Realizability algebras II : new models of ZF + DC

Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the cardinals of which are strictly decreasing + other similar properties of R. These results seem not to have been previously obtained by forcing.


Introduction
The technology of classical realizability was developed in [15,18] in order to extend the proof-program correspondence (also known as Curry-Howard correspondence) from pure intuitionistic logic to the whole of mathematical proofs, with excluded middle, axioms of ZF, dependent choice, existence of a well ordering on P(N), . . .We show here that this technology is also a new method in order to build models of ZF and to obtain relative consistency results.The main tools are : • The notion of realizability algebra [18], which comes from combinatory logic [2] and plays a role similar to a set of forcing conditions.The extension from intuitionistic to classical logic was made possible by Griffin's discovery [7] of the relation between the law of Peirce and the instruction call-with-current-continuation of the programming language SCHEME.In this paper, we only use the simplest case of realizability algebra, which I call standard realizability algebra ; somewhat like the binary tree in the case of forcing.
• The theory ZF ε [13] which is a conservative extension of ZF, with a notion of strong membership, denoted as ε.The theory ZF ε is essentially ZF without the extensionality axiom.We note an analogy with the Fraenkel-Mostowski models with "urelements" : we obtain a non well orderable set, which is a Boolean algebra denoted ‫,2ג‬ all elements of which (except 1) are empty.But we also notice two important differences : • The final model of ZF + ¬ AC is obtained directly, without taking a suitable submodel.
• There exists an injection from the "pathological set" ‫2ג‬ into R, and therefore R is also not well orderable.We show the consistency, relatively to the consistency of ZF, of the theory ZF + DC (dependent choice) with the following properties : there exists a sequence (X n ) n∈N of infinite subsets of R, the "cardinals" of which are strictly increasing (this means that there is an injection but no surjection from X n to X n+1 ), and such that X m ×X n is equipotent with X mn for m, n ≥ 2 ; there exists a sequence of infinite subsets of R, the "cardinals" of which are strictly decreasing.More detailed properties of R in this model are given in theorems 5.5 and 5.9.As far as I know, these consistency results are new, and it seems they cannot be obtained by forcing.But, in any case, the fact that the simplest non trivial realizability model (which I call the model of threads) has a real line with such unusual properties, is of interest in itself.Another aspect of these results, which is interesting from the point of view of computer science, is the following : in [18], we introduce read and write instructions in a global memory, in order to realize a weak form of the axiom of choice (well ordering of R).
Therefore, what we show here, is that these instructions are indispensable : without them, we can build a realizability model in which R is not well ordered.

Standard realizability algebras
The structure of realizability algebra, and the particular case of standard realizability algebra are defined in [18].They are variants of the usual notion of combinatory algebra.Here, we only need the standard realizability algebras, the definition of which we recall below : We have a countable set Π 0 which is the set of stack constants.We define recursively two sets : Λ (the set of terms) and Π (the set of stacks).Terms and stacks are finite sequences of elements of the set : Π 0 ∪ {B, C, E, I, K, W, cc, ς, k, (, ), [, ], .} which are obtained by the following rules : • B, C, E, I, K, W, cc, ς are terms (elementary combinators) ; • each element of Π 0 is a stack (empty stacks) ; • if ξ, η are terms, then (ξ)η is a term (this operation is called application) ; • if ξ is a term and π a stack, then ξ .π is a stack (this operation is called push) ; • if π is a stack, then k[π] is a term.A term of the form k[π] is called a continuation.From now on, it will be denoted as k π .A term which does not contain any continuation (i.e. in which the symbol k does not appear) is called proof-like.

