1.1 Realizability algebras

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Realizability algebras III : some examples

Jean-Louis Krivine

Université Paris-Diderot - C.N.R.S.

April 14, 2016

Introduction

The notion ofrealizability algebra, which was introduced in [17, 18], is a tool to study the proof-program correspondence and to build new models of set theory, which we callrealiz- ability models of ZF.

It is a variant of the well known notion ofcombinatory algebra, with a new instructioncc, and a new type for theenvironments.

Thesets of forcing conditions, in common use in set theory, are (very) particular cases of realizability algebras ; and the forcing models of ZF are very particular cases of realizability models.

We show here how to extend an arbitrary realizability algebra, by means of a certain set of conditions, so that the axiom DC of dependent choice is realized.

In order to avoid introducing new instructions, we use an idea of A. Miquel [19].

This technique has applications of two kinds : 1. Construction of models of ZF + DC.

When the initial realizability algebra is not trivial (that is, if we are not in the case of forcing or equivalently, if the associated Boolean algebraג^{2 is}⁶⁼{0, 1}), then we always obtain in this way a model of ZFwhich satisfies DC + there is no well ordering of R.

By suitably choosing the realizability algebras, we can get, for instance, the relative consis- tency over ZF of the following two theories :

i) ZF + DC + there exists an increasing function i7→X_i, from the countable atomless Boolean algebraBintoP(R) such that :

X₀={0} ;i6=0⇒X_i is uncountable ; X_i∩X_j=X_i_∧j ;

if i∧j=0 then X_i_∨_jis equipotent withX_i×X_j ; X_i×X_i is equipotent withX_i ;

there exists a surjection fromX₁ontoR;

if there exists a surjection fromX_j ontoX_i, theni≤j ;

if i,j 6=0,i∧j=0, there is no surjection from X_i]X_j (disjoint union) ontoX_i×X_j ; more generally, ifA⊂Band if there exists a surjection from S

j∈AX_j ontoX_i, theni ≤j for some j∈A.

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In particular, there exists a sequence of subsets ofR, the cardinals of which are not compa- rable, and also a sequence of subsets ofR, the cardinals of which are strictly decreasing.

ii) ZF + DC + there existsX ⊂Rsuch that :

X is uncountable and there is no surjection fromX ontoℵ1

(and therefore, every well orderable subset ofX is countable) ; X×X is equipotent withX ;

there exists a total order onX, every proper initial segment of which is countable ; there exists a surjection fromX×ℵ1ontoR;

there exists an injection fromℵ1(thus also fromX×ℵ1) intoR.

2. Curry-Howard correspondence.

With this technique of extension of realizability algebras, we can obtain a program from any proof, in ZF + DC, of an arithmetical formulaF, which is aλc-term, that is, aλ-term con- tainingcc,but no other new instruction.

This is a notable difference with the method given in [14, 15], where we use the instruction quoteand which is, on the other hand, simpler and not limited to arithmetical formulas.

It is important to observe that the program we get in this way does not really depend on the given proof of DC→F in ZF, but only on theprogram P extracted from this proof, which is a closedλc-term. Indeed, we obtain this program by means of an operation ofcompilation applied to P (look at the remark at the end of the introduction of [17]).

Finally, apart from applications 1 and 2, we may notice theorem 27, which gives an interest- ing property ofeveryrealizability model : as soon as the Boolean algebraג2 is not trivial (i.e.

if the model is not a forcing model), there exists a non well orderable individual.

Acknowledgements. I want to thank the referees, especially the second one for numerous pertinent and helpful remarks and suggestions.

1 Generalities

1.1 Realizability algebras

It is a first order structure, which is defined in [17]. We recall here briefly the definition (with a little change) and some essential properties :

Arealizability algebraA is made up of three sets :Λ(the set ofterms),Π(the set ofstacks), Λ ? Π(the set ofprocesses) with the following operations :

(ξ,η)7→(ξ)ηfromΛ²intoΛ(application) ; (ξ,π)7→ξ

.

πfromΛ×ΠintoΠ(push) ;

(ξ,π)7→ξ?πfromΛ×ΠintoΛ ? Π(process) ; π7→k_πfromΠintoΛ(continuation).

There are, inΛ, distinguished elements B,C,I,K,W,cc, calledelementary combinatorsorin- structions.

Notation.The term (. . . (((ξ)η1)η2) . . .)ηnwill be also written as (ξ)η1η2. . .ηnorξη1η2. . .ηn. For instance : ξηζ=(ξ)ηζ=(ξη)ζ=((ξ)η)ζ.

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We define a preorder onΛ ? Π, denoted by Â, which is calledexecution; ξ?πÂξ⁰?π⁰is read as :the processξ?πreduces toξ⁰?π⁰.

.

η

.

ζ

.

πÂξ?(η)ζ

.

π.

cc_?_ξ

.

πÂξ?k_π

.

π. k_π?ξ

.

$Âξ?π.

