1 The BBC realizability algebra

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Bar recursion in classical realisability : dependent choice and well ordering of R

Jean-Louis Krivine

June 30, 2016

Introduction

This paper is about the bar recursion operator [9], in the context of classical realizability [6, 7]. It is a sequel to the three papers [1, 2, 10]. We use the definitions and notations of the theory of classical realizability as expounded in [5, 6, 7].¹

In [1], S. Berardi, M. Bezem and T. Coquand have shown that a form of the bar recursion operator can be used, in a proof-program correspondence, to interpretthe axiom of dependent choicein proofs ofΠ⁰₂-formulas of arithmetic. Their work was enhanced and adapted to the theory of domains by U. Berger and P. Oliva in [2]. In [10], T. Streicher has shown, by using the bar recursion operator of [2], that the models of ZF, associated with realizability algebras [5, 7] obtained from usual models ofλ-calculus (Scott domains, coherent spaces, . . . ), satisfy the axiom of dependent choice.

We give here a proof of this result, but for a realizability algebra which is built following the presentation of [1], which we call the BBC-algebra.

In section 1, we define and study this algebra ; we define also the bar recusion operator, which is a closedλ-term.

In sections 2 and 3, which are very similar, we show that this operator realizes the axiom of countable choice (CC), then the axiom of dependent choix (DC). The proof is a little simpler for CC.

In section 4, we deduce from this result, using results of [8] that, in the model of ZF associated with this realizability algebra,every real (more generally, every sequence of ordinals) is constructible.

The formulas “Ris well ordered” and “Continuum Hypothesis” are therefore realized in these models by a closedλc-term (i.e. aλ-term containing the control instructionccof Felleisen- Griffin).

We show also thatevery true formula of analysis is realized by a closedλc-term.

Using these new results, we show how to obtain a program (closedλc-term) from any proof of an arithmetical formula in the theory ZF + “Dependent choice” + “Every real is constructible”

(and therefore “Well ordering ofR” and “Continuum Hypothesis”).

1Articles [5, 6, 7, 8] are available at www.pps.univ-paris-diderot.fr/∼krivine/

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Remark. In [6], we obtain programs from proofs in ZF + “Dependent choice”, using another realizability algebra (thethread model). We cannot use Continuum Hypothesis in this context but, on the other hand, we are not limited to proofs of arithmetical formulas.

1 The BBC realizability algebra

The definition and general properties of a realizability algebraare given at the beginning of [5]. It consists of a set oftermsΛ, a set ofstacksΠand a set ofprocessesΛ?Π. Every closed λ-term is interpreted as a term.

In this paper, we consider a particular realizability algebraBB=(Λ,Π,⊥⊥), which we call the BBC algebrabecause it is a reformulation of the programming language of [1]. It is defined as follows :

• The set ofprocessesΛ?ΠisΛ×Π.

• The set oftermsΛis the smallest set which contains the followingconstants of term: B,C,I,K,W(Curry’s combinators),cc(Felleisen-Griffin instruction), A(abort instruction), p,q₀, . . . ,q_N (variables)whereNis a fixed integer ;

and is such that :

ifξ,η∈Λthen (ξ)η∈Λ (application);

with each sequenceξi(i∈N) of closed elements ofΛ(i.e. which contain no variable p,q₀, . . . ,q_N) is associated, in a one-to-one (and well founded) way, a constant of term denoted byV

iξi.

Therefore, each termξ∈Λis a finite sequence of constants of term and parentheses.

Λis defined by an induction of lengthℵ1and is of cardinality 2^ℵ⁰.

Notations.The application (. . . ((ξ1)ξ2) . . .)ξnis often written (ξ1)ξ2. . .ξnor evenξ1ξ2. . .ξn. The finite sequenceq₀, . . . ,q_N will be often written~q.

• The set ofstacksΠis defined as follows : a stackπis a finite sequencet₀

.

^{. . .}

.

^tn−1

.

π0with t₀, . . . ,t_n−1∈Λ; it is terminated by the symbolπ0which represents theempty stack.