Notation.
For sake of brevity, the term (. . .(((ξ)η 1 )η 2 ) . ..)η n will be also denoted as (ξ)η 1 η 2 . . .η n or ξη 1 η 2 . . .η n , if the meaning is clear.For example : ξηζ = (ξ)ηζ = (ξη)ζ = ((ξ)η)ζ.We now choose a recursive bijection from Λ onto N, which is written ξ −→ n ξ .We put σ = (BW )(B)B (the characteristic property of σ is given below).For each n ∈ N, we define n ∈ Λ recursively, by putting : 0 = KI ; n + 1 = (σ)n ; n is the n-th integer and σ is the successor in combinatory logic.We define a preorder relation ≻ on Λ ⋆ Π.It is the least reflexive and transitive relation such that, for all ξ, η, ζ ∈ Λ and π, ̟ ∈ Π, we have : For instance, with the definition of 0 and σ given above, we have : Finally, we have a subset ⊥ ⊥ of Λ ⋆ Π which is a final segment for this preorder, which means that : In other words, we ask that ⊥ ⊥ has the following properties : Remark.Thus, the only arbitrary elements in a standard realizability algebra are the set Π 0 of stack constants and the set ⊥ ⊥ of processes.
c-terms and λ-terms.
We call c-term a term which is built with variables, the elementary combinators B, C, E, I, K, W , cc, ς and the application (binary function In [18], it is shown that this definition is correct.This allows us to translate every λ-term into a c-term.In the following, almost every c-term will be written as a λ-term.The fundamental property of this translation is given by theorem 1.1, which is proved in [18] : Theorem 1.1.Let t be a c-term with the only variables x 1 , . . ., Remark.The property we need for the term σ (the successor (to prove theorem 4.12).Therefore, by theorem 1.1, we could define σ = λnλf λx(nf )(f )x.The definition we chose is much simpler.

The formal system
We write formulas and proofs in the language of first order logic.This formal language consists of : • individual variables x, y, . . .; • function symbols f, g, . . .; each one has an arity, which is an integer ; function symbols of arity 0 are called constant symbols.
• relation symbols ; each one has an arity ; relation symbols of arity 0 are called propositional constants.We have two particular propositional constants ⊤, ⊥ and three particular binary relation symbols ε / , / ∈, ⊆.The terms are built in the usual way with individual variables and function symbols.
Remark.We use the word "term" with two different meanings : here as a term in a first order language, and previously as an element of the set Λ of a realizability algebra.I think that, with the help of the context, no confusion is possible.
The atomic formulas are the expressions R(t 1 , . . ., t n ), where R is a n-ary relation symbol, and t 1 , . . ., t n are terms.Formulas are built as usual, from atomic formulas, with the only logical symbols →, ∀ : • each atomic formula is a formula ; The usual logical symbols are defined as follows : More generally, we shall write ∃x{F 1 , . . ., F k } for ∀x(F 1 , . . ., F k → ⊥) → ⊥.We shall sometimes write F for a finite sequence of formulas F 1 , . . ., F k ; Then, we shall also write F → G for F 1 , . . ., F k → G and ∃x{ F } for ∀x( F → ⊥) → ⊥.A ↔ B is the pair of formulas {A → B, B → A}.The rules of natural deduction are the following (the A i 's are formulas, the x i 's are variables of c-term, t, u are c-terms, written as λ-terms) : . ., x n : A n ⊢ t : ∀x A where x is an individual variable which does not appear in A 1 , . . ., A n . 5. x 1 : A 1 , . . ., x n : A n ⊢ t : ∀x A ⇒ x 1 : A 1 , . . ., x n : A n ⊢ t : A[τ /x] where x is an individual variable and τ is a term.6.