We are also given a subset⊥⊥ofΛ ? Πsuch that :

ξ?πÂξ⁰?π⁰,ξ⁰?π⁰∈ ⊥⊥ ⇒ ξ?π∈ ⊥⊥.

Given two processesξ?π,ξ⁰?π⁰, the notationξ?πÂÂξ⁰?π⁰means : ξ?π∉ ⊥⊥ ⇒ξ⁰?π⁰∉ ⊥⊥.

Therefore, obviously, ξ?πÂξ⁰?π⁰ ⇒ ξ?πÂÂξ⁰?π⁰.

Finally, we choose a set of terms QP_A ⊂Λ, containing the elementary combinators :

B,C,I,K,W,ccand closed by application. They are called theproof-like terms of the algebraA. We write also QP instead of QP_A if there is no ambiguity aboutA.

The algebraA is calledcoherent if, for every proof-like term θ∈QP_A, there exists a stackπ such thatθ?π∉ ⊥⊥.

Remark.Thesets of forcing conditionscan be considered as degenerate cases of realizability algebras, if we present them in the following way : an inf-semi-latticeP, with a greatest element1and an initial segment⊥⊥ofP (the set offalseconditions). Two conditionsp,q∈P are calledcompatibleif their g.l.b.p∧qis not in⊥⊥.

We get a realizability algebra if we setΛ=Π=Λ ? Π=P ;B=C=I=K=W=cc=1and QP={1} ; (p)q =p

.

^q =p?q=p∧q andkp=p. The preorder pÂq is defined asp ≤q, i.e. p∧q=p. The condition of coherence is1∉ ⊥⊥.

1.2 c -terms and λ -terms

The terms of the language of combinatory algebra, which are built with variables, the constant symbolsB,C,I,K,W,ccand the application (binary operation), will be calledcombina- tory termsorc-terms, in order to distinguish them from the terms of the algebraA, which are elements ofΛ.

Each closedc-term (i.e. without variable) takes a value in the algebraA, which is a proof-like term ofA.

Let us callatomac-term of length 1, i.e. a constant symbolB,C,I,K,W,ccor a variable.

Lemma 1. Everyc-term t can be written, in a unique way, in the form t=(a)t₁. . .t_kwhere a is an atom and t₁, . . . ,t_karec-terms.

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Immediate, by recurrence on the length oft.

Q.E.D.

The result of thesubstitution of ξ1, . . . ,ξn∈Λto the variables x₁, . . . ,x_nin ac-termt, is aterm (i.e. an element ofΛ) denoted byt[ξ1/x₁, . . . ,ξn/x_n] or, more briefly,t[~ξ/~x].

The inductive definition is : a[~ξ/~x]=ξi ifa=x_i(1≤i≤n) ;

a[~ξ/~x]=aifais an atom6=x₁, . . . ,x_n; (t u)[~ξ/~x]=(t[~ξ/~x])u[~ξ/~x].

Given ac-termtand a variablex, we define inductively ont, a newc-term denoted by λ x t, which does not containx. To this aim, we apply the first possible case in the following list : 1. λ x t=(K)tiftdoes not containx.

2. λ x x=I.

3. λ x(t)u=(C λ x t)uifudoes not containx.

4. λ x(t)x=tiftdoes not containx.

5. λ x(t)x=(W) λ x t(iftcontainsx).

6. λ x(t)(u)v= λ x(((B)t)u)v (ifuvcontainsx).

Lemma 2. This rewriting is finite for every c-term.

For eachc-termt, we define inductively as follows the integersl[t],ν0[t],ν1[t],ν2[t] : l[a]=1 for every atoma;l[(t)u]=l[t]+l[u]+2 ;

ν0[a]=0 for every atoma;ν0[(t)u]=l[u] ;

ν1[x]=1 ;ν1[a]=0 for every atoma6=x;ν1[(t)u]=ν1[t]+ν1[u] ;

ν2[t]=0 ifxis not int;ν2[x]=1 ;ν2[(t)u]=ν2[u] ifxappears inu;ν2[(t)u]=ν2[t]+l[u]+1 ifxis not inu(and thenxappears int).

l[t] is the length oft;

ν0[t] is the length of the argument, whentis an application ; ν1[t] is the number of occurrences ofxint;

ν2[t] is the distance to the end of the termt, of the last occurrence ofxint.

Rule 5 strictly decreasesν1(t) ; rule 3 does not changeν1[t] and strictly decreasesν2[t] ; rule 6 does not changeν1[t] andν2[t] and strictly decreasesν0[t].

Therefore, after a finite number of applications of rules 3, 5 and 6, we must apply one of the rules 1, 2 or 4, and the rewriting stops.

Q.E.D.

Given ac-termtand a variablex, we now define thec-termλx t by setting : λx t= λ x(I)t.

π

Âa[ξ/x,~η/~y]?t₁[ξ/x,~η/~y]

.

^{. . .}

.

^tk−1[ξ/x,~η/~y]

.

^tk[~η/~y]πby the induction hypothesis

≡a[ξ/x,~η/~y]?t₁[ξ/x,~η/~y]

.