For each stackπ, thecontinuationk_πis a term which is defined by recurrence : k_π₀=A; kt

.

π=`tk_π, with`t =((C)(B)CB)torλkλx(k)(x)t.

Thus, if the stackπist₀

.

^{. . .}

.

^tn−1

.

π0, we have k_π=(`t0) . . . (`tn−1)Aorλx(A)(x)t₀. . .t_n₋₁. Theinteger nis defined as follows :

0=(K)Iorλxλy y;n+1=(σ)n withσ=(BW)(C)(B)BBorλnλfλx(f)(n)f x.

The relation ofexecutionÂis the least preorder onΛ?Πdefined by the following rules (with ξ,η,ζ∈Λ,π∈Πandn∈N) :

1. (ξ)η?πÂξ?η

.

πÂξ?π0;(abort)or(delete the stack) 9. V

iξi?n

.

πÂξn?π; (oracle)

There is no execution rule for p andq_i which are, therefore, stop instructions ; p plays a special role because of the definition of⊥⊥below.

Whenξ,η∈Λ, we setξÂηiff (∀π∈Π)(ξ?πÂη?π).

• Execution of processes.

For every processξ?π, at most one among the rules 1 to 9 applies. By iterating these rules, we obtain thereductionor theexecutionof the processξ?π. This execution stops if and only if the stack is insufficient (rules 2 to 8) or does not begin with an integer (rule 9) or else if the process has the formp?$orq_i?$.

• Definition of⊥⊥.

We set⊥⊥ ={ξ?π∈Λ?Π; (∃$∈Π)(ξ?πÂp?$)}.

• Proof-like terms.

LetPL₀be the countable set of terms built with the constantsB,C,I,K,W,cc and the application. It is the smallest possible set ofproof-like terms.

We shall also consider the setPLofclosed terms (i.e. with no occurrence of p,~q) which is much bigger : it containsAand the oraclesV

iξiand is therefore of cardinality 2^ℵ⁰. Lemma 1. BBis a coherent realizability algebra.

B

Bis a realizability algebra:

It remains to check thatk_π?ξ

.

$Âξ?π, which is done by recurrence onπ: ifπ=π0, it is rule 8 ;

A coherent realizability algebra is useful in order to givetruth valuesto formulas of ZF. In fact, we use a theory called ZF_ε[6] which is a conservative extension of ZF.

This theory has an additional strong membership relation symbolεwhich is not extensional.

For each closed formula F of ZF_ε, we define two truth values, denotedkFk and |F|, with kFk ⊂Πand|F| ⊂Λ, with the relationξ∈ |F| ⇔(∀π∈ kFk)(ξ?π∈ ⊥⊥).

The relationξ∈ |F|is also writtenξ||−F and reads“the termξrealizes the formula F ”.

All the necessary definitions are given in [5, 6, 7].

The following lemma 2 is a useful property of the BBC realizability algebraBB.

Lemma 2. For all formulas A,B of ZF_ε, and all terms ξ∈Λ, we have : ξ||−A→B iff (∀η∈Λ)(η||−A⇒ξη||−B).

Indeed, by the general definition of ||−, we have :

(ξ||−A→B)⇔(∀η||−A)(∀π∈ kBk)(ξ?η

.

π∈ ⊥⊥).

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Now, by the above definition of⊥⊥, it is clear that (ξ?η

.

π∈ ⊥⊥)⇔(ξη?π∈ ⊥⊥) from which the result follows.

Q.E.D.

Classical realizability is an extension of forcing. As in forcing, we start with an ordinary model M of ZFC (or even ZF + V = L) which we call theground model, and we build arealizability modelN which satisfies ZF_εin the following sense :

M andN have the same domain, but neither the same language, nor the same truth values.

The language ofN has the additional binary symbol εof strong membership. The truth values ofN are not 0, 1 as forM, but are taken inP(Π) endowed with a suitable structure of Bolean algebra [5, 7]. We say thatN satisfies a formula F iff there is a proof-like termθwhich realizesF or equivalently, if the truth value ofF is the unit of the Boolean algebraP(Π).

