of classical realizability models of ZF

On the structure

of classical realizability models of ZF

Jean-Louis Krivine

I.R.I.F. University Paris-Diderot, CNRS krivine@irif.fr

May 23, 2015 updated April 20, 2018

Introduction

In [6, 7, 9], we have introduced the technique of classical realizability, which permits to extend the Curry-Howard correspondence between proofs and programs [5], to Zermelo- Fraenkel set theory. The models of ZF we obtain in this way are calledrealizability models; this technique is an extension of the method of forcing, in which the ordered sets (sets of conditions) are replaced with more complex first order structures calledrealizability alge- bras. These structures are refinements of the well knowncombinatory algebras[3], with the call/ccinstruction of [4].

We show here that every realizability modelN of ZF contains a transitive submodel, which has the same ordinals asN , and which is an elementary extension of the ground model. It follows that the constructible universe of a realizability model is an elementary extension of the constructible universe of the ground model (a trivial fact in the particular case of forcing, since these classes areidentical).

We obtain this result by showing the existence of an ultrafilter on thecharacteristic Boolean algebraג2 of the realizability model, which is defined in [7, 9].

From this result, it follows that theShoenfield absoluteness theoremapplies to realizability models and therefore that :

Every closedΣ¹₃formula which is true in the ground model is realized by a closedλc-term.

Another application is given in [8] : thebar-recursion operatorwas defined and studied in [1, 2, 11] where it is shown that it realizes theaxiom of dependent choice.

In [8] it is shown, by means of the results of the present paper, that every closed formula of analysis (i.e. Σ¹_n orΠ¹_n) which is true in the ground model, is realized by a closedλc-term containing this operator ; and that the same is true for the axiom :Ris well-ordered.

Acknowledgements. I want to thank the referees, especially the first one for numerous per- tinent and helpful remarks and suggestions.

Background and notations

We use here the basic notions and notations of the theory ofclassical realizability, which was developed in [6, 7, 9].¹

We consider a modelM of ZF + V = L, which we call theground model²and, inM, arealiz- ability algebraA =(Λ,Π,Λ ? Π, QP,⊥⊥).

Λis the set ofterms, Πis the set ofstacks,Λ ? Πis the set ofprocesses, QP⊂Λis the set of proof-like terms, and⊥⊥is a distinguished subset ofΛ ? Π.

They satisfy the axioms ofrealizability algebra, which are given in [6] or [9].

In the modelM, we use the language of ZF with the binary relation symbols∉,⊂and func- tion symbols, which we shall define when needed, by means of formulas of ZF.

We can now build (see [6]) therealizability modelN , which has the same set of individuals asM, the truth value set of which isP(Π), endowed with a suitable Boolean algebra struc- ture (not the usual one for the powerset).

The language of this model has three binary relation symbolsε/ ,∉,⊂, and the same function symbols as the modelM, with the same interpretation.

Theformulasare built as usual, from atomic formulas,with the only logical symbols⊥,→,∀.

εis called thestrong membership relation;∈is called theweakorextensional membership relation.

The formula∀z(xε/z→yε/z) is writtenx=y; it is thestrongorLeibniz equality.

The formulax⊂y∧y⊂xis writtenx'y; it is theweakorextensional equality.

Notations.We shall write :

¬FforF→ ⊥;F1, . . . ,Fn→FforF1→(. . .→(Fn→F) . . .) ;

∃x Ffor¬∀x¬F;∃x{F1, . . . ,Fn} for¬∀x(F1, . . . ,Fn→ ⊥).

We shall often use the notation~xfor a finite sequencex₁, . . . ,x_n; for instance, we shall writeF[~x] for F[x1, . . . ,xn].

By means of the completeness theorem, we obtain fromN an ordinary modelN⁰, with truth values in {0, 1}. The set of individuals ofN ⁰generallystrictly containsN .

The elements ofN ⁰are calledindividuals of N ⁰or evenindividuals of N . The individuals are generally denoted bya,b,c, . . . ,a₀,a₁, . . .

In [6] or [7], we define a theory ZF_ε, written in this language. The axioms forεare essentially the same as the axioms for∈in ZF (sometimes in an unusual form),without extensionality.

For instance, the infinity axiom is the following scheme :

∀~z∀a∃b©

aεb, (∀xεb)(∃y F[x,y,~z]→(∃yεb)F[x,y,~z])ª for every formulaF[x,y,z₁, . . . ,z_n].

The axioms for∈,⊂are a kind of coinductive definition fromε:

∀x∀y(x∈y↔(∃zεy)x'z) ;∀x∀y(x⊆y↔(∀zεx)z∈y).

We show that ZF_εis aconservative extensionof ZF, and that the modelN satisfies the axioms of ZF_ε, which means that each one of these axioms isrealized by a proof-like term.

1The papers [6, 7, 9, 8] are available at http://www.pps.univ-paris-diderot.fr^/∼krivine/

2In fact, it suffices thatM satisfy thechoice principle CP, which is written as follows, in the language of ZF with a new binary relation symbol/:“/ is a well ordering relation onM”.

It is well known that, in every countable model of ZFC, we can define such a binary symbol, so as to get a model of ZF + CP. Thus, ZF + CP is aconservative extensionof ZFC.

Given a termξ∈Λand a closed formulaF[a₁, . . . ,a_n] in the language of ZF_ε, with parameters a1, . . . ,aninN (or, which is the same, inM), we shall write :

ξ||−F[a₁, . . . ,a_n] in order to say that the termξrealizesF[a₁, . . . ,a_n].

The truth value of this formula is a subset ofΠ, denoted bykF[a₁, . . . ,a_n]k. We write ||−F in order to say thatF is realized by some proof-like term.

Thus, the modelN ⁰satisfies ZF_ε; therefore, inN⁰, we can define a model of ZF, denotedN_∈⁰, in which equality is interpreted by extensional equivalence.

The general properties of the realizability models are described in [9] ; we shall use the defi- nitions and notations of this paper.

In what follows, unless otherwise stated, each formula of ZF_εmust be interpreted inN (its truth value is a subset ofΠ) or, if one prefers, inN ⁰ (then its truth value is 0 or 1). If the formula must be interpreted inM, (in that case, it does not contains the symbol6ε) it will be explicitly stated.

Function symbols

Notations.The formula ∀z(zε/y→zε/x) is denoted by x⊆y(strong inclusion) ; the formula x⊆y∧y⊆x is denoted by x∼=y(strong extensional equivalence).