^{. . .}

.

^tk−1[ξ/x,~η/~y]

.

^tk[ξ/x,~η/~y]

.

πsincexis not int_k. In rules 4 and 5, we havet_k=x, i.e.t=(u)x.

π.

π

πÂt[ξ/x,~η/~y]?π.

Immediate by lemma 4 and the definition ofλx t which is λ x(I)t.

Q.E.D.

We can now prove theorem 3 by induction onn; the casen=0 is trivial.

1.3 The formal system

We write formulas and proofs in the language of first order logic. This formal language con- sists of :

• individual variables x,y, . . . ;

• function symbols f,g, . . . of various arities ; function symbols of arity 0 are calledconstant symbols.

• relation symbols; there are three binary relation symbols : ε/ ,∉,⊂.

The terms of this first order language will be called`-terms; they are built in the usual way with individual variables and function symbols.

Remark.Thus, we use four expressions with the wordterm: term,c-term,λ-term and`-term.

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Theatomic formulasare the expressions>,⊥,tε/u,t∉u,t⊂u, wheret,uare`-terms.

Formulasare built as usual, from atomic formulas,with the only logical symbols →,∀:

• each atomic formula is a formula ;

• if A,Bare formulas, then A→B is a formula ;

• if Ais a formula andxan individual variable, then∀x Ais a formula.

Notations.LetA₁, . . . ,A_n,A,Bbe formulas. Then : A→ ⊥ is written ¬A;

A1→(A2→ · · · →(An→B)· · ·) is written A1,A2, . . . ,An→B ;

¬A₁, . . . ,¬A_n→ ⊥ is written A₁∨. . .∨A_n; (A₁, . . . ,A_n→ ⊥)→ ⊥ is written A₁∧. . .∧A_n;

¬∀x(A1, . . . ,An→ ⊥) is written ∃x{A1, . . . ,An}.

Therules of natural deductionare the following (the Ai’s are formulas, thexi’s are variables ofc-term,t,uarec-terms, written asλ-terms) :

1.x1:A1, . . . ,xn:An`xi:Ai.

2.x₁:A₁, . . . ,x_n:A_n`t:A→B, x₁:A₁, . . . ,x_n:A_n`u:A ⇒ x₁:A₁, . . . ,x_n:A_n`t u:B. 3.x₁:A₁, . . . ,x_n:A_n,x:A`t:B ⇒ x₁:A₁, . . . ,x_n:A_n`λx t :A→B.

4.x1:A1, . . . ,xn:An`t:A ⇒ x1:A1, . . . ,xn:An`t:∀x A wherexis an individual variable which does not appear inA₁, . . . ,A_n.

5.x₁:A₁, . . . ,x_n:A_n`t:∀x A ⇒ x₁:A₁, . . . ,x_n:A_n`t: A[τ/x] wherexis an individual variable andτis a`-term.

6.x₁:A₁, . . . ,x_n:A_n`cc: ((A→B)→A)→A (law of Peirce).

7.x₁:A₁, . . . ,x_n:A_n`t:⊥ ⇒ x₁:A₁, . . . ,x_n:A_n`t:A for every formulaA.

1.4 Realizability models

We formalize set theory with the first order language described above. We write, in this language, the axioms of a theory named ZF_ε, which are given in [18].

The usual set theory ZF is supposed written with the only relation symbols∉,⊂. Then, ZF_εis aconservative extensionof ZF, which is proved in [18].

Remark. In ZF_ε, we have two membership relations : ∈which theweak orextensionalone, andε which is thestrongone and does not satisfy extensionality.

Therefore, we have also two relations of inclusion : x⊂y ≡ ∀z(z∉ y→z ∉x) (∈-inclusion or ZF- inclusion) andx⊆y≡ ∀z(zε/y→zε/x) (ε-inclusion). Finally, we have three notions of equality : the weakest one is (x∼y)≡x⊂y∧y⊂x(∈-equality) ; the strongest one is (x=y)≡ ∀z(xε/z→yε/z) (Leibniz equality) ; the middle one is (x'y)≡x⊆y∧y⊆x(ε-equality).

Let us consider acoherentrealizability algebraA, defined in a modelM of ZF + V = L, which is called theground model. The elements ofM will be calledindividuals(in order to avoid the wordset, as far as possible).

We defined, in [18], arealizability model, denoted byNA (or evenN , if there is no ambiguity about the algebraA).

It has the same domain (the same individuals) asM and the interpretation of the function symbols is the same as inM.

Each closed formulaF of ZF_ε with parameters inM, hastwo truth valuesinN , which are

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denoted bykFk(which is a subset ofΠ) and|F|(which is a subset ofΛ).

Here are their definitions :

|F|is defined immediately fromkFk as follows :

ξ∈ |F| ⇔ (∀π∈ kFk)ξ?π∈ ⊥⊥.

