The Herbrand Topos

Benno van den Berg

Amsterdam, 14 October 2013

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Me

Category theory, categorical logic, topos theory

Constructivism

Proof theory

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What is a constructive proof?

Theorem (Euclid)

There are infinitely many prime numbers.

Proof.

First of all, 2 is prime. Now suppose p1, . . . , pn is a list of prime numbers.Consider u = p1 . . . pn + 1 and let v be the smallest divisor of u biggerthan 1. Then v is prime and different from all pi .

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) = 2e is rational. Contradiction!

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What is a constructive proof?

Theorem (Skolem-Mahler-Lech)

The indices of the null elements of a linear recurrence sequence are theunion of a finite set and finitely many arithmetic progressions.

Theorem

There is a non-principal ultrafilter on the natural numbers, a collection Uof 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, andeven with this axiom we cannot write down a formula in the language ofset theory defining a specific non-principal ultrafilter.

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BrouwerBrouwer: No ineffective proofs!

In 1908 he wrote a paper “Over de onbetrouwbaarheid der logischeprincipes”. He concluded that traditional logic carries some of the blamefor the non-constructive nature of classical mathematics. He objected inparticular 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:NN → 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.

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Topos theory

Does this still make sense?

We need a notion of an alternative mathematical universe in which onlythe laws of intuitionistic logic hold. A beautiful such notion is provided bycategory theory and topos theory in particular (see next semester’s lectureby Jaap van Oosten).

Topos theory has its origins in algebraic geometry and the theory ofsheaves (Grothendieck). Surprisingly, there is a sheaf topos in whichBrouwer’s theorem saying that all functions f :R→ R are continuous, isvalid.

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Effective topos

But then there is also a topos in which all f :N→ N are computable andin which Church’s Thesis holds: the effective topos due to Martin Hyland.

The effective topos is used in computer science to provide models ofprogramming 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 ofarithmetic.

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What’s this talk about?

There is topos closely related to the effective topos in which there arenonstandard models of arithmetic: the Herbrand topos. In this toposChurch’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 suchthat (∀x ∈ N) (∃y ≤ f (x))ϕ(x , y).

Unfortunately, I will not be able to show you the topos. But I will give youa subcategory of the Herbrand topos, the Herbrand assemblies, which willstill give you the consistency of Bounded Church’s Thesis with theexistence of nonstandard models.

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Notation

For A ⊆ N, put

Pfin(A) = {n ∈ N : n codes a finite set all whose elements belong to A},

and

T (A) = {α ∈ P(Pfin(A)) : α is non-empty and upwards closed },

where upwards closed means

m ∈ α,m ⊆ n =⇒ n ∈ α.

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Herbrand assemblies

Herbrand assemblies

A Herbrand assembly is a triple (X ,A, α) where X is a set, A is a subset ofN and α is a function X → TA. A morphism of Herbrand assembliesf : (X ,A, α)→ (Y ,B, β) is a function f : X → Y for which there is acomputable function ϕ such that:

1 the function ϕ is defined on every n ∈ Pfin(A) and its result belongsto Pfin(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 oneinterpret logic in this category?

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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 manis an island. You exist only through, and you are completely determinedby, your connections with others. You are nothing more than the sum ofyour relationships.

Can you formulate usual set-theoretic constructions in such relationalterms?

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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 Herbrandassemblies.

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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-Chevalleycondition.

But what does this have to do with logic?

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Categorical analysis of logic

When I say logic, I mean multi-sorted logic with empty sorts and in whichthe 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 thelogical operations like disjunction and conjunction.

4 for every term f : Y → X a substitution operation sending predicateson X to predicates on Y .

5 both adjoints (Lawvere: quantifiers as adjoints), satisfyingBeck-Chevalley.

So now it’s obvious, right?

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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) = Pfin(N).

These are not isomorphic.

N is the natural numbers object in the Herbrand assemblies, while M is anonstandard model.

What are the maps f : N → N in the Herbrand assemblies?

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Some suggestions for further reading

B. van den Berg – The Herbrand topos. Mathematical Proceedings ofthe 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. CambridgeUniversity Press, Cambridge, 1987.

A.S. Troelstra and D. van Dalen – Constructivism in mathematics.An introduction. 2 volumes, Studies in Logic and the Foundations ofMathematics, 121 and 123. North-Holland Publishing Co.,Amsterdam, 1988.

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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. CambridgeUniversity Press, Cambridge, 2003.

On categorical logic:

A.M. Pitts – Categorical logic. Handbook of logic in computerscience, 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 relatedcategories. Lecture Notes in Mathematics, Vol. 611. Springer-Verlag,Berlin-New York, 1977.

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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 firstintroduction 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. ElsevierB. V., Amsterdam, 2008.

P.T. Johnstone – Sketches of an elephant: a topos theorycompendium. 2 volumes, Oxford Logic Guides, 43 and 44. TheClarendon Press, Oxford University Press, New York, 2002.

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