and Y X are infinite. - The splitting number s is the smallest cardinality of any family S of subsets of N such that every infinite subset of N is split by some element of S. Recall the Bolzano-Weierstrass Theorem: Any bounded sequence of real numbers has a convergent subsequence. Recall that we can extend the Bolzano-Weierstrass theorem to say: For any many countably bounded sequence of real numbers xk = (xkn)n ∈ N there is a single infinite set A ⊆ N such that the subsequences indexed by A, (xkn)n ∈ A all converge. If one tries to extend the result for uncountably many sequences s above is the first cardinal for which the analogous result fail. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 20 / 31
  • 21. Cardinal characteristics The structure of PV Proposition[Rangel] The object (R, R, =) is maximal amongst objects of PV. Given any object A = (U, X, α) of PV we know both U and X have cardinality small than |R|. In particular this means that there is an injective function ϕ: U → R. (let ψ be its left inverse, i.e ψ(ϕu) = u) Since α is a relation α ⊆ U × X over non-empty sets, if one accepts the Axiom of Choice, then for each such α there is a map f : U → X such that for all u in U, uαf(u). Need a map Φ: R → X such that U α X ⇑ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr) R ϕ ? = R 6 Φ Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 21 / 31
  • 22. Cardinal characteristics The structure of PV Axiom of Choice is essential Given cardinality fn ϕ: U → R, let ψ be its left inverse, ψ(ϕu) = u, for all u ∈ U. Need a map Φ: R → X such that U α X ⇑ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr) R ϕ ? = R 6 Φ Let u and r be such that ϕu = r and define Φ as ψ ◦ f. Since ϕu = r can apply Φ to both sides to obtain Φ(ϕ(u)) = Φ(r). Substituting Φ’s definition get f(ψ(ϕu))) = Φ(r). As ψ is left inverse of ϕ (ψ(ϕu) = u) get fu = Φ(r). Now the definition of f says for all u in U uαfu, which is uαΦr holds, as desired. (Note that we did not need the cardinality function for X.) Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 22 / 31
  • 23. Cardinal characteristics The structure of PV Proposition[Rangel] The object (R, R, =) is minimal amongst objects of PV. This time we use the cardinality function ϕ: X → R for X. We want a map in PV of the shape: R = R ⇑ ∀r ∈ R, ∀x ∈ X α(Φr, x) → (r = ϕx) U Φ ? α X 6 ϕ Now using Choice again, given the relation α ⊆ U × X we can fix a function g: X → U such that for any x in X g(x) is such that ¬g(x)αx. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 23 / 31
  • 24. Cardinal characteristics The structure of PV Axiom of Choice is essential Given cardinality fn ϕ: X → R, let ψ: R → X be its left inverse, ψ(ϕx) = x, for all x ∈ X. Need a map Φ: R → U such that R = R ⇑ ∀r ∈ R, ∀x ∈ X α(Φr, x) → ¬(ϕx = r) U Φ ? α X 6 ϕ Exactly the same argument goes through. Let r and x be such that ϕx = r and define Φ as ψ ◦ g. Since ϕx = r can apply Φ to both sides to obtain Φ(ϕ(x)) = Φ(r). Substituting Φ’s definition get g(ψ(ϕx))) = Φ(r). As ψ is left inverse of ϕ, (ψ(ϕx) = x) get gx = Φ(r). But the definition of g says for all x in X ¬gxαx, which is ¬α(Φr, x) holds. Not quite the desired, unless you’re happy with RAA. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 24 / 31
  • 25. Cardinal characteristics More structure of GSets Given objects A and B of GSets we can consider their tensor products. Actually two different notions of tensor products were considered in GSets, but only one has an associated internal-hom. (Blass considered also a mixture of the two tensor products of GSets, that he calls a sequential tensor product.) Having a tensor product with associated internal hom means that we have an equation like: A ⊗ B → C ⇐⇒ A → (B → C) Can we do the same for PV? Would it be useful? The point is to check that the extra conditions on PV objects are satisfied. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 25 / 31
  • 26. Computational Complexity Computational Complexity 101 A decision problem is given by a set of instances of the problem, together with a set of positive instances. The problem is to determine, given an instance whether it is a positive one or not. Examples: 1. Instances are graphs and positive instances are 3-colorable graphs; 2. Instances are Boolean formulas and positive instances are satisfiable formulas. A many-one reduction from one decision problem to another is a map sending instances of the former to instances of the latter, in such a way that an instance of the former is positive if and only its image is positive. An algorithm computing a reduction plus an algorithm solving the latter decision problem can be combined in an algorithm solving the former. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 26 / 31
  • 27. Computational Complexity Computational Complexity 101 A search problem is given by a set of instances of the problem, a set of witnesses and a binary relation between them. The problem is to find, given an instance, some witness related to it. The 3-way colorability decision problem (given a graph is it 3-way colorable?) can be transformed into the 3-way coloring search problem: Given a graph find me one 3-way coloring of it. Examples: 1. Instances are graphs, witnesses are 3-valued functions on the vertices of the graph and the binary relation relates a graph to its proper 3-way colorings 2. Instances are Boolean formulas, witnesses are truth assignments and the binary relation is the satisfiability relation. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 27 / 31
  • 28. Computational Complexity Computational Complexity 101 A many-one reduction from one search problem to another is a map sending instances of the former to instances of the latter, in such a way that an instance of the former is positive if and only its image is positive. An algorithm computing a reduction plus an algorithm solving the latter decision problem can be combined in an algorithm solving the former. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 28 / 31
  • 29. Computational Complexity Cardinals from Analysis - If f and g are functions N → N, we say that f dominates g if for all except finitely many n’s in N, f(n) ≤ g(n). - The dominating number d is the smallest cardinality of any family D contained in NN such that every g in NN is dominated by some f in D. - The bounding number b is the smallest cardinality of any family B ⊆ NN such that no single g dominates all the members of B. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 29 / 31
  • 30. Computational Complexity Connecting Cardinals to the Category Have the following inequalities: ω ≤ s ≤ d ≤ 2ω ω ≤ b ≤ r ≤ rσ ≤ 2ω The proofs of these inequalities use the category of Galois-Tukey connections. Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 30 / 31

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