Afunctionalon the ground modelM is a formulaF(~x,y) of ZF with parameters inM, such thatM |= ∀~x∃!y F(~x,y). Denoting such a functional byf, we writey=f(~x) forF(~x,y).

SinceM andN have the same domain, all the functionals defined onM are also defined onN andthey satisfy the same equationsand even the sameHorn formulasi.e. formulas of the form∀~x(f₁(~x)=g₁(~x), . . . ,f_n(~x)=g_n(~x)→ f(~x)=g(~x)).

A particularly useful binary functional onM (and thus also on N ) is theapplication, denoted by app, which is defined as follows : app(f,x)={y; (x,y)∈f}.

We shall often writef[x] for app(f,x). This allows to consider each set inM (and inN ) as a unary functional.

Remark.We can define a setf inMby givingf[x] for everyx, provided that there exists a setXsuch thatf[x]= ;for allx∉X : takef =S

x∈X{x}×f[x].

In the ground modelM, every function is defined in this way but in general,this is false inN.

Quantifiers restricted to N

In [7], we defined the quantifier∀x^int, by setting : k∀x^intF[x]k =S

n∈Nk{n}→F[n]}k ={n

.

π;n∈N,π∈ kF[n]k}, so that we have : ξ||− ∀x^intF[x] ⇔ ξn||−F[n] for alln∈N;

and it is shown that it is a correct definition of the restricted quantifier toN.

Indeed the equivalence ∀x^intF[x]↔ ∀x(int[x]→F[x]) is realized by a closedλ-term inde- pendent ofF, called astorage operator.

^t by replacing, ink_π, the last occurrence ofAby`tA.

Lemma 5. Ifξ?π∈ ⊥⊥, then ξ⁰?π⁰∈ ⊥⊥and ξ⁰?π⁰

.

t∈ ⊥⊥, whereξ⁰?π⁰is obtained by replac- ing, in ξ?π, some occurrences of Aby(`u)A=ku

.

π0 and some occurrences of the variabless q₀, . . . ,q_N by t₀, . . . ,t_N ; t₀, . . . ,t_N,t,u are arbitrary terms.

Remark.In particular, it follows thatξ?π0∈ ⊥⊥ ⇒ξ||− ⊥.

Proof by recurrence on the length of the execution ofξ?π∈ ⊥⊥by means of rules 1 to 9. We consider the last used rule. There are two non trivial cases :

• Rule 7 (execution ofcc) ; we must showcc?ξ⁰

^t.

• Rule 8 (execution ofA) ; we must show (`u)A?ξ⁰

.

π⁰

.

t∈ ⊥⊥.

π.

Lemma 6. If ξ?π∈ ⊥⊥, then :

• ξ⁰?π⁰∈ ⊥⊥, whereξ⁰?π⁰is obtained by substituting arbitrary terms for the non efficient occurrences ofp.

• ξ⁰?π⁰∉ ⊥⊥and indeedξ⁰?π⁰Âq₀?$, whereξ⁰?π⁰is obtained by substituting q₀for the efficient occurrence ofp, and arbitrary terms for the non efficient occurrences ofp.

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The proof is immediate, by recurrence on the length of the reduction ofξ?πby means of rules 1 to 9 : consider the last used rule.

Q.E.D. Corollary 7.

If ξ||− >,⊥ → ⊥and ξ||− ⊥,> → ⊥, then ξ||− >,> → ⊥, and thus : λx(x)I I ||− ¬∀x^ג²(x6=0,x6=1→ ⊥)and W ||− ∀x^ג²¡

∀y^ג²(y6=0,y6=x→y^x),^x6=0→ ⊥¢ . Remark. These two formulas express that the Boolean algebraג2, which is defined in [7, 8], is non trivial and even atomless. Intuitively,ג2 represents thetype of Booleansof the realizability modelN. It is called thecharacteristic Boolean algebraofN.