We shall write ξ||−F (read“ξrealizes F ”) for ξ∈ |F|.

kFkis now defined by recurrence on the length ofF :

• F is atomic ;

thenFhas one of the forms >,⊥,aε/b,a⊂b,a∉bwhere a,bare parameters inM. We set : k>k = ;; k⊥k =Π; kaε/bk ={π∈Π; (a,π)∈b}.

ka⊂bk,ka∉bkare defined simultaneously by induction on (rk(a)∪rk(b), rk(a)∩rk(b)) (rk(a) being the rank ofainM).

ka⊂bk =[

c

{ξ

.

π; ξ∈Λ,π∈Π, (c,π)∈a,ξ||−c∉b} ; ka∉bk =[

c

{ξ

.

ξ⁰

.

π;ξ,ξ⁰∈Λ,π∈Π, (c,π)∈b,ξ||−a⊂c,ξ⁰||−c⊂a}.

• F ≡A→B ; then kFk ={ξ

.

π; ξ||−A,π∈ kBk}.

• F ≡ ∀x A: then kFk =[

a

kA[a/x]k.

The following theorem, proved in [18], is an essential tool : Theorem 6(Adequacy lemma).

Let A₁, . . . ,A_n,A be closed formulas of ZF_ε, and suppose that x₁:A₁, . . . ,x_n:A_n`t:A.

If ξ1||−A₁, . . . ,ξn||−A_n then t[ξ1/x₁, . . . ,ξn/x_n]||−A. In particular, if `t:A, then t||−A.

LetF be a closed formula of ZF_ε, with parameters inM. We say thatNA realizes F or thatF is realized inN_A (which is written N_A ||−F or even ||−F), if there exists a proof-like termθ such that θ||−F.

It is shown in [18] thatall the axioms of ZF_εare realized inN_A, and thus also all the axioms of ZF.

Remarks. NA is not a Tarski model of ZF_ε, because the truth values of formulas are not 0 or 1 but subsets ofΠ. We can obtain Tarski models by using the completeness theorem, or else an ultrafilter on the set of truth values equipped with a suitable structure of Boolean algebra (see [18]). Note that, doing this, we introduce new individuals in the model, i.e. individuals which are not inM.

An individual a of M is, in general, not related with its interpretation inNA. For instance, if no element ofais an ordered pair (b,π) withπ∈Π, thenais interpreted as;inNA.

Indeed, we havek∀x(xε/a)k = ; = k>k.

Definitions.Given a set of termsX ⊂Λand a formulaF, we shall use the notationX →F as anextended formula; its truth value is kX →Fk ={ξ

.

π; ξ∈X,π∈ kFk}.

Two formulas F[x₁, . . . ,x_n] and G[x₁, . . . ,x_n] of ZF_ε will be called interchangeable if the formula∀x₁. . .∀x_n(F[x₁, . . . ,x_n]↔G[x₁, . . . ,x_n]) is realized.

That is, for instance, the case if kF[a₁, . . . ,an]k = kG[a1, . . . ,an]k or also if kF[a₁, . . . ,a_n]k = k¬¬G[a₁, . . . ,a_n]k for everya₁, . . . ,a_n∈M.

The following lemma gives a useful example :

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Lemma 7. For every formula A, define ^¬A⊂Λby ^¬A={k_π;π∈ kAk}.

Then ¬A→B and ^¬A→B are interchangeable, for every formula B .

We have immediately k_π||− ¬A for everyπ∈ kAk. Therefore,k^¬A→Bk ⊂ k¬A→Bkand it follows that I_||−(¬A→B)→(^¬A→B).

Conversely, letξ,η∈Λ,ξ||−^¬A→B,η||− ¬Band letπ∈ kAk.

We haveξk_π||−B, thus (η)(ξ)k_π||− ⊥and therefore (η)(ξ)k_π?π∈ ⊥⊥. It follows thatθ?ξ

.

η

.

π∈ ⊥⊥withθ=λxλy(cc)λk(y)(x)k.

Finally, we have shown thatθ||−(^¬A→B)→(¬B→A), from which the result follows.

Q.E.D.

1.5 Equality and type-like sets

The formula x=y is, by definition, ∀z(xε/z→yε/z)(Leibniz equality).

Ift,u are`-terms andF is a formula of ZF_ε, with parameters inM, we define the formula t=u,→F. When it is closed, its truth value is :

kt=u,→Fk = k>k = ; if M |=t6=u;kt=u,→Fk = kFk if M |=t=u. The formula t=u,→ ⊥ is written t6=u.

The formula t₁=u₁,→(t₂=u₂,→ · · ·,→(t_n=u_n,→F)· · ·) is written : t₁=u₁,t₂=u₂, . . . ,t_n=u_n,→F.

The formulas t=u→F and t=u,→F are interchangeable, as is shown in the : Lemma 8.

i) C I I ||− ∀x∀y¡

(x=y→F)→(x=y,→F)¢

; ii) C I _{||− ∀}x∀y¡

(x=y,→F)→(x=y→F)¢ . i) Trivial.

ii) Leta,bbe individuals ; letξ||−a=b,→F,η||−a=bandπ∈ kFk. We show thatη?ξ

.