We recall that⊂and'are the symbols of inclusion and of extensional equivalence of ZF : x⊂y≡ ∀z(z∉y→zε/x) ; x'y≡(x⊂y∧y⊂x).

Function symbols associated with axioms of ZF

In this section, we define a function symbol for each of the following axioms of ZF_ε: comprehension, pairing, union, power set and collection.

Comprehension.

For each formulaF[y,~z] of ZF_ε, (where~zis a finite sequence of variablesz₁, . . . ,zn) we define, inM, a symbol of function of arityn+1, denoted provisionally by Compr_F(x,~z), (Compr is an abbreviation forComprehension) by setting :

Compr_F(a,~c)={(b,ξ

.

It was shown in [9] (and it is easily checked) that we have : kbε/ Compr_F(a,~c)k = kF[b,~c]→bε/ak. Thus, we have : I ||− ∀x∀y∀~z(yε/ Compr_F(x,~z)→(F[y,~z]→yε/x)) ; I ||− ∀x∀y∀~z((F[y,~z]→yε/x)→yε/ Compr_F(x,~z)).

Therefore, instead of Compr_F(x,~z), we shall use for this function symbol, the more intuitive notation {yεx; F[y,~z]}, in whichyis abound variable.

Pairing.

We define the following binary function symbol :

pair(x,y)={zε{x,y}×Π; (z=x)∨(z=y)}.

It is easily checked that we have the desired property :

||− ∀x∀y∀z(zεpair(x,y)↔z=x∨z=y).

Remark.We could also define a symbol pair(x,y), with this property, directly inM, as follows : pair(x,y)={(x, 1

.

.

In the sequel, when working inN , we shall use the (natural) abbreviations : {x,y} for pair(x,y) ; (x,y) for pair(pair(x,x), pair(x,y)).

Union and power set.

We define below two unary function symbolsS

xandP(x), such that :

||− ∀x∀z(zεS

x↔(∃yεx)zεy).

||− ∀x(∀yεP(x))(∀zεy)(zεx) ; ||− ∀x∀y(∃y⁰εP(x))∀z(zεy⁰↔zεx∧zεy).

Theorem 1. LetV,Qbe the unary function symbols defined inM as follows : V(a)=Cl(a)×Π and Q(a)=P(Cl(a)×Π)×Π

where Cl(a)is the transitive closure of a. Then, we have : i) I ||− ∀x∀y∀z(zεy,zε/V(x)→yε/x).

ii) I _{||− ∀}x∀~z¡

{yεx;F[y,~z]}εQ(x)¢

for every formula F[x,~z]of ZF_ε. i) Leta,b,c be individuals inM, ξ,η∈Λandπ∈Πsuch that : ξ||−cεb,η||−cε/V(a) andπ∈ kbε/ak; we have therefore (b,π)∈a.

We must showξ?η

.

We show thatkcε/bk ⊂ kcε/V(a)k: indeed, ifρ∈ kcε/bk, then we have (c,ρ)∈b. But we have (b,π)∈aand thusc∈Cl(a) and it follows thatkcε/V(a)k =Π.

Therefore,η||−cε/b; by hypothesis onξ, we haveξ?η

.

ii) Leta,~cbe individuals inM ; we must show I _||−AεQ(a), whereA={yεa;F[y,~c]}.

We have A={(b,ξ

.

Q.E.D.

We can now define the function symbolsS

andP by setting :

Sx={zεV(x) ; (∃yεx)zεy} ; P(x)={yεQ(x) ; y⊆x}.

Collection.

We shall use in the following, function symbols associated with a strong form of thecollection scheme.

In order to define these function symbols, it is convenient to decompose them, which is done in theorems 2, 3 and 4.

Theorem 2. For each formula F(x,~z)of ZF_ε, we have :

||− ∀~z¡

∃x F(x,~z)→(∃xεφF(~z))F(x,~z)¢

; ||− ∀~z(∀xεφF(~z))F(x,~z) whereφF is a function symbol defined inM.

We show λx(x)I||− ∀x(xεΦF(~z)→F(x,~z))→ ∀x F(x,~z) where the function symbolΦF is defined as follows :

By means of the collection scheme inM, we define a function symbolΨ(~z) such that : k∀x F(x,~z)k =S

x∈Ψ(~z)kF(x,~z)k and we set ΦF(~z)=Ψ(~z)×Π. Letξ||− ∀x(xεΦF(~z)→F(x,~z)) and π∈ k∀x F(x,~z)k.

Then π∈ kF(x,~z)kfor somex∈Ψ(~z), and therefore I ||−xεΦF(~z) andξ?I

.

Therefore, by replacingF with¬F, we have ||− ∃x F(x,~z)→(∃xεΦ_¬F(~z))F(x,~z).

Thus, we only need to set φF(~z)={xεΦ¬F(~z) ;F(x,~z)}.

Q.E.D.

Theorem 3. For every formula F(y,~z)of ZF_ε, we have :

||− ∀~z¡

∃x∀y(F(y,~z)→yεx)→ ∀y(F(y,~z)↔yεγF(~z))¢ whereγF is a function symbol defined inM.

By theorem 2, we have :

||− ∀~z¡

∃x∀y(F(y,~z)→yεx)→(∃xεφ(~z))∀y(F(y,~z)→yεx)¢ whereφis a function symbol. Therefore we have, by definition of Sφ(~z) :

||− ∀~z³

∃x∀y(F(y,~z)→yεx)→ ∀y(F(y,~z)→yεSφ(~z))´ . Now, we only need to set γF(~z)={yεSφ(~z) ;F(y,~z)} (comprehension scheme).

Q.E.D.

When the hypothesis ∃x∀y(F(y,~z)→yεx) is satisfied, we say thatthe formula F(y,~z)defines a set.

For the function symbolγF(~z), we shall use the more intuitive notation {y;F(y,~z)}, wherey is a bound variable.

Theorem 4.

Let f(x,~z)be a(n+1)-ary function symbol (defined inM). Then, we have :

||− ∀a∀y∀~z¡

yεφf(a,~z)↔(∃xεa)(y=f(x,~z))¢ whereφf is a(n+1)-ary function symbol.

We define, inM, the symbolφf as follows :

Leta₀,y₀,~z₀be fixed individuals inM ; we set φf(a₀,~z₀)={(f(x,~z₀),π) ; (x,π)∈a₀}.