π0∈ ⊥⊥for everyt,u∈Λ, again by lemma 6.

The last two assertions follow from the fact that :

k∀x^ג²(x6=0,x6=1→ ⊥)k = k>,⊥ → ⊥k ∪ k⊥,> → ⊥kand therefore :

|∀x^ג²(x6=0,x6=1→ ⊥)| = |>,> → ⊥|.

Q.E.D.

Theorem 8. For every sequenceξi ∈Λ(i∈N), there exists φ∈Λsuch that :

• φiÂξifor every i ∈N;

• for every U∈Λsuch that Uφ||− ⊥, there exists k ∈Nsuch that Uψ||− ⊥for every ψ∈Λ such that ψi Âξi for every i<k.

Remark. Theorem 8 will be used in order to show properties of the bar recursion operator. In fact, the following weaker formulation is sufficient :

For every sequenceξi∈Λ(i∈N) and everyU∈Λsuch that :

(∀k∈N)(∃ψ∈Λ){Uψ6||− ⊥, (∀i<k)(ψiÂξi)}

there exists φ∈Λsuch thatUφ6||− ⊥and (∀i∈N)(φiÂξi).

In the particular case of forcing, this is exactly thedecreasing chain condition: every decreasing sequence of (non false) conditions has a lower bound (which is non false).

We setηi=λpλ~qξi; thus, we haveηi∈PLandηip~qÂξi. Letη=V

iηi andφ=λx(η)xp~q. Thus, we haveη∈PLandφi Âξi. We may assume that the constantηdoes not appear inU. By the inductive definition ofV

i, it does not appear either inξi.

Now, we haveUφ||− ⊥ ⇔U?φ

.

π0∈ ⊥⊥(lemma 5). During the execution of the process U?φ

.

π0, the constantηarrives in head position a finite number of times, always throughφ (since it is deleted each time it arrives in head position), therefore as follows :

• We want aλ-termχsuch that :

χk f ziÂf i if i<k;χk f ziÂz if i ≥k.

Therefore, we set :

χ=λkλfλzλi((i<k)(f)i)z where the boolean (i<k) is defined by :

(i<k)=((k A)λd0)(i A)λd1 with0=λxλy yorK I,1=λxλy xorKandA=λxλy y xorC I.

The termχk f is a representation, inλ-calculus, of the finite sequence (f0,f1, . . . ,f k−1).

• We want aλ-termΨsuch that :

Ψg uk f Â(u)(χk f)(g)λz(Ψg uk⁺)(χ)k f z

wherek⁺=((BW)(C)(B)BB)korλfλx(f)(k)f xis the successor of the integerk. Thus, we set : Ψ=λgλu(Y)λhλkλf(u)(χk f)(g)λz(hk⁺)(χ)k f z.

whereYis the Turing fix point operator : Y=X X with X =λxλf(f)(x)x f or (W)(B)(BW)(C)B. The termΨwill be called thebar recursion operator.

2 Realizing countable choice

Theaxiom of countable choiceis the following formula : (CC) ∀n∃x F[n,x]→ ∃f∀n^intF[n,f[n]]

whereF[n,x] is an arbitrary formula of ZF_ε(see [6]), with parameters and two free variables.

The notationf[n] stands for app(f,n) (the functional app has been defined above).

Remark. This is a strong form of countable choice which shows that, in the realizability modelN, every countable sequence has the formn7→f[n] for somef. This will be used in section 4.

Theorem 9. λgλu(Ψ)g u0 0 ||−CC.

The axiom of countable choice is therefore realized in the model of ZF associated with the BBC realizability algebra (in fact, it is sufficient that the realizability algebra satisfies the property formulated in the remark following theorem 8).

We write the axiom of countable choice as follows :

(CC) ∀n¬∀x¬F[n,x],∀f¬∀n^intF[n,f[n]]→ ⊥

LetG,U∈Λbe such that G||− ∀n¬∀x¬F[n,x] and U ||− ∀f¬∀n^intF[n,f[n]].

