The Herbrand Topos Benno van den Berg Amsterdam, 14 October 2013 1 - PowerPoint PPT Presentation

The Herbrand Topos Benno van den Berg Amsterdam, 14 October 2013 1 / 19

Me Category theory, categorical logic, topos theory Constructivism Proof theory 2 / 19

What is a constructive proof? Theorem (Euclid) There are infinitely many prime numbers. Proof. First of all, 2 is prime. Now suppose p 1 , . . . , p n is a list of prime numbers. Consider u = p 1 . . . p n + 1 and let v be the smallest divisor of u bigger than 1. Then v is prime and different from all p i . Theorem Either e + π or e − π is irrational. Proof. Suppose both e + π and e − π are irrational. Then also ( e + π ) + ( e − π ) = 2 e is rational. Contradiction! 3 / 19

What is a constructive proof? Theorem (Skolem-Mahler-Lech) The indices of the null elements of a linear recurrence sequence are the union of a finite set and finitely many arithmetic progressions. Theorem There is a non-principal ultrafilter on the natural numbers, a collection U of sets of natural numbers such that: (i) U does not contain any finite set; (ii) for any subset A of N , either A or its complement belongs to U ; (iii) if two sets belong to U , then so does their intersection. But note: the axiom of choice is needed to prove the second result, and even with this axiom we cannot write down a formula in the language of set theory defining a specific non-principal ultrafilter. 4 / 19

Brouwer Brouwer: No ineffective proofs! In 1908 he wrote a paper “Over de onbetrouwbaarheid der logische principes”. He concluded that traditional logic carries some of the blame for the non-constructive nature of classical mathematics. He objected in particular to the Law of Excluded Middle ( ϕ ∨ ¬ ϕ ). Heyting: constructive (or intuitionistic) logic. 5 / 19

Brouwer goes further Brouwer actually argued for principles contradicting usual mathematics: Theorem? All functions f : R → R are continuous. And so are functions Φ: N N → N (for the Baire and discrete topology, respectively). Acceptance of such principles makes you an intuitionist . Other constructivists have argued for: Theorem? All functions f : N → N are computable. And if ( ∀ x ∈ N ) ( ∃ y ∈ N ) ϕ ( x , y ), then there is a computable f : N → N such that ( ∀ x ∈ N ) ϕ ( x , f ( x )) (“Church’s Thesis”). Acceptance of such principles makes you a Russian-style constructivist . 6 / 19

Topos theory Does this still make sense? We need a notion of an alternative mathematical universe in which only the laws of intuitionistic logic hold. A beautiful such notion is provided by category theory and topos theory in particular (see next semester’s lecture by Jaap van Oosten). Topos theory has its origins in algebraic geometry and the theory of sheaves (Grothendieck). Surprisingly, there is a sheaf topos in which Brouwer’s theorem saying that all functions f : R → R are continuous, is valid. 7 / 19

Effective topos But then there is also a topos in which all f : N → N are computable and in which Church’s Thesis holds: the effective topos due to Martin Hyland. The effective topos is used in computer science to provide models of programming languages (for example, the Calculus of Constructions). In the effective topos there are no nonstandard models of arithmetic, because: Theorem (McCarty) Church’s Thesis implies that there are no nonstandard models of arithmetic. 8 / 19

What’s this talk about? There is topos closely related to the effective topos in which there are nonstandard models of arithmetic: the Herbrand topos. In this topos Church’s Thesis fails (of course), but the following principle holds: Bounded Church’s thesis If ( ∀ x ∈ N ) ( ∃ y ∈ N ) ϕ ( x , y ), then there is a computable f : N → N such that ( ∀ x ∈ N ) ( ∃ y ≤ f ( x )) ϕ ( x , y ). Unfortunately, I will not be able to show you the topos. But I will give you a subcategory of the Herbrand topos, the Herbrand assemblies , which will still give you the consistency of Bounded Church’s Thesis with the existence of nonstandard models. 9 / 19

Notation For A ⊆ N , put P fin ( A ) = { n ∈ N : n codes a finite set all whose elements belong to A } , and T ( A ) = { α ∈ P ( P fin ( A )) : α is non-empty and upwards closed } , where upwards closed means m ∈ α, m ⊆ n = ⇒ n ∈ α. 10 / 19