π∈ ⊥⊥. Letc={(b,π)} ; by hypothesis onη, we haveη||−aε/c→bε/c. Sinceπ∈ kbε/ck, it suffices to show thatξ||−aε/c. This is clear ifa6=b, sincekaε/ck = ;in this case.

Ifa=b, thenξ||−F, by hypothesis onξ, thusξ?π∈ ⊥⊥; butkaε/ck ={π} in this case, and thereforeξ||−aε/c.

Q.E.D.

We setג^X ⁼^X^×Πfor every individualX ofM ; we define the quantifier∀x^ג^X as follows : k∀x^ג^XF[x]k =S

a∈XkF[a]k. Of course, we set ∃x^ג^XF[x]≡ ¬∀x^ג^X¬F[x].

The quantifier∀x^ג^X has the intended meaning, which is that the formulas ∀x^ג^XF[x] and

∀x(xεג^X ^→^F[x]) are interchangeable. This is shown by the : Lemma 9.

C I_{||− ∀}x^ג^XF[x]→ ∀x^ג^X¬¬F[x]; cc||− ∀x^ג^X¬¬F[x]→ ∀x^ג^XF[x];

k∀x^ג^X¬¬F[x]k = k∀x(¬F[x]→xε/ג^X)k. Immediate.

Q.E.D.

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Eachfunctional f :Mⁿ →M, defined inM by a formula of ZF with parameters, gives a function symbol, that we denote also by f, and which has the same interpretation in the realizability modelN_A.

Proposition 10.

Let t,t₁, . . . ,t_n,u,u₁, . . . ,u_n be `-terms, built with variables x₁, . . . ,x_k and function symbols ofM.

i) If M |= ∀x₁. . .∀x_k(t₁=u₁, . . . ,t_k=u_k→t=u), then : I_{||− ∀}x₁. . .∀x_k(t₁=u₁, . . . ,t_k=u_k,→t=u).

ii) If M |=(∀x1∈X₁) . . . (∀xk∈X_k)(t₁=u₁, . . . ,t_k=u_k→t=u), then : I||− ∀x₁^ג^X¹. . .∀x^ג_k^X^k(t₁=u₁, . . . ,t_k=u_k,→t=u).

Trivial.

Q.E.D.

Proposition 11. If f :X₁×· · ·×X_n→Y is a function inM, its interpretation inNAis a function f :ג^X1×· · ·×ג^Xⁿ^→ג^{Y .}

Indeed, let f⁰,f⁰⁰:Mⁿ→M be any two functionals which are extensions of the function f to the whole ofMⁿ. By proposition 10(ii), we have :

I ||− ∀x₁^ג^X¹. . .∀x^ג_k^X^k(f⁰(x₁, . . . ,x_k)=f⁰⁰(x₁, . . . ,x_k)).

Q.E.D.

An important example is the set 2={0, 1} equipped with the trivial boolean functions, written

∧,∨,¬. The extension toN_A of these operations gives a structure of Boolean algebra onג^2.

It is called thecharacteristic Boolean algebraof the modelN_A. Conservation of well-foundedness

Theorem 12 says that every well founded relation in the ground modelM, gives a well founded relation in the realizability modelN .

Theorem 12.

Let f :M²→2be a functional defined in the ground modelM such that f(x,y)=1is a well founded relation onM. Then, for every formula F[x]of ZF_εwith parameters inM :

Y||− ∀y¡

∀x(f(x,y)=1,→F[x])→F[y]¢

→ ∀y F[y]

with Y=A A and A=λaλf(f)(a)a f (or A=(W)(B)(BW)(C)B).

Let us fixb∈X and let ξ||− ∀y¡

∀x(f(x,y)=1,→F[x])→F[y]¢

. We show, by induction onb, following the well founded relationf(x,y)=1, thatY?ξ

.

π∈ ⊥⊥for everyπ∈ kF[b]k.

Thus, suppose that π∈ kF[b]k; since Y?ξ

.

πÂξ?Yξ

.

π, we need to show that ξ?Yξ

.

π∈ ⊥⊥. By hypothesis, we have ξ||− ∀x(f(x,b)=1,→F[x])→F[b] ; thus, it suffices to show that : Yξ||−f(a,b)=1,→F[a] for every a∈X. This is clear if f(a,b)6=1, by definition of ,→. Iff(a,b)=1, we must show Y_ξ_||−F[a], i.e. Y_?ξ

.

$∈ ⊥⊥for every $∈ kF[a]k. But this follows from the induction hypothesis.

Q.E.D. Remarks.

i) If the functionf is only defined on a setXin the ground modelM, we can apply theorem 12 to the

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extensionf⁰off defined byf⁰(x,y)=0 if (x,y)∉X².

This shows that, in the realizability modelN, the binary relation f(x,y)=1 is well founded onגX. ii) We can use theorem 12 to show that the axiom of foundation of ZF_εis realized inNA.