Then, we have immediately ky₀ε/φf(a₀,~z₀)k = k∀x(y₀=f(x,~z₀),→xε/a₀)k. Therefore :

||− ∀x(y₀=f(x,~z₀),→xε/a₀)↔y₀ε/φf(a₀,~z₀) which gives the desired result.

Q.E.D.

Remark.Theconnective,→is defined in [7, 9]. It is equivalent to→but simpler to realize. Its hypoth- esis must be a strong equality.

For the function symbolφf(a,~z), we shall use the more intuitive notation {f(x,~z) ; xεa}, wherexis a bound variable. We call itimage of a by the function f(x).

Miscellaneous symbols

In the following, we shall use some function symbols, the definition and properties of which are given in [9]. We simply recall their definition below.

• The unary function symbolג, defined inM byג^x=x×Π.

For any individualEofM, therestricted quantifier∀x^גE is defined in [7] or [9] by : k∀x^גEF[x]k =S

x∈EkF[x]k and we have ||− ∀x^גEF[x]↔ ∀x(xεג^E→F[x]).

In the realizability modelN , the formulaxεג^E may be intuitively understood as“x is of type E ”. For instance,ג2 may be considered asthe type of booleansandגNasthe type of integers.

• the function symbols∧,∨,¬, with domains {0, 1}×{0, 1} and {0, 1}, and values in {0, 1}, are defined inM by means of the usual truth tables.

These functions define, inN , a structure of Boolean algebra onג^2.

We call it thecharacteristic Boolean algebraof the realizability modelN .

• a binary function symbol with domain {0, 1}×M, denoted by (α,x)7→αx, by setting : 0x= ;; 1x=x.

In the modelN , the domain of this function isג²×N .

• a binary function symbol twith domainM×M, by settingxty=x∪y.

Remark.The extension of this function to the modelN is notthe union∪, which explains the use of another symbol.

Lemma 5(Linearity).

Let f be a binary function symbol, defined inM. Then, we have : i) I _{||− ∀α}^ג2_∀x∀y(αf(x,y)=αf(αx,y)).

ii) Moreover, if f(;,;)= ;, then : I ||− ∀α^ג2∀α^0ג2∀x∀y∀x⁰∀y⁰¡

α∧α⁰=0,→f(αxtα⁰x⁰,αytα⁰y⁰)=αf(x,y)tα⁰f(x⁰,y⁰)¢ . It suffices to check :

for (i) the two casesα=0, 1 ;

for (ii) the three cases (α,α⁰)=(0, 0), (0, 1), (1, 0) ; which is is trivial.

Q.E.D.

Symbols for characteristic functions

LetR(x₁, . . . ,x_n) be ann-ary relation defined inM. Its characteristic function, with values in {0, 1}, will be denoted by〈R(x₁, . . . ,x_n)〉. Therefore, we have : M|= ∀~x(R(~x)↔ 〈R(~x)〉 =1).

Therefore, in the realizability modelN , the function symbol〈R(~x)〉takes its values inג^2.

The theorem 8 below shows that, if a binary relation y≺xis well founded inM, then the relation〈y≺x〉 =1 is well founded inN .

Well founded relations

In this section, we study properties of well founded relations inN . All the results obtained here are, of course, trivial in ZF. The difficulties come from the fact that the relationεof strong membership does not satisfy extensionality.

Given a binary relation≺, an individualais saidminimal for≺if we have ∀x¬(x≺a).

The binary relation≺is calledwell foundedif we have :

∀X¡

∀x(∀y(y≺x→yε/X)→xε/X)→ ∀x(xε/X)¢ .

The intuitive meaning is that each non empty individualX has anε-element minimal for≺.

Theorem 6 shows that this also true for non empty classes.

Theorem 6.

If the relation x≺y is well founded then, for every formula F[x,~z]of ZF_ε, we have :

∀~z¡

∀x(∀y(y≺x→F[y,~z])→F[x,~z])→ ∀x F[x,~z]¢ .

Proof by contradiction ; we consider, inN , an individualaand a formulaG[x] such that :

(1) G[a] ;∀x¡

G[x]→ ∃y{G[y],y≺x}¢ .

We apply the axiom scheme of infinity of ZF_ε:

(2) ∃b©

aεb, (∀xεb)¡

∃y H(x,y)→(∃yεb)H(x,y)¢ª

by settingH(x,y)≡G[x]∧G[y]∧y≺x. LetX ={xεb;G(x)} ; by (1) and (2), we getaεX. We obtain a contradiction with the hypothesis, by showing (∀xεX)(∃yεX)(y≺x) : supposexεbandG[x] ; by (2), we have :

∃y{G[x],G[y],y≺x}→(∃yεb){G[x],G[y],y≺x}.

ByG[x] and (1), we have∃y{G[x],G[y],y≺x}.

Therefore, we have (∃yεb){G[y],y≺x}, hence the result.

Q.E.D.

Therefore, in order to show∀x F[x], it suffices to show∀x¡

∀y(y≺x→F[y])→F[x]¢ .

Then, we say that we have shown∀x F[x]by induction on x, following the well founded rela- tion≺.

Theorem 7. The binary relation x∈y is well founded.

We must show ∀x(∀y(y∈x→yε/X)→xε/X)→ ∀x(xε/X).

We apply theorem 6 to the well founded relationxεyand the formulaF[x]≡x∉X. This gives : ∀x(∀y(yεx→y∉X)→x∉X)→ ∀x(x∉X).

Now, we have immediately ||−x∉X →xε/X. Thus, it remains to show :

||− ∀x(∀y(y∈x→yε/X)→xε/X)→ ∀x(∀y(yεx→y∉X)→x∉X).

But we havex∉X ≡ ∀x⁰(x⁰'x→x⁰ε/X). Therefore, we need to show :

||− ∀x(∀y(y∈x→yε/X)→xε/X),∀y(yεx→y∉X),x⁰'x→x⁰ε/X ; it is enough to show :

||− ∀y(yεx→y∉X),x⁰'x→ ∀y(y∈x⁰→yε/X).

Now, from x⁰'x,y∈x⁰, we deducey∈x. Thus, there is somey⁰'ysuch thaty⁰εx.

Then, from∀y(yεx→y∉X), we deducey⁰∉X, and therefore yε/X.

Q.E.D.

For instance, in the following, we shall use the fact that, if there is an ordinalρsuch thatF[ρ], then there exists a least such ordinal, for any formulaF[ρ] written in the language of ZF_ε. This follows from theorem 7.