The theory ZF ε
We write below a set of axioms for a theory called ZF ε .Then : • We show that ZF ε is a conservative extension of ZF.
• We define the realizability models and we show that each axiom of ZF ε is realized by a proof-like c-term, in every realizability model.It follows that the axioms of ZF are also realized by proof-like c-terms in every realizability model.We write the axioms of ZF ε with the three binary relation symbols ε / , / ∈, ⊆.Of course, x ε y and x ∈ y are the formulas x ε / y → ⊥ and x / ∈ y → ⊥.The notation x ≃ y → F means x ⊆ y, y ⊆ x → F .Thus x ≃ y, which represents the usual (extensional) equality of sets, is the pair of formulas {x ⊆ y, y ⊆ x}.We use the notations (∀x ε a)F (x) for ∀x(¬F (x) → x ε / a) and (∃x ε a) F (x) for ¬∀x( F (x) → x ε / a).For instance, (∃x ε y) t ≃ u is the formula ¬∀x(t ⊆ u, u ⊆ t → x ε / y).
Remark.The usual statement of the axiom of infinity is the particular case of this scheme, where a is ∅, and F (x, y) is the formula y ≃ x ∪ {x}.
Let us show that ZF ε is a conservative extension of ZF.First, it is clear that, if ZF ε ⊢ F , where F is a formula of ZF (i.e.written only with / ∈ and ⊆), then ZF ⊢ F ; indeed, it is sufficient to replace ε / with / ∈ in any proof of ZF ε ⊢ F .Conversely, we must show that each axiom of ZF is a consequence of ZF ε .
Proof.i) Using the foundation axiom, we assume ∀x(x ε a → x ⊆ x), and we must show a ⊆ a ; therefore, we add the hypothesis x ε a.It follows that x ⊆ x, then x ≃ x, and therefore : ∃y{x ≃ y, y ε a}, that is to say x ∈ a.Thus, we have ∀x(x ε a → x ∈ a), and therefore a ⊆ a. ii) Just shown.
Proof.Call F (a), F ′ (a) these two formulas.We show F (a) by foundation : thus, we suppose (∀x ε a)F (x) and we first show F ′ (a) : by hypothesis, we have a ⊆ y, z ∈ a ; thus, there exists a ′ such that z ≃ a ′ and a ′ ε a, and thus F (a ′ ).From a ′ ε a and a ⊆ y, we deduce a ′ ∈ y.From z ≃ a ′ and a ′ ∈ y, we deduce z ∈ y by F (a ′ ).Then, we show F (a) : by hypothesis, we have y ≃ a, a ∈ z, thus a ≃ y ′ and y ′ ε z for some y ′ .In order to show y ∈ z, it is sufficient to show y ≃ y ′ .Now, we have y ≃ a, a ≃ y ′ , and thus y ′ ⊆ a, a ⊆ y.From F ′ (a), we get ∀z(z ∈ a → z ∈ y) ; from y ′ ⊆ a, we deduce y ′ ⊆ y by lemma 3.3.We have also y ⊆ a, a ⊆ y ′ .From F ′ (a), we get ∀z(z ∈ a → z ∈ y ′ ) ; from y ⊆ a, we deduce y ⊆ y ′ by lemma 3.3.
With corollary 3.2, we obtain : It is now easy to deduce the equality and extensionality axioms of ZF : Remark.This shows that ≃ is an equivalence relation which is compatible with the relations ∈ and ⊆ ; but, in general, it is not compatible with ε.It is the equality relation for ZF ; it will be called extensional equivalence.
Notation.The formula ∀z(z ε / y → z ε / x) will be written x ⊂ y.The ordered pair of formulas x ⊂ y, y ⊂ x will be written x ∼ y.By theorem 3.1, we get ZF ε ⊢ ∀x∀y(x ⊂ y → x ⊆ y).Thus ⊂ will be called strong inclusion, and ∼ will be called strong extensional equivalence.
From x ∈ b, a ≃ b, we deduce x ∈ a and therefore (by axiom 0), x ′ ε a for some x ′ ≃ x.Finally, we get x ′ ε b for some x ′ .Thus x ′ ε a and F [x ′ ].From x ≃ x ′ and x ′ ε a, we deduce x ∈ a.Since ⊆ and ∈ are compatible with ≃, it is the same for F ; thus, we obtain F [x].
Conversely, if we have F [x] and x ∈ a, we have x ≃ x ′ and x ′ ε a for some x ′ .Since F is compatible with ≃, we get F [x ′ ], thus x ′ ε b and x ∈ b.
• Union axiom : ∀a∃b∀x∀y(x ∈ a, y ∈ x → y ∈ b).From x ∈ a we have x ≃ x ′ and x ′ ε a for some x ′ ; we have y ∈ x, therefore y ∈ x ′ , thus y ≃ y ′ and y ′ ε x ′ for some y ′ .From axiom 4 of ZF ε , x ′ ε a and y ′ ε x ′ , we get y ′ ε b ; therefore y ∈ b, by y ≃ y ′ .
• Power set axiom : ∀a∃b∀x∃y{y ∈ b, ∀z(z ∈ y ↔ (z ∈ a ∧ z ∈ x))} Given a, we obtain b by axiom 5 of ZF ε ; given x, we define x ′ by the condition : for every formula F [x, y, x 1 , . . ., x n ] written with the only relation symbols / ∈, ⊆.From x ∈ a and ∃y F [x, y], we get x ≃ x ′ , x ′ ε a for some x ′ , and thus ∃y F [x ′ , y] since F is compatible with ≃.From axiom scheme 6 of ZF ε , we get (∃y ε b)F [x ′ , y], and therefore (∃y for every formula F [x, y, x 1 , . . ., x n ] written with the only relation symbols / ∈, ⊆.Same proof.