We setH=ΨGU and we have to show thatH0 0||− ⊥. In fact, we shall show that H0ξ||− ⊥ for everyξ∈Λ.

Lemma 10. Let k∈Nand φ∈Λbe such that (∀i <k)∃ai(φi ||−F[i,ai]).

If H kφ6||− ⊥, then there exist a set a_kand a term ζk,φ∈Λsuch that : ζk,φ||−F[k,a_k] and (H k⁺)(χ)kφζk,φ6||− ⊥.

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Defineηk,φ=λz(H k⁺)(χ)kφz, so thatH kφÂ(U)(χkφ)(G)ηk,φ.

If ηk,φ||− ∀x¬F[k,x] then, by hypothesis onG, we haveGηk,φ||− ⊥. Let us check that : (χkφ)(G)ηk,φ||− ∀n^intF[n,f_k[n]]

wheref_kis defined by : f_k[i]=a_i ifi<k(i.e.i ∈k) ;f_k[i]= ;ifi∉k.

Indeed, if we set φ⁰=(χkφ)(G)ηk,φ, we have :

φ⁰iÂφi ||−F[i,a_i] fori<kandφ⁰iÂ(G)ηk,φ||− ⊥fori≥k, and thereforeφ⁰i||−F[i,;].

By hypothesis onU, it follows that (U)(χkφ)(G)ηk,φ||− ⊥, in other wordsH kφ||− ⊥.

Thus, we have shown that, if H kφ6||− ⊥, thenηk,φ6||− ∀x¬F[k,x], which gives immediately the desired result.

Q.E.D.

Letφ0∈Λbe such thatH0φ06||− ⊥. By means of lemma 10, we defineφk+1∈Λanda_krecur- sively onk, by setting φ_k+1=χkφkζk,φk.

By definition ofχ, we haveφk+1iÂζ_k,φ_k fori≥k.

Then, we show easily, by recurrence onk:

φk+1iÂφi+1i Âζi,φi ||−F[i,a_i] fori≤k; H kφk 6||− ⊥. Therefore, we can define :

a functionf of domainNsuch that f[i]=a_i for everyi∈N; and, by theorem 8, a termφ∈Λsuch thatφkÂζk,φk for allk∈N.

Therefore, we haveφi ||−F[i,f[i]] for everyi∈N, that is to say φ||− ∀n^intF[n,f[n]].

By hypothesis onU, it follows that Uφ||− ⊥. Therefore, by theorem 8, applied to the se- quenceξi=ζi,φi, there exists an integerksuch thatUψ||− ⊥, for every termψ∈Λsuch that ψiÂζi,φi fori <k.

Thus, in particular, we have (U)(χkφk)ξ||− ⊥for everyξ∈Λ.

Now, by definition ofH, we have H kφk Â(U)(χkφk)ξ withξ=(G)λz(H k⁺)(χ)kφkz, and therefore H kφk||− ⊥, that is a contradiction.

Thus, we have shown that H0φ0||− ⊥for everyφ0∈Λ.

Q.E.D.

3 Realizing dependent choice

Theaxiom of dependent choiceis the following formula : (DC) ∀x∃y F[x,y]→ ∃f∀n^intF[f[n],f[n+1]]

whereF[x,y] is an arbitrary formula of ZF_ε, with parameters and two free variables.

The notationf[n] stands for app(f,n) as defined above.

Theorem 11. λgλu(Ψ)g u0 0||− DC.

The axiom of dependent choice is therefore realized in the model of ZF associated with the BBC realizability algebra (or, more generally, with any realizability algebra satisfying the property formulated in the remark after theorem 8).

The proof of theorem 11 is almost the same as theorem 9.

We write theaxiom of dependent choiceas follows :

(DC) ∀x¬∀y¬F[x,y],∀f¬∀n^intF[f[n],f[n+1]]→ ⊥.

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LetG,U∈Λbe such that G||− ∀x¬∀y¬F[x,y] andU ||− ∀f¬∀n^intF[f[n],f[n+1]].