Preservation of well-foundedness

Theorem 8. Let ≺be a well founded binary relation defined in the ground modelM. Then, the relation 〈y≺x〉 =1is well founded inN . In fact, we have :

Y||− ∀X¡

∀x(∀y(〈y≺x〉 =1,→yε/X)→xε/X)→ ∀x(xε/X)¢ where Y₌(λxλf(f)(x)x f)λxλf(f)(x)x f (Turing fixpoint combinator).

Letξ∈Λbe such that ξ||− ∀x(∀y(〈y≺x〉 =1,→ yε/X₀)→xε/X₀), X₀being any individual inM. We setF[x]≡(∀π∈ kxε/X0k)(Y?ξ

.

Since ≺is a well founded relation, it suffices to show ∀x¡

∀y(y≺x→F[y])→F[x]¢

, or equiv- alently¬F[x₀]→(∃y≺x₀)¬F[y], for any individualx₀.

By the hypothesis¬F[x0], there existsπ0∈ kx0ε/X0ksuch thatY?ξ

.

.

By hypothesis onξ, we deduce Y_{ξ6||− ∀}y(〈y≺x₀〉 =1,→yε/X₀).

Thus, there existsy0≺x0such thatYξ6||−y0ε/X0.

Therefore, we have (∃π∈ ky₀ε/X₀k)(Y?ξ

.

Q.E.D.

Definition of a rank function

Definition.Afunction with domain Dis an individualφsuch that : (∀zεφ)(∃xεD)∃y(z=(x,y)) ; (∀xεD)∃y((x,y)εφ) ;

∀x∀y∀y⁰((x,y)εφ, (x,y⁰)εφ→y=y⁰).

Letφbe a function with domainDandF[y,~z] a formula of ZF_ε. Then, the formula :

∃y{(x,y)εφ,F[y,~z]} is denoted by F[φ(x),~z].

Remark.Beware, despite the same notationφ(x), it is not a function symbol.

By means of theorem 3, we define the binary function symbol Im by setting : Im(φ,D)={y; (∃xεD) (x,y)εφ}.

Whenφis a function with domainD, we shall use, for Im(φ,D), the more intuitive notation {φ(x) ;xεD}, which we callimage of the functionφ.

LetD⁰⊆D, that is∀x(xε/D→xε/D⁰) ;a restriction ofφto D⁰is, by definition, a functionφ⁰ with domainD⁰such that φ⁰⊆φ.

For instance, {zεφ; (∃xεD⁰)∃y(z=(x,y))} is a restriction ofφtoD⁰. Ifφ⁰₀,φ⁰₁are both restrictions ofφtoD⁰, thenφ⁰₀∼=φ⁰₁.

Definition.

A binary relation ≺is calledranked, if we have∀x∃y∀z(z≺x→zεy), in other words : the minorants of any individual form a set.

By theorem 3, if the relation≺is ranked and defined by a formulaP[x,y,~u] of ZF_εwith pa- rameters~uinN , we have :

N |= ∀x∀y(x≺y↔xεf(y,~u)),for some symbol of function f , defined inM. In what follows, we suppose that≺is a rankedtransitivebinary relation.

A functionφwith domain {x; x≺a} will be calleda-inductive for ≺, if we have : φ(x)'{φ(y) ; y≺x} for everyx≺a. In other words :

(∀x≺a)(∀y≺x)φ(y)∈φ(x) ; (∀x≺a)(∀zεφ(x))(∃y≺x)z'φ(y).

Ifφisa-inductive for≺, we set O(φ,a)={φ(x) ; x≺a} (image ofφ).

Lemma 9. Letφ,φ⁰be two functions, a-inductive for≺. Then : i) φ(x)'φ⁰(x)for every x≺a.

ii) O(φ,a)'O(φ⁰,a).

iii) (∀x≺a)On(φ(x)); O(φ,a)is an ordinal, calledordinal ofφ.

i) Proof by induction onφ(x), following∈: ifuεφ(x), thenu'φ(y) withy≺x.

Sinceφ(y)∈φ(x), we have φ(y)'φ⁰(y) by the induction hypothesis ; thereforeφ(y)∈φ⁰(x) andφ(x)⊂φ⁰(x).

Conversely, ifuεφ⁰(x), thenu'φ⁰(y) with y≺x. Thus, we haveφ(y)∈φ(x), and therefore φ(y)'φ⁰(y) by the induction hypothesis ; thereforeu∈φ(x) andφ⁰(x)⊂φ(x).

ii) Immediate, by (i).

iii) We show On(φ(x)) by induction onφ(x), for the well founded relation∈:

Ifuεφ(x), we haveu'φ(y) withy≺x; therefore, we have On(u) by the induction hypothe- sis. Ifvεu, thenvεφ(y), thereforev'φ(z) withz≺y; thereforev∈φ(x).

It follows thatφ(x) is a transitive set of ordinals, thus an ordinal.

Then,O(φ,a) is also a transitive set of ordinals, and therefore an ordinal.

Q.E.D.

Lemma 10. Ifφis a-inductive for ≺, and if b≺a, then every restrictionψofφto the domain {x;x≺b}is a b-inductive function for ≺.

Indeed, we have,ψ(x)=φ(x)'{φ(y) ; y≺x}'{ψ(y) ; y≺x}.

Q.E.D.

By means of theorem 2, we define a unary function symbolΦ, such that :

∀x(∀f εΦ(x))(f is ax-inductive function) ;

∀x∀f³

f is ax-inductive function→ ∃f(f εΦ(x))´ .

In other words,Φ(x) is a set ofx-inductive functions, which is non void if there exists at least one such function.

Finally, we define the unary function symbol Rk, using theorem 4, by setting : Rk(x)=S

{O(f,x) ; f εΦ(x)}

(the symbolS

is defined after theorem 1).

Therefore, Rk(x) is the union of the ordinals of thex-inductive functions in the setΦ(x).

Since all these ordinals are extensionally equivalent, by lemma 9(ii), their union Rk(x) is also an equivalent ordinal.

Remarks.

If there exists nox-inductive function, then Rk(x) is void.

The function symbolsO,Φ, Rk have additional arguments, which are the parameters~uof the formula P[x,y,~u] which defines the relationy≺x.

We suppose now that≺is a ranked transitive relation, which iswell founded. It is therefore a strict ordering.

Lemma 11. Every restriction of Rkto the domain{x;x≺a}is an a-inductive function for ≺. Proof by induction ona, following ≺.

Let f be a restriction of Rk to the domain {x ; x≺a} and let x≺a. We must show that f(x)'{f(y) ; y≺x}, in other words, that we have :

Rk(x)'{Rk(y) ; y≺x}.