Indeed, let us definef :M²→2 by settingf(x,y)=1⇔ ∃z((x,z)∈y). The binary relationf(x,y)=1 is obviously well founded inM. Now, we have I||− ∀x∀y(f(x,y)6=1→xε/y) becauseπ∈ kxε/yk ⇒ f(x,y)=1. Thus, the relationxεy is stronger than the relation f(x,y)=1, which is well founded inNA by theorem 12.

1.6 Integers

Letφ,α∈Λandn∈N; we define (φ)ⁿα∈Λby setting (φ)⁰α=α; (φ)ⁿ⁺¹α=(φ)(φ)ⁿα.

Forn∈N, we define n=(σ)ⁿ0 with 0=KIand σ=(BW)(B)B; nis “the integern” andσthe “successor” in combinatory logic.

.

π) ;n∈N,π∈Π} ; it is shown below thatNA is the set of integers of the realizability modelN_A.

We define the quantifier ∀x^intas follows :

k∀x^intF[x]k ={n

.

π;n∈N,π∈ kF[n]k}.

which means intuitively :

k∀x^intF[x]k = k∀n^ג^N({n}→F[n])k.

The formulas∀x^intF[x] and∀x(xεNA →F[x]) are interchangeable, as is shown in the : Lemma 13.

λxλnλy(y)(x)n||− ∀xîntF[x]→ ∀xînt¬¬F[x]; λxλn(cc)(x)n||− ∀xînt¬¬F[x]→ ∀xîntF[x]; k∀xînt¬¬F[x]k = k∀x(¬F[x]→xε/NA)k. Immediate.

Q.E.D. Lemma 14.

i) K_{||− ∀}x(xε/גN→xε/NA).

ii) λx(x)0||−0ε/N_A → ⊥; λfλx(f)(σ)x||− ∀y^ג^N((y+1)ε/N_A→yε/N_A).

iii) I||− ∀x^int¡

∀y^ג^N(F[y]→F[y+1]),F[0]→F[x]¢

for every formula F[x]of ZF_ε. i) and ii) Immediate.

iii) Letn∈N,φ||− ∀y^ג^N(F[y]→F[y+1]),α||−F[0] etπ∈ kF[n]k. We must show : n?φ

.

α

.

π∈ ⊥⊥i.e., by lemma 15, (φ)ⁿα?π∈ ⊥⊥.

But it is clear, by recurrence onn, that (φ)ⁿα||−F[n] for everyn∈N.

Q.E.D.

Lemma 14(i) shows thatN_A is a subset ofג^N.

But it is clear thatגNcontains 0 and is closed by the functionn7→n+1.

Now, by lemma 14(ii) and (iii),NA is the smallest subset ofגNwhich contains 0 and is closed by the functionn7→n+1. Therefore :

N_A is the set of integers of the modelN_A.

The following lemmas 15 and 16 will be used in section 3 (proof of lemma 39).

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Lemma 15.

.

π for everyα,δ,ν,φ∈Λandπ∈Π. For instance :Ω=(K)(K)I;Σ=(B)(BW)(B)B.

Then, for every n∈N,α,δ,ζ,φ∈Λandπ∈Π: i) (Σ)ⁿΩ ?δ

.

φ

.

α

.

πÂ(φ)ⁿα?π.

2 The characteristic Boolean algebra ג ²

2.1 Function symbols

Let us now define the principal function symbols commonly used in the sequel. As explained before, they are defined in the ground modelM and are immediately extended to the realizability modelNA.

• The projections pr₀:X×Y →X and pr₁:X×Y →Y defined by : pr₀(x,y)=x,pr₁(x,y)=y

give, inN_A, abijectionfrom ג^(X^×Y^{) onto} ג^X^×ג^Y^.

• We define, inM, the function app :Y^X×X →Y (readapplication) by setting : app(f,x)=f(x) forf ∈Y^X andx∈X.

This gives, inNA, an application app :ג^(Y^X⁾×ג^X →ג^Y^. We shall writef(x) for app(f,x).

Theorem 17.

If X 6= ;, then I _{||− ∀}f^ג^(Y^X⁾∀g^ג^(Y^X⁾¡

∀x^ג^X(app(f,x)=app(g,x))→ f =g¢

. In other words : I||− the functionappgives an injection fromג^(Y^X⁾^into ⁽ג^Y⁾^ג^X^.

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Letf,g ∈Y^X,ξ||− ∀x^ג^X(app(f,x)6=app(g,x)→ ⊥) andπ∈ kf 6=g → ⊥k. We must showξ?π∈ ⊥⊥. We choosea∈X ; thenξ||−(f(a)6=g(a)→ ⊥).

Iff =g, we havekf(a)6=g(a)→ ⊥k = kf 6=g→ ⊥k = k⊥ → ⊥k. Hence the result.

Iff 6=g, we could chooseasuch thatf(a)6=g(a).