Herbrand assemblies Herbrand assemblies A Herbrand assembly is a triple ( X , A , α ) where X is a set, A is a subset of N and α is a function X → TA . A morphism of Herbrand assemblies f : ( X , A , α ) → ( Y , B , β ) is a function f : X → Y for which there is a computable function ϕ such that: 1 the function ϕ is defined on every n ∈ P fin ( A ) and its result belongs to P fin ( B ). 2 if x ∈ X and n ∈ α ( x ), then ϕ ( n ) ∈ β ( f ( x )). But why is this a bit like the category of sets? And how could one interpret logic in this category? 11 / 19

Ubuntu Ubuntu philosophy in the words of Bruce Bartlett A thing is a thing only in the way that it relates to other things. No man is an island. You exist only through, and you are completely determined by, your connections with others. You are nothing more than the sum of your relationships. Can you formulate usual set-theoretic constructions in such relational terms? 12 / 19

Some categorical notions terminal object, initial object product, sum, exponential subobjects pullback natural numbers object All these notions exist both in the category of sets and in the Herbrand assemblies. 13 / 19

Logic Both in the category of sets and in the Herbrand assemblies, we have: if X is an object, then Sub ( X ) is a Heyting algebra. if f : Y → X , then the pullback functor f ∗ : Sub ( X ) → Sub ( Y ) preserves this structure. This pullback functor has both adjoints satisfying the Beck-Chevalley condition. But what does this have to do with logic? 14 / 19

Categorical analysis of logic When I say logic , I mean multi-sorted logic with empty sorts and in which the sorts are closed under products. Then I have: 1 a collections of sorts. 2 a collections of terms, going from one sort to another. 3 for every sort a collection of predicate of that sort closed under the logical operations like disjunction and conjunction. 4 for every term f : Y → X a substitution operation sending predicates on X to predicates on Y . 5 both adjoints (Lawvere: quantifiers as adjoints), satisfying Beck-Chevalley. So now it’s obvious, right? 15 / 19

The nonstandard model There are two Herbrand assemblies: N = ( N , N , γ ) and M = ( N , N , δ ), with γ ( n ) = { m : n ∈ m } , δ ( n ) = P fin ( N ) . These are not isomorphic. N is the natural numbers object in the Herbrand assemblies, while M is a nonstandard model. What are the maps f : N → N in the Herbrand assemblies? 16 / 19

Some suggestions for further reading B. van den Berg – The Herbrand topos. Mathematical Proceedings of the Cambridge Philosophical Society , volume 155, issue 2 (2013), pp. 361-374. On constructivism: D. Bridges and F. Richman – Varieties of constructive mathematics . London Mathematical Society Lecture Note Series, 97. Cambridge University Press, Cambridge, 1987. A.S. Troelstra and D. van Dalen – Constructivism in mathematics. An introduction. 2 volumes, Studies in Logic and the Foundations of Mathematics, 121 and 123. North-Holland Publishing Co., Amsterdam, 1988. 17 / 19

Some suggestions for further reading On category theory: S. Awodey – Category theory . Second edition. Oxford Logic Guides, 52. Oxford University Press, Oxford, 2010. F.W. Lawvere and R. Rosebrugh – Sets for mathematics . Cambridge University Press, Cambridge, 2003. On categorical logic: A.M. Pitts – Categorical logic. Handbook of logic in computer science , Vol. 5, 39–128, Handb. Log. Comput. Sci., 5, Oxford Univ. Press, New York, 2000. M. Makkai and G.E. Reyes – First order categorical logic. Model-theoretical methods in the theory of topoi and related categories. Lecture Notes in Mathematics, Vol. 611. Springer-Verlag, Berlin-New York, 1977. 18 / 19

Some suggestions for further reading On topos theory: R. Goldblatt – Topoi. The categorial analysis of logic. Second edition. Studies in Logic and the Foundations of Mathematics, 98. North-Holland Publishing Co., Amsterdam, 1984. S. Mac Lane and I. Moerdijk – Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the 1992 edition. Universitext. Springer-Verlag, New York, 1994. J. van Oosten – Realizability: an introduction to its categorical side. Studies in Logic and the Foundations of Mathematics, 152. Elsevier B. V., Amsterdam, 2008. P.T. Johnstone – Sketches of an elephant: a topos theory compendium. 2 volumes, Oxford Logic Guides, 43 and 44. The Clarendon Press, Oxford University Press, New York, 2002. 19 / 19