Letψbe any restriction of Rk to the domain {y; y≺x}. By the induction hypothesis,ψis a x-inductive function for≺.

We now show that Rk(x)'{Rk(y) ; y≺x} :

i) IfuεRk(x), thenuεO(φ,x) for some functionφwhich isx-inductive for≺,provided that there exists such a function. Now, there exists effectively one, otherwise Rk(x) would be void.

Therefore, by definition ofO(φ,x), we haveu=φ(y) withy≺x. But Rk(y)'φ(y), sinceφ,ψ are bothx-inductive functions for≺, andψ(y)=Rk(y) (lemma 9(i)).

Therefore, we haveu'Rk(y), withy≺x.

ii) Conversely, if y ≺x, then Rk(y)=ψ(y). Let φεΦ(x) ; thenφ,ψ arex-inductive for ≺; thereforeφ(y)'ψ(y) (lemma 9(i)).

Now φ(y)εO(φ,x), and thereforeφ(y)εRk(x) by definition of Rk(x).

It follows that Rk(y)=ψ(y)∈Rk(x).

Q.E.D.

Theorem 12. We have Rk(x)'{Rk(y) ; y≺x}for every x.

Proof by induction onx, following≺; letψbe any restriction of Rk to the domain {y; y≺x}.

By lemma 11,ψis ax-inductive function for≺.

Then, we finish the proof, by repeating paragraphs (i) and (ii) of the proof of lemma 11.

Q.E.D.

Rk is called the rank functionof the ranked, well founded and transitive relation ≺. Rk(x) is, for everyx, a representative of the ordinal of anyx-inductive function for ≺.

The values of the rank function Rk form an initial segment of On, which we shall callthe image of Rk. It is therefore,either an ordinal, or the whole of On.

Proposition 13. Let≺0,≺1be two ranked transitive well founded relations, and f a function such that ∀x∀y(x≺0y→f(x)≺1f(y)).

IfRk0, Rk1are their rank functions, then we have∀x¡

Rk0(x)≤Rk1(f(x))¢

, and the image of Rk₀is an initial segment of the image ofRk₁.

We show immediately∀x¡

Rk₀(x)≤Rk₁(f(x))¢

by induction following≺0. Hence the result, since the image of a rank function is an initial segment of On.

Q.E.D.

An ultrafilter on ג ²

In all of the following, we write y<xfory ∈Cl(x) inM, where Cl(x) denotes the transitive closure ofx. It is a strict well founded ordering (many other such orderings would do the job, for instance the relation rank(y)<rank(x)).

The binary function symbol〈y<x〉is therefore defined inN , with values inג^2.

By theorem 8, the binary relation〈y<x〉 =1 is well founded inN .

Theorem 14. ||−There exists an ultrafilterDonג2, which is defined as follows : D={αεג^{2 ;} the relation〈y<x〉 ≥αis well founded}.

Remark.By lemma 5, the formula〈y<x〉 ≥αmay be written〈αy<αx〉 =α. The formulaαεD, which we shall also writeD[α], is therefore :

D[α]≡ ∀X¡

∀x(∀y(〈y<x〉 ≥α,→yε/X)→xε/X)→ ∀x(xε/X)¢ Remark.We have :

D[1]≡ ∀X¡

∀x(∀y(〈y<x〉 =1,→yε/X)→xε/X)→ ∀x(xε/X)¢ . D[0]≡ ∀X((;ε/X→ ;ε/X)→ ;ε/X).

We have immediately : λx xI_{||− ¬D}[0] ;Y_||−D[1] ; I ||− ∀α^ג2∀β^ג2¡

α≤β,→(D[α]→D[β])¢

(more precisely : kD[1]k ⊂ kD[0]k).

Therefore, in order to prove theorem 14, it suffices to show :

||− ∀α^ג2∀β^ג2¡

α∧β=0,→(D[α∨β]→D[α]∨D[β])¢

; see theorem 15 ;

||− ∀α^ג2∀β^ג2¡

α∧β=0,→(D[α],D[β]→ ⊥)¢

; or even only :

||− ∀α^ג2(D[α],D[¬α]→ ⊥) ; see theorem 22.

Q.E.D.

Notation.Forαεג2, we shall write x<_αy for 〈x<y〉 ≥α.

Theorem 15.

i) ||− ∀α^ג2∀β^ג2¡

α∧β=0,→(D[α∨β]→D[α]∨D[β])¢ . ii) ||− ∀α^ג2∀β^ג2¡

D[α∨β]→D[α]∨D[β]¢ .

i) Letα,βεג2 be such thatα∧β=0,¬D[α],¬D[β]. We have to show ¬D[α∨β].

By hypothesis onαandβ, there exists individualsa0,A (resp. b0,B) such thata0εA (resp.

b₀εB) andA(resp.B) has no minimalε-element for<_α(resp. for<_β). We set : c₀=αa₀tβb₀ andC={αxtβy;xεA,yεB}.

Therefore, we havec₀εC ; it suffices to show thatC has no minimalε-element for<_α∨β. Let cεC,c=αatβb, with aεA,bεB. By hypothesis onA,B, there existsa⁰εA andb⁰εB such thata⁰<_αa,b⁰<_βb. If we setc⁰=αa⁰tβb⁰, we havec⁰εC, as needed. We also have :

〈c⁰=a⁰〉 ≥α,〈a⁰<a〉 ≥α,〈c=a〉 ≥α; it follows that〈c⁰<c〉 ≥α.

In the same way, we have〈c⁰<c〉 ≥βand therefore, finally,〈c⁰<c〉 ≥α∨β.

ii) We setβ⁰=β∧(¬α) ; we haveα∧β⁰=0 andα∨β⁰=α∨β. Therefore, we have : D[α∨β]→D[α]∨D[β⁰]. Now, we have β⁰≤βand thereforeD[β⁰]→D[β].

Q.E.D. Lemma 16.

i) I ||− ∀x∀y(〈x<y〉 6=1→xε/y).

ii) IfM |=u∈v, then I ||−uεג^v.

iii) I_{||− ∀}x∀y∀α^ג2¡

〈x<y〉 ≥α,→αxεג^Cl({y})¢. iv) ||− ∀x∀y¡

〈x<y〉 =1↔xεג^Cl(y)¢.

i) Leta,b be two individuals. Letξ||− 〈a<b〉 6=1,π∈ kaε/bk; then (a,π)∈b and therefore

〈a<b〉 =1 andξ||− ⊥; thusξ?π∈ ⊥⊥.

ii) Indeed, we havekuε/ג^vk ={π∈Π; (u,π)∈v×Π}=Π. iii) Letα∈{0, 1} anda,b∈M such that〈a<b〉 ≥α.