We setH=ΨGU and we have to show thatH0 0||− ⊥. In fact, we shall show that H0ξ||− ⊥ for everyξ∈Λ.

Lemma 12.

Let a0, . . . ,a_kbe a finite sequence inM and φ∈Λbe such that (∀i<k)(φi ||−F[ai,ai+1]).

If H kφ6||− ⊥, then there existζ∈Λand a_k₊₁inM such that : ζ||−F[a_k,a_k+1]and(H k⁺)(χ)kφζ6||− ⊥. Defineηk,φ=λz(H k⁺)(χ)kφz, so thatH kφÂ(U)(χkφ)(G)ηk,φ.

If ηk,φ||− ∀y¬F[a_k,y] then, by hypothesis onG, we haveGηk,φ||− ⊥. We check that : (χkφ)(G)ηk,φ||− ∀n^intF[f_k(n),f_k[n+1]]

wheref_kis defined by f_k[i]=ai fori≤k(i.e.i∈k+1) ; f_k[i]= ;fori∉k+1.

Indeed, if we set φ⁰=(χkφ)(G)ηk,φ, we have :

φ⁰iÂφi ||−F[a_i,a_i+1] fori <kandφ⁰iÂ(G)ηk,φ||− ⊥fori≥k.

Therefore, we haveφ⁰i ||−F[f_k[i],f_k[i+1]] for everyi∈N.

By hypothesis onU, it follows that (U)(χkφ)(G)ηk,φ||− ⊥, that isH kφ||− ⊥.

Thus, we have shown that, ifH kφ6||− ⊥, thenηk,φ6||− ∀y¬F[a_k,y], which gives immediately the desired result.

Q.E.D.

Letφ0∈Λbe such thatH0φ06||− ⊥and leta₀= ;. Using lemma 12, we defineφk+1∈Λand a_k₊₁inM recursively onk, by setting φk+1=χkφkζk,φk, whereζk,φk is given by lemma 12, where we setφ=φk. By definition ofχ, we haveφ_k+1iÂζ_k,φ_k fori ≥k.

Then, we show easily, by recurrence onk:

φk+1iÂφi+1iÂζi,φi ||−F[a_i,a_i₊₁] fori≤k; H kφk6||− ⊥.

Therefore, we can define :

a functionf of domainNsuch that f[i]=a_i for everyi∈N;

and, by means of theorem 8, a termφ∈Λsuch thatφkÂζk,φk for everyk∈N.

Thus, we haveφi ||−F[f[i],f[i+1]] for everyi∈N, that is to say φ||− ∀n^intF[f[n],f[n+1]].

By hypothesis onU, it follows that Uφ||− ⊥. Therefore, by theorem 8, applied to the se- quenceξi =ζi,φi, there exists an integerksuch thatUψ||− ⊥, for every termψ∈Λsuch that ψiÂζi,φi fori <k.

Thus, in particular, we have (U)(χkφk)ξ||− ⊥for everyξ∈Λ.

But, by definition ofH, we have H kφk Â(U)(χkφk)ξ with ξ=(G)λz(H k⁺)(χ)kφkz, and therefore H kφk||− ⊥, that is a contradiction.

Thus, we have shown that H0φ0||− ⊥for everyφ0∈Λ.

Q.E.D.

4 A well ordering on R

In this section, we use the notations and the results of [7] and [8].

IfF is a closed formula of ZF_ε, the notation ||−F means that there exists a proof-like term θ∈PL₀(i.e. a closedλc-term) such that θ||−F.

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In section 2, we have realized the axiom of countable choice (CC). We replaceF[n,x] with int(n)→F[n,x] and we add a parameterφ; we obtain :

||− ∀φ¡

∀n^int∃x F[n,x,φ]→ ∃f∀n^intF[n,f[n],φ]¢ for every formulaF[n,x,φ] of ZF_ε.

In particular, takingφε2^NandF[n,x,φ]≡(x=φ(n))∧(x=0∨x=1) (i.e. (n,x)εφ∧(x=0∨x=1)), we find :

||−(∀φε2^N)∃f∀n^int¡

(f[n]=φ(n))∧(f[n]=0∨f[n]=1)¢ . For any setf in the ground modelM, letg={x; f[x]=1}.