Then,kf(a)6=g(a)→ ⊥k = kf 6=g → ⊥k = k> → ⊥k. Hence the result.

Q.E.D.

• Let sp :M →{0, 1} (readsupport) the unary function symbol defined by : sp(;)=0 ; sp(x)=1 ifx6= ;.

In the realizability modelN_A, we have sp :N →ג^2.

• Let P: {0, 1}×M →M (readprojection) the binary function symbol defined by : P(0,x)= ;;P(1,x)=x.

In the realizability modelN_A, we have P:ג²×N →N. In the following, we shall write i xinstead ofP(i,x).

Whent,uare`-terms with values inג2, we writet≤ufort∧u=t.

Proposition 18.

i) I _{||− ∀}i^ג²∀x(i(j x)=(i∧j)x).

ii) I ||− ∀i^ג²∀x(i x=x ^sp(x)≤i).

iii) If ; ∈E , then I||− ∀i^ג²∀x^ג^E(i xεג^E).

iv) If f :Mⁿ→M is a function symbol such that f(;, . . . ,;)= ;, then : I ||− ∀j^ג²∀x1. . .∀xn

¡j f(x1, . . . ,xn)=f(j x1, . . . ,j xn)¢ . v) I ||− ∀i^ג²∀x¡

i6=1→ ∀y(yε/i x)¢

and therefore K²I||− ∀i^ג²∀x¡

i6=1→ ∀y(y∉i x)¢ . Trivial.

Q.E.D.

Remark. Proposition 18(v) shows that, in the realizability modelN, every non empty individual has support 1.

Because of property (iv), we shall define, as far as possible, each function symbolf inM, so that to havef(;, . . . ,;)= ;.

• Thus, let us change the ordered pair (x,y) by setting (;,;)= ;. Then, we have : I _{||− ∀}i^ג²∀x∀y¡

i(x,y)=(i x,i y)¢ .

• We define the binary function symbol t:M²→M by setting : atb=a∪b.

Remark. The extension toN of this operation is neither the union for the∈-membership, nor the union for theε-membership.

The operationגi

LetE∈M be such that; ∈E. InM, we defineגⁱ^E^forⁱ^∈2 by setting : ג0E=ג{;}={;}×Π;ג1E=ג^E=E×Π.

In this way, we have now definedגiEinN , for everyiεג^2.

Proposition 19.

i) I ||− ∀i^ג²∀x∀y(i(xty)=i xti y).

ii) I _{||− ∀}i^ג²∀j^ג²∀x((i∨j)x=i xtj x).

iii) I||− ∀i^ג²∀j^ג²∀x∀y∀z(i∧j=0,z=i xtj y,→i z=i x).

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I _{||− ∀}i^ג²∀j^ג²∀x∀y∀z(i∧j=0,z=i xtj y,→j z=j y).

iv) I||− ∀i^ג²∀j^ג²∀x^ג^E∀y^ג^E∀z¡

i∧j=0,z=i xtj y,→zεגⁱ∨jE¢ . Trivial.

Q.E.D. Proposition 20.

If; ∈E,E⁰, the following formulas are realized : i) גiE increases with i . In particular,גiE⊆ג^{E .} ii) The ε-elements of גiE are the i x for xεג^{E .}

iii) The ε-elements of גiE are those of גE such that sp(x)≤i . iv) The only ε-element common toגiE andג1−iE is;.

v) If i∧j=0, then the application x7→(i x,j x)is a bijection fromגi∨jE ontoגiE×גjE . The inverse function is (x,y)7→xty.

vi) גi(E×E⁰)=גiE×גiE⁰.

We check immediately i), ii), iii), iv) below : i) I ||− ∀i^ג²∀j^ג²∀x(i∧j =i,→(xε/ג^j^E^→^x^ε^/גⁱ^E)).

ii) I||− ∀i^ג²∀x^ג^E(i xεגiE) ;I ||− ∀i^ג²∀x^ג^E(i x6=x→xε/גiE).

iii) I _{||− ∀}i^ג²∀x^גÊ(xε/גiE→i∧sp(x)6=sp(x)) ;I_{||− ∀}i^ג²∀x^גÊ(i∧sp(x)6=sp(x)→xε/גiE) ; iv) I ||− ∀i^ג²∀x^גÊ∀y^גÊ(i x=(1−i)y,→i x= ;).

v) By proposition 19(ii), we have i xtj x=(i∨j)x=x if xεגi∨jE.

By proposition 19(iii,iv), ifx,yεג^E, there exists zεגi∨jE such thati z=i x,j z=j y, namely z=i xtj y.

vi) By proposition 18(iv), we have I||− ∀i^ג²∀x∀y(i(x,y)=(i x,i y)).

Q.E.D.

Proposition 21. Let E,E⁰∈M be such that; ∈E,E⁰and E is equipotent with E⁰. Then :

||− ∀i^ג²(גiE is equipotent withגiE⁰).