Ifα=0, we must show I_{||− ;ε}ג^Cl({y}) which follows from (ii).

Ifα=1, then〈a<b〉 =1, that isa∈Cl(b), thereforea∈Cl({b}).

From (ii), it follows that I ||−aεג^Cl({b}).

iv) Indeed, ifa,bare individuals ofM, we have trivially : k〈a<b〉 6=1k = kaε/ג^Cl(b)k.

Q.E.D.

Lemma 17. The well founded relation〈x<y〉 =1is ranked, and its rank function Rhas for image the whole of On.

Lemma 16(iv) shows that this relation is ranked.

Letρbe an ordinal andran individual'ρ. We show, by induction onρ, thatR(r)≥ρ.

Indeed, for everyρ⁰∈ρ, there existsr⁰εr such thatr⁰'ρ⁰. We have R(r⁰)≥ρ⁰by induction hypothesis, and〈r⁰<r〉 =1 from lemma 16(i). Therefore, we haveρ⁰∈R(r) by definition ofR, and finallyR(r)≥ρ. This shows that the image ofRis not bounded in On. Since it is an initial segment, it is the whole of On.

Q.E.D.

Theorem 18. Let F(x,y)be a formula of ZF_ε, with parameters. Then, we have : I ||− ∀x∀y¡

∀$^גΠF(x,f(x,$))→F(x,y)¢

for some function symbol f , defined dansM, with domainM×Π.

Since the ground modelM satisfies V = L (or only thechoice principle), we can define, inM, a function symbolf such that :

∀x∀y(∀$∈Π)¡

$∈ kF(x,y)k →$∈ kF(x,f(x,$))k¢ . Leta,bbe individuals,ξ||− ∀$^גΠF(a,f(a,$)) andπ∈ kF(a,b)k.

Thus, we have π∈ kF(a,f(a,π))k, and thereforeξ?π∈ ⊥⊥.

Q.E.D.

Definitions.Letabe any individual ofN andκan ordinal (therefore,κis not an individual ofN , but an equivalence class for').

Afunctionorapplicationfromκintoais, by definition, a binary relationR(ρ,x) such that :

∀x∀x⁰(∀ρ,ρ⁰∈κ)¡

R(ρ,x),R(ρ⁰,x⁰),ρ'ρ⁰→x=x⁰)¢

; (∀ρ∈κ)(∃xεa)R(ρ,x).

It is aninjectionif we have ∀x(∀ρ,ρ⁰∈κ)¡

R(ρ,x),R(ρ⁰,x)→ρ'ρ⁰¢ . Asurjectionfromaontoκis a functionf of domainasuch that : (∀ρ∈κ)(∃xεa)f(x)'ρ.

Theorem 19.

For any individual a, there exists an ordinalκ, such that there is no surjection from a ontoκ.

Let f be a surjection froma onto an ordinalρ. We define a strict ordering relation ≺f by setting x≺f y⇔xεa∧yεa∧f(x)<f(y). It is clear that this relation is well founded, that f is ana-inductive function, and thatO(f,a)'ρ.

We may consider this relation as a subset ofa×a.

By means of the axioms of union, power set and collection given above (theorems 1 to 4), we define an ordinalκ0, which is the union of theO(f,a) for all the functions f which are a-inductive for some well founded strict ordering relation ona.

In fact, we consider the set :

B(a)={XεP(a×a) ; X is a well founded strict ordering relation ona}.

Then, we setκ0=S

{O(f,a) ; XεB(a),f εΦ(X,a)}.

In this definition, we use the function symbolΦ, defined after lemma 10, which associates with each well founded strict ordering relation X ona, anon voidset ofa-inductive func- tions for this relation.

Then, there exists no surjection fromaontoκ0+1.

Q.E.D.

We denote by∆the first ordinal ofN such that there is no surjection fromגΠonto∆: for every functionφ, there existsδ∈∆such that∀x^גΠ(φ(x)6'δ).

For eachαεג2, we denote byNαthe class defined by the formula x=αx.

Lemma 20. Letα0,α1εג^{2, α}0∧α1=0and R₀ (resp. R₁) be a functional relation of domain N_α0 (resp.N_α1) with values inOn. Then, either R₀, or R₁, is not surjective onto∆.

Proof by contradiction : we suppose thatR₀andR₁are both surjective onto∆. We apply theorem 18 to the formulaF(x₀,x₁)≡ ¬(R0(α0x₀)'R₁(α1x₁)), and we get :

∀x₀¡

∃x₁(R₀(α0x₀)'R₁(α1x₁))→ ∃$^גΠ(R₀(α0x₀)'R₁(α1f(x₀,$)))¢ wheref is a suitable function symbol (thereforedefined inM).

Replacingx₀withα0x₀, we obtain :

∀x₀¡

∃x₁(R₀(α0x₀)'R₁(α1x₁))→ ∃$^גΠ(R₀(α0x₀)'R₁(α1f(α0x₀,$)))¢ .

But, by lemma 5(i), we have α1f(α0x,$)=α1f(α1α0x,$)=α1f(;,$). It follows that :

∀x0

¡∃x1(R0(α0x0)'R1(α1x1))→ ∃$^גΠ(R0(α0x0)'R1(α1f(;,$)))¢ . By hypothesis, we have (∀ρ∈∆)∃x₀∃x₁(ρ'R₀(α0x₀)'R₁(α1x₁)). It follows that : (∀ρ∈∆)∃x₀∃$^גΠ¡

ρ'R₀(α0x₀)'R₁(α1f(;,$))¢

; therefore, we have : (∀ρ∈∆)∃$^גΠ¡

ρ'R₁(α1f(;,$))¢ .

Therefore, the function$7→R₁(α1f(;,$)) is a surjection fromג^Πonto∆. But this is a con- tradiction with the definition de∆.

Remark.We should writef(α0,α1,x₀,$) instead off(x₀,$), since the function symbolf depends on the four variablesα0,α1,x0,$. In fact, it depends also on the parameters which appear inR0,R1. The proof does not change.

Q.E.D.

Corollary 21. Letα0,α1εג^{2,α0∧α1=0, and≺0,≺1}be two well founded ranked strict ordering relations with respective domainsNα0,Nα1. Let Rk₀,Rk₁be their rank functions. Then, either the image ofRk₀, or that ofRk₁is an ordinal <∆.

