On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions 

On a forcing model for non-standard arithmetic 

Benno van den Berg, TU Darmstadt, Germany 

Aug 5, 2009 

Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic


Classical forcing 

Let P be a preorder. 

Then one has the following forcing clauses: 

p −φ ∧ ψ ⇔ p −φ and p −ψ 

p −φ ∨ ψ ⇔ {q ≤ p : q −φ or q −ψ} is dense below p 

p −φ → ψ ⇔ for all q ≤ p, q −φ implies q −ψ 

p −∃xφ(x) ⇔ {q ≤ p : there exists an a such that q −φ(a)} 

is dense below p 

p −∀xφ(x) ⇔ for all q ≤ p and for all a, q −φ(a) 



Coverages 

This can be generalised using the notion of a coverage. 

A sieve on p is a downwards closed subset of ↓ p. A coverage Cov 

is a function that assigns to every p ∈ P a collection Cov(p) of 

sieves on p, in such a way that three axioms are satisfied: 

1 ↓ p ∈ Cov(p); 

2 if q ≤ p and S ∈ Cov(p), then S∩ ↓ q ∈ Cov(q); 

3 if S ∈ Cov(p) and Rq ∈ Cov(q) for every q ∈ S, then 

∪q∈SRq ∈ Cov(p). 

Example: the ¬¬-coverage. 

Cov(p) = {S ⊆↓ p : S is downwards closed and dense below p}. 



Covering semantics 

For a coverage Cov the forcing clauses read as follows: 

p −φ ∧ ψ ⇔ p −φ and p −ψ 

p −φ ∨ ψ ⇔ {q ≤ p : q −φ or q −ψ} ∈ Cov(p) 

p −φ → ψ ⇔ for all q ≤ p, q −φ implies q −ψ 

p −∃xφ(x) ⇔ {q ≤ p : there exists an a such that q −φ(a)} 

∈ Cov(p) 

p −∀xφ(x) ⇔ for all q ≤ p and for all a, q −φ(a) 

Warning: For a general coverage it is no longer the case that 

classical logic is forced! But the laws of intuitionistic logic are. 



Topos-theoretic perspective 

A presheaf on P is a functor P op → Sets. 

So it consists of a family of sets {X (p) : p ∈ P} together with a 

restriction operation assigning to every element x ∈ X (p) and 

q ≤ p an element x · q ∈ X (q). 

Presheaves form a topos PSh(P). Every topos has an “internal 

logic”: the internal logic of categories of presheaves is Kripke 

semantics (for intuitionistic logic). 



Topos-theoretic perspective, continued 

A presheaf is a sheaf (relative to a coverage Cov) if it satisfies: 

Sheaf axiom 

For every S ∈ Cov(p) and family of elements {xq ∈ X (q) : q ∈ S} 

that is compatible (i.e. satisfies p ≤ q ∈ S ⇒ xq · p = xp), there is 

a unique element x ∈ X (p) such that x · q = xq for all q ∈ S. 

Sheaves also form a topos Sh(P, Cov). The internal logic of 

categories of sheaves is precisely covering semantics. The internal 

logic of Sh(P, ¬¬) is classical forcing. 



The work of Krivine 

In his (unpublished) paper 

Structures de réalisabilité, RAM et ultrafiltre sur N, 

Krivine constructs a forcing model in which there exists an 

(explicit) non-principal ultrafilter on the natural numbers. 

How does this work? 



The forcing model 

Let P be the preorder consisting of infinite subsets of N, preordered 

by: 

p ≤ q ⇔ ∃n ∀k ≥ n (k ∈ p → k ∈ q), 

and let E = Sh(P, ¬¬). The object of truth values in E is 

Ω(p) = {closed sieves on p}, 

where a sieve S is closed, if S dense below q implies q ∈ S. The 

power set PN = Ω N is given by 

Ω N (p) = {X : N → Ω(p)}. 



The ultrafilter 

So suppose X ∈ Ω N (p), i.e., X is a function N → Ω(p). 

The ultrafilter is given by: p −X ∈ U iff 

{q ≤ p : q ∈ X (n) for all n ∈ q} is dense below p. 