We have trivially I ||− 〈n∈g〉 = 〈f[n]=1〉.² It follows that : ||− ∀f∃g∀n¡

(f[n]=0∨f[n]=1)→f[n]= 〈n∈g〉¢ . We have shown that : ||−(∀φε2^N)∃g∀n^int(φ(n)= 〈n∈g〉).

Now, in [8], we have built an ultrafilterD :ג²→2 on the Boolean algebra ג2, with the following property : the modelN , equipped with the binary relationsD(〈x∈y〉),D(〈x=y〉), is a model of ZF, denotedM_D, which is an elementary extension of the ground modelM. Moreover,M_D is isomorphic to a transitive submodel ofN (considered as a model of ZF), which contains every ordinal ofN .

M_Dsatisfies the axiom of choice, because we suppose thatM|=ZFC.

If we suppose thatM |= V = L, thenM_D is isomorphic to the classL^N of constructible sets ofN .

For everyφ:N→2, we have obviouslyD(φ(n))=φ(n). It follows that :

||−(∀φε2^N)∃g∀n^int¡

φ(n)=D〈n∈g〉¢ .

This shows that the subset ofNdefined byφis in the modelM_D: indeed, it is the elementg of this model.

We have just shown thatN andM_Dhave the same reals.

Therefore,Ris well ordered inN , and we have : ||−(Ris well ordered).

Moreover, if the ground modelM satisfies V = L, we have : ||− (every real is constructible).

Therefore, the continuum hypothesis is realized.

Since the modelsN andMDhave the same reals, every formula of analysis (closed formula with quantifiers restricted toNorR) has the same truth value inM_D,M orN.

It follows that :

For every formula F of analysis, we have M |=F if and only if ||−F . In particular, we have ||−F or ||− ¬F.

References

[1] S. Berardi, M. Bezem and T. Coquand.On the computational content of the axiom of choice.J. Symb. Logic 63, 2 (1998) p. 600-622.

[2] U. Berger and P. Oliva.Modified bar recursion and classical dependent choice.

Proc. Logic Colloquium 2001 - Springer (2005) p. 89-107.

2The notationsג^{2 and}〈F〉whereFis a closed formula of ZF, with parameters in the realizability modelN, are defined in [7, 8].ג2 is called thecharacteristic Boolean algebra ofN. We have〈F〉εג^2.

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[3] E. Engeler.Algebras and combinators.Algebra Universalis, vol. 13, 1 (1981) p. 389-392.

[4] T. Griffin.A formulæ-as-type notion of control.

Conf. record 17th A.C.M. Symp. on Principles of Progr. Languages (1990).

[5] J.-L. Krivine.Realizability algebras : a program to well orderR.

Logical Methods in Computer Science vol. 7, 3:02 (2011) p. 1-47.

[6] J.-L. Krivine.Realizability algebras II : new models of ZF + DC.

Logical Methods in Computer Science, vol. 8, 1:10 (2012) p. 1-28.

[7] J.-L. Krivine.Realizability algebras III: some examples.

http://arxiv.org/abs/1210.5065 (2013)³. To appear in Math. Struct. Comp. Sc.

[8] J.-L. Krivine.On the structure of classical realizability models of ZF.

http://arxiv.org/abs/1408.1868 (2014). To appear in Proceedings Types ’2014.

[9] C. Spector.Provably recursive functionals of analysis : a consistency proof of analysis by an extension of principles in current intuitionistic mathematics.

Recursive function theory : Proc. Symp. in pure math. vol. 5, Amer. Math. Soc. Provi- dence, Rhode Island, 1962, p. 1-27.

[10] T. Streicher. A classical realizability model arising from a stable model of untyped λ- calculus.http://arxiv.org/abs/1407.1547 (2013).

3Articles [5, 6, 7, 8] are available atwww.pps.univ-paris-diderot.fr/∼krivine/