Letφbe, inM, a bijection fromEontoE⁰, such thatφ(;)= ;. Thenφis, inN, a bijection fromגÊôntoגÊ⁰. But we have immediately :I ||− ∀i^ג²∀x^גÊ(φ(i x)=iφ(x)). This shows thatφ is a bijection fromגiEontoגiE⁰.

Q.E.D.

2.2 Some general theorems

Theorems 22 to 30, which are shown in this section, are validin every realizability model.

In the ground modelM, which satisfies ZF + V = L, we denote by κthe cardinal ofΛ∪Π∪N (which we shall also call thecardinal of the algebraA) and byκ₊=P(κ) the power set ofκ.

Theorem 22.

Let ∀~x∀y F[~x,y]be a closed formula of ZF_εwith parameters inM (where~x=(x₁, . . . ,x_n)).

Then, there exists inM, a functional fF :κ×Mⁿ→M such that :

i) If ~a,b∈M andξ||−F[~a,b], then there existsα∈κsuch that ξ||−F[~a,f_F(α,~a)].

ii) C I ||− ∀~x∀y¡

F[~x,y]→ ∃ν^ג^κF[~x,f_F(ν,~x)]¢ .

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i) Letξ7→ αξ be an injection from Λinto κ. Using the principle of choice in M (which satisfies V = L), we can define a functional fF :κ×Mⁿ→M such that, inM, we have :

∀~x∀y(∀ξ∈Λ)¡

ξ||−F[~x,y]⇒ξ||−F[~x,f_F(α_ξ,~x)]¢ .

ii) Letξ||−F[~a,b],η||− ∀ν^ג^κ¬F[~a,f_F(ν,~a)] andπ∈Π. Thus, we have η||− ¬F[~a,f_F(α_ξ,~a)] ; by definition off_F, we have ξ||−F[~a,f_F(αξ,~a)]. Therefore η?ξ

.

π∈ ⊥⊥, and C I_?_ξ

.

η

.

π∈ ⊥⊥.

Q.E.D.

Remark.The functionfFis a kind ofweak choice functionfor the relationF[~x,y] ; given~x, it does not give exactly a witnessysuch thatF[~x,y], but a family indexed byגκin which there is such a witness.

Subsets ofגκ₊

Theorem 23. Let ∀x∀y∀z F[x,y,z]be a closed formula of ZF_ε, with parameters inM. Then, there exists, inM,a functionalβF :M →κ₊such that :

W||− ∀z¡

∀x∀y∀y⁰(F[x,y,z],F[x,y⁰,z]→y=y⁰)→ ∀i^ג²∀x(F[x,iβF(z),z]→sp(x)≥i)¢ . By theorem 22(i), there exists, inM, a functional g:κ×M²→M such that :

(∗) Ifa,b,c∈M and ξ||−F[a,b,c], there existsα∈κsuch that ξ||−F[a,g(α,a,c),c].

Using the principle of choice inM, we define a functionalβF :M →κ+such that : for everyα∈κandc∈M, we haveβF(c)6=g(α,;,c).

This is possible sinceκ₊is of cardinal>κ.

Now let : a,c∈M, i∈{0, 1}, φ||− ∀x∀y∀y⁰(F[x,y,c],F[x,y⁰,c],y6=y⁰→ ⊥), ξ||−F[a,iβF(c),c],η||−sp(a)i6=i and π∈Π.

π∈ ⊥⊥. We set b=iβF(c) and therefore, we have ξ||−F[a,b,c].

Thus, by (∗), we have ξ||−F[a,g(α,a,c),c] for someα∈κ.

Let us show thatkb6=g(α,a,c)k ⊂ ksp(a)i6=ik; there are three possible cases : Ifi =0, thenksp(a)i6=ik = k06=0k =Π, hence the result.

Ifi =1 anda6= ;, thenksp(a)i6=ik = k16=1k =Π, hence the result.

Ifi =1 anda= ;, then :

kb6=g(α,a,c)k = kiβF(c)6=g(α,a,c)k = kβF(c)6=g(α,;,c)k = k>k = ;, by definition ofβF, hence the result.

It follows that η||−b6=g(α,a,c). Now, we have seen that : ξ||−F[a,b,c] and ξ||−F[a,g(α,a,c),c].

Therefore, by hypothesis onφ, we have φ?ξ

.

ξ

.

η

.

π∈ ⊥⊥.

Q.E.D.

Remark.Theorem 23 is crucial for the applications in the sequel. A perhaps more intuitive reformu- lation is given in corollary 24(i).

Corollary 24. If ; ∈E , then the following formulas are realized : i) ∀i^ג²(there is no surjection from S

{גjE; jεג^2,^j6≥i}ontoגiκ+).

ii) ∀i^ג²∀j^ג²(if there exists a surjection fromג^j^{E onto}גⁱ^κ+then j≥i).

iii) ∀i^ג²∀j^ג²¡

i,j6=0,i∧j=0→(there is no surjection from גiκ₊]גjκ₊onto גiκ₊×גjκ₊)¢ . Remarks.

]is the symbol fordisjoint union.