One easily verifies that E believes this to be a filter, which does 

not contain the finite sets. So the main issue is to show that it is 

in fact an ultrafilter. 



Key combinatorial lemma 

The proof that U is an ultrafilter relies on the following lemma: 

Lemma 

Every infinitely descending sequence of elements 

x0 ≥ x1 ≥ x2 ≥ . . . in P has a lower bound. 

Proof. 

Suppose x0 ≥ x1 ≥ x2 ≥ . . . is an infinitely descending sequence in 

P. For every n the set 

i≤n xi is infinite, so one may construct a 

sequence of elements an ∈ 

i≤n xi with an = ai for all i < n. Then 

the set y = {a0, a1, a2, . . .} is a lower bound for the sequence. 

Nota bene: In a similar fashion one can show that P does not 

satisfy the countable chain condition! 



Model for non-standard arithmetic 

We now try to compute the model of non-standard arithmetic this 

non-principal ultrafilter leads to. 

The natural numbers in E are what Bell calls “mixed natural 

numbers”: at level p, they consist of an anti-chains 

{qi ≤ p : i ∈ I } in P together with a natural number ni for each 

i ∈ I . (Where such anti-chains can be uncountable!) 

This makes the structure of N → N in E rather complicated. 

However using two lemmas we can keep things simple. 



First useful fact 

Lemma 

The ordinary functions N → N lie dense in the object playing the 

rôle of N → N in E, and therefore validity of statements 

concerning N → N in E can be reduced to validity of statements 

concerning ordinary functions N → N. 

Proof. 

Use the key combinatorial lemma. 



Second useful fact 

Lemma 

If S lies dense below p, then for all but finitely many n ∈ p there is 

a q ∈ S such that n ∈ q. 

Proof. 

Suppose not, et cetera. 



The forcing clauses 

One is led to the following forcing clauses, where non-standard 

natural numbers are interpreted as functions f : N → N: 

p −f = g ⇔ for all but finitely many n ∈ p, f (n) = g(n) 

p −f ≤ g ⇔ for all but finitely many n ∈ p, f (n) ≤ g(n) 

. . . 



Being standard 

The model believes that a function f : N → N represents a 

standard natural number, if it is “locally constant”. More precisely: 

p −st(f ) ⇔ f ↾ p is bounded. 

In short, we end up with the model introduced by Avigad in: 

J. Avigad – Weak theories of nonstandard arithmetic and analysis, 

in: Reverse mathematics 2001, Lect. Notes Log. 21, pp. 19–46. 



Intermediate intuitionistic models 

Question: are there models of intuitionistic non-standard 

arithmetic such that what we saw above can be seen as the result 

of combining the intuitionistic model with a double negation 

translation? 

I believe this can be done in at least two ways. 



Intuitionistic model 1 

We work in PSh(P). This means: forcing clauses for the atomic 

formulas as above and the clauses for the interpretation of logical 

connectives as in Kripke semantics. 

Interesting fact: 

Lemma 

In PSh(P) the double negation shift holds: 

PSh(P) |= ∀n ∈ N(¬¬P(n)) → ¬¬∀nP(n). 

Proof. 

Use the key combinatorial lemma. 



Intuitionistic model 2 

Let B be the powerset of N, ordered by inclusion, and consider the 

ideal I of finite sets in N. I determines a coverage on B: 

S ∈ Cov(c) ⇔ ∃i ∈ I , b1, . . . , bn ∈ S : b1 ∨ . . . ∨ bn ∨ i = c, 

which we will also denote by I . In E = Sh(B, I ) the following 

defines an filter on N: 

p −X ∈ U ⇔ {q ≤ p : q ∈ X (n) for all n ∈ q} ∈ Cov(p). 

Forcing clauses for the atomic formulas are as above and the 

clauses for the interpretation of logical connectives are as in 

covering semantics. 



Questions 

1 What holds in these intuitionistic models? Which one is 

“better”? 

2 Jeremy Avigad and his student Henry Townser have used the 

classical model for proof-mining purposes. Is a direct 

functional interpretation of the language of non-standard 

arithmetic possible?

On a forcing model for non-standard arithmetic

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