In order to be able to define the rank functions Rk₀, Rk₁, we consider the relations≺⁰₀,≺⁰₁, with domain the whole ofN , defined by x≺⁰_i y≡(x=αix)∧(y=αiy)∧(x≺i y) fori=0, 1.

These strict ordering relations are well founded and ranked.

Their rank functions Rk⁰₀, Rk⁰₁take the value 0 outsideNα0,Nα1respectively : indeed, all the individuals outsideN_αi are minimal for≺⁰_i.

By lemma 20, one of them, Rk⁰₀for instance, is not surjective onto∆.

Since the image of any rank function is an initial segment of On, the image of Rk0is an ordi- nal <∆.

Q.E.D. Theorem 22.

i) ||− ∀α^ג₀²∀α^ג₁²(α0∧α1=0,→(D[α0],D[α1]→ ⊥)).

ii) ||− ∀α^ג₀²∀α^ג₁²(D[α0],D[α1]→D[α0∧α1]).

i) InN , letα0,α1εג2 be such thatα0∧α1=0 and the relations〈x<y〉 ≥α0,〈x<y〉 ≥α1be well founded. Therefore, we haveα0,α16=0 (and thus,α0,α16=1).

Therefore, the relationsx≺i y≡(x=αix)∧(y=αiy)∧(〈x< y〉 =αi) fori =0, 1, are well founded strict orderings.

From lemma 16(iii), it follows that these relations areranked.

Now, by lemma 5, we have : ||− ∀x∀y∀α^ג2(〈x<y〉 =1→ 〈αx<αy〉 =α).

But, by lemma 17, the rank function of the well founded relation〈x<y〉 =1 has for image the whole of On. Therefore, by proposition 13, the same is true for the rank functions of the well founded strict order relationsx≺0yandx≺1y.

But this contradicts corollary 21.

ii) We haveα0≤(α0∧α1)∨(¬α1). Therefore, byD[α0] and theorem 15, we haveD[α0∧α1] or D[¬α1]. ButD[¬α1] is impossible, byD[α1] and (i).

Q.E.D.

Corollary 23. D[α]is equivalent with each one of the following propositions :

i) There exists a well founded ranked strict ordering relation ≺with domain Nα, the rank

function of which has an image ≥∆.

ii) There exists a function with domainN_αwhich is surjective onto∆.

D[α]⇒(i) :

By definition ofD[α], the binary relation (x=αx)∧(y=αy)∧(〈x<y〉 =α) is well founded.

By lemma 16(iii), this relation is ranked. We have seen, in the proof of theorem 22, that the image of its rank function is the whole of On.

(i)⇒(ii) : obvious.

(ii)⇒D[α] :

SinceDis an ultrafilter, it suffices to show¬D[¬α]. But, (ii) andD[¬α] contradict lemma 20.

Q.E.D. Theorem 24.

If ג²is non trivial, there exists no set, which is totally ordered byε, the ordinal of which is≥∆. Letαεג^{2,α6=0, 1 andX} be a set which is totally ordered by ε, and equipotent with∆.

Then, we show that the applicationx7→αxis an injection fromX intoN_α: Indeed, by lemma 16(i), we havexεy→ 〈x<y〉 =1 and, by lemma 5, we have :

〈x< y〉 =1→ 〈αx<αy〉 =α. Therefore, ifx,yεX and x6= y, we have, for instance xεy, therefore〈αx<αy〉 =αand thereforeαx6=αysinceα6=0.

Thus, there exists a function with domainN_αwhich is surjective onto∆. The same reason- ing, applied to¬αgives the same result for¬α. But this contradicts lemma 20.

Q.E.D.

Remark.Theorem 24 shows that it is impossible to define Von Neumann ordinals inN, withεinstead of∈, unlessג2 is trivial, i.e. the realizability model is, in fact, a forcing model.

The model M

For each formulaF[x₁, . . . ,x_n] of ZF, we have defined, in the ground model M, an n-ary function symbol with values in {0, 1}, denoted by〈F[x₁, . . . ,xn]〉, by setting, for any individu- alsa₁, . . . ,a_nofM : 〈F[a₁, . . . ,a_n]〉 =1 ⇔ M |=F[a₁, . . . ,a_n].

InN , the function symbol〈F[x₁, . . . ,x_n]〉takes its values in the Boolean algebraג^2.

We define, inN , two binary relations∈Dand=D, by setting :

(x∈Dy)≡D[〈x∈y〉] ; (x=Dy)≡D[〈x=y〉].

The classN, equipped with these relations, will be denotedM_D.

For each formulaF[~x,y] of ZF, withn+1 free variablesx₁, . . . ,xn,y, we can define, by means of thechoice principleinM, ann-ary function symbol f_F, such that :

M |= ∀~x¡

F[~x,f_F(~x)]→ ∀y F[~x,y]¢

; fF is called theSkolem functionof the formulaF[~x,y].

Lemma 25.

i) I ||− ∀~x∀y¡

〈∀y F[~x,y]〉 ≤ 〈F[~x,y]〉¢ ii) I ||− ∀~x∀y¡

〈∀y F[~x,y]〉 = 〈F[~x,fF(~x)]〉¢ .

Trivial.

Q.E.D.

For each formulaF[~x] of ZF, we define, by recurrence onF,a formula of ZF_ε, which has the same free variables, and that we denote M_D|=F[~x] (read :M_Dsatisfies F[~x]).

• F is atomic :

(MD|=x₁∈x₂) is x₁∈Dx₂; (MD|=x₁=x₂) is x₁=Dx₂; (MD|= ⊥) is ⊥.

• F ≡F₀→F₁: then (M_D|=F) is the formula (M_D|=F₀)→(M_D|=F₁).

• F[~x]≡ ∀y G[~x,y] : then (M_D|=F[~x]) is the formula ∀y(M_D|=G[~x,y]).

Lemma 26. For each formula F[~x]of ZF, we have ||− ∀~x³

(M_D|=F[~x])↔D〈F[~x]〉´ . Proof by recurrence on the length ofF. IfF is atomic, we have :

I ||− ∀~x³

(MD|=F[~x])→D〈F[~x]〉

´

and I ||− ∀~x³

D〈F[~x]〉 →(MD|=F[~x])´ because (M_D|=F[~x]) is identical withD〈F[~x]〉.

IfF ≡F₀→F₁, the formula (M_D|=F)↔D〈F〉is : ((MD|=F₀)→(MD|=F₁))↔D〈F₀→F₁〉.

SinceDis an ultrafilter, this formula is equivalent with :

((M_D|=F₀)→(M_D|=F₁))↔(D〈F₀〉 →D〈F₁〉), which is a logical consequence of : (MD|=F₀)↔D〈F₀〉 and (MD|=F₁)↔D〈F₁〉.

Hence the result, by the recurrence hypothesis.

IfF[~x]≡ ∀y G[~x,y], letf_G(~x) be the Skolem function ofG.

Then, we have (M_D|= ∀y G[~x,y])≡ ∀y(M_D|=G[~x,y]), and therefore : I ||−(M_D|= ∀y G[~x,y])→(M_D|=G[~x,f_G(~x)]).

Therefore, by the recurrence hypothesis, we have :

||−(M_D|= ∀y G[~x,y])→D〈G[~x,fG(~x)]〉.

Applying lemma 25(ii), we obtain ||−(M_D|= ∀y G[~x,y])→D〈∀y G[~x,y]〉. Conversely, by lemma 25(i), we have ||− ∀y¡

D〈∀y G[~x,y]〉 →D〈G[~x,y]〉¢ . Therefore, applying the recurrence hypothesis, we obtain :

||−D〈∀y G[~x,y]〉 → ∀y(M_D|=G[~x,y]), and thus, by definition of (M_D|= ∀y G[~x,y]) :

||−D〈∀y G[~x,y]〉 →(MD|= ∀y G[~x,y]).

We can manage the caseF[~x]≡ ∀y G[~x,y] without using Skolem functions, and therefore without assuming thatM |=AC. We have to show ||− ∀y(M_D|=G[~x,y])↔D〈∀y G[~x,y]〉.

By the recurrence hypothesis, we have ||− ∀y(M_D|=G[~x,y]↔D〈G[~x,y]〉).

Thus it suffices to show ||− ∀~x(∀yD〈G[~x,y]〉 ↔D〈∀y G[~x,y]〉). In fact, we show : k∀yD〈G[~a,y]〉k = kD〈∀y G[~a,y]〉kfor every~a∈M.

IfM |= ∀y G[~a,y], we have〈∀y G[~a,y]〉 =1 and〈G[~a,b]〉 =1 for everyb∈M. Thusk∀yD〈G[~a,y]〉k = kD〈∀y G[~a,y]〉k = kD(1)k.

IfM |= ¬∀y G[~a,y], we have〈∀y G[~a,y]〉 =0 and〈G[~a,b]〉 =0 for someb ∈M. Moreover

〈G[~a,c]〉 =0 or 1 for anyc∈M. Therefore, we have : i) kD〈∀y G[~a,y]〉k = kD(0)k;

ii) k∀yD〈G[~a,y]〉k =S

ckD〈G[~a,c]〉kand thuskD(0)k ⊂ k∀yD〈G[~a,y]〉k ⊂ kD(0)k ∪ kD(1)k.

It follows thatk∀yD〈G[~a,y]〉k = kD(0)kbecausekD(1)k ⊂ kD(0)k.

Q.E.D.

Theorem 27. MDis an elementary extension of the ground modelM.

Remark.Theorem 27 is, in fact, true foranyultrafilter onג2, with the same proof,whenM satisfies the principle of choice.

LetF[~a] be a closed formula of ZF, with parametersa₁, . . . ,a_ninM.

IfM |=F[~a], we have〈F[~a]〉 =1 (by definition), and therefore, of course, ||−D〈F[~a]〉. Therefore, by lemma 26, we have ||−(M_D|=F[~a]).

IfM 6|=F[~a], thenM |= ¬F[~a] ; therefore, we have ||−(M_D|= ¬F[~a]).

Q.E.D.

Theorem 28. Let @be a well founded binary relation, defined in the ground modelM. Then the relationD〈x@^y〉is well founded in the realizability modelN.

Remark.Theorem 28 is an improvement on theorem 8.

Notations.We shall write x@D y for 〈x@^y〉εD.

Recall thatx<ymeansx∈Cl(y) ; and that x<_αymeans〈x<y〉 ≥α, forαεג^2.

We define, in the model M, a binary relation@@on the class {0, 1}×M by setting, for any α,α⁰∈{0, 1} anda,a⁰inM :

(α⁰,a⁰)@@⁽α,a)⇔(α⁰<α)∨(α=α⁰=0∧a⁰<a)∨(α=α⁰=1∧a⁰@^a).

The relation@@^{is the}ordered direct sumof the relations @^,<. It is easily shown that it iswell founded inM.

The binary function symbol associated with this relation, of domain {0, 1}×M and values in {0, 1}, is given by :

〈(α⁰,a⁰)@@⁽α,a)〉 =(¬α0∧α)∨(¬α0∧¬α∧〈a⁰<a〉)∨(α0∧α∧〈a⁰@^a〉).

This definition gives, inN, a binary function symbol with arguments inג²×N , and values inג^2.

By theorem 8,the binary relation〈(α⁰,a⁰)@@(α,a)〉 =1is well founded inN . Proof of theorem 28.

Proof by contradiction : we assume that the binary relation @D is not well founded.

Thus, there existsa₀,A₀such thata₀εA₀andA₀has no minimalε-element for @D. We define, inN , the classX of ordered pairs (α,x), such that :

There exists X such that xεX and X has no minimalε-element, neither for@Dnor for<_¬α. Therefore, the formulaX(α,x) is :

αεג²∧ ∃Xn

xεX, (∀uεX){(∃vεX)(v@Du), (∃wεX)(w<¬αu)}o .

If (α,x) is inX, then we haveD(α) : indeed, the set X is non void and has no minimalε- element for<¬α. Therefore, we have¬D(¬α), and thusD(α), sinceDis an ultrafilter.

We obtain the desired contradiction by showing that the classX is non void and has no minimal element for the binary relation 〈(α⁰,x⁰)@@⁽α,x)〉 =1.

The ordered pair (1,a₀) is inX : indeed, we havex<0xfor everyx, and thereforeA₀has no minimalε-element for<0.

Now let (α,a) be inX ; we search for (α⁰,a⁰) inX such that 〈(α⁰,a⁰)@@⁽α,a)〉 =1.

By hypothesis on (α,a), there exists Asuch thataεAandAhas no minimalε-element, nei- ther for @D nor for <_¬α. Thus, there exists a⁰,a¹εA such that we have D〈a⁰@ ^{a〉 and}