Realizability models of ZF ε
As usual in relative consistency proofs, we start with a model M of ZFC, called the ground model or the standard model.In particular, the integers of M are called the standard integers.
The elements of M will be called individuals.In the sequel, the model M will be our universe, which means that every notion we consider is defined in M. In particular, the realizability algebra (Λ, Π, ⊥ ⊥) is an individual of M. We define a realizability model N , with the same set of individuals as M.But N is not a model in the usual sense, because its truth values are subsets of Π instead of being 0 or 1.Therefore, although M and N have the same domain (the quantifier ∀x describes the same domain for both), the model N may (and will, in all non trivial cases) have much more individuals than M, because it has individuals which are not named.In particular, it will have non standard integers.
Remark.This is a great difference between realizability and forcing models of ZF.In a forcing model, each individual is named in the ground model ; it follows that integers, and even ordinals, are not changed.
For each closed formula F with parameters in M, we define two truth values : The following theorem is an essential tool : Theorem 4.1 (Adequacy lemma).Let A 1 , . . ., A n , A be closed formulas of ZF ε , and suppose that We need to prove a (seemingly) more general result, that we state as a lemma : be formulas of ZF ε , with z = (z 1 , . . ., z k ) as free variables, and suppose that Proof.By recurrence on the length of the derivation of We consider the last used rule.1.
. This case is trivial.2. We have the hypotheses : which is the desired result.3. We have the hypotheses : Now, by the induction hypothesis, we have u[ ξ/ x, η/y] || − C[ a/ z], which gives the result.4. We have the hypotheses : . Thus, we take an arbitrary set b in M and we show t But this follows from the hypothesis on ξ i , because z 1 is not free in the formulas A i . 5. We have the hypotheses :  Realized formulas and coherent models.In the ground model M, we interpret the formulas of the language of ZF : this language consists of / ∈, ⊆ ; we add some function symbols, but these functions are always defined, in M, by some formulas written with / ∈, ⊆.We suppose that this ground model satisfies ZFC.The value, in M, of a closed formula F of the language of ZF, with parameters in M, is of course 1 or 0. In the first case, we say that M satisfies F , and we write M |= F .In the realizability model N , we interpret the formulas of the language of ZF ε , which consists of ε / , / ∈, ⊆ and the same function symbols as in the language of ZF.The domain of N and the interpretation of the function symbols are the same as for the model M. The value, in N , of a closed formula F of ZF ε with parameters (in M or in N , which is the same thing) is an element of P(Π) which is denoted as F , the definition of which has been given above.Thus, we can no longer say that N satisfies (or not) a given closed formula F .But we shall say that N realizes F (and we shall write N || − F ), if there exists a proof-like term θ such that θ || − F .We say that two closed formulas , but the converse is far from being true.The model N allows us to make relative consistency proofs, since it is clear, from the adequacy lemma (theorem 4.1), that the class of formulas which are realized in N is closed by deduction in classical logic.Nevertheless, we must check that the realizability model N is coherent, i.e. that it does not realize the formula ⊥.We can express this condition in the following form : For every proof-like term θ, there exists a stack π ∈ Π such that θ ⋆ π / ∈ ⊥ ⊥.When the model N is coherent, it is not complete, except in trivial cases.This means that there exist closed formulas F of ZF ε such that N || − F and N || − ¬F .In the same way, we have : Therefore, we have a / ∈ b = ∀z(a ⊆ z, z ⊆ a → z ε / b) , so that : Notation.We shall write ξ for a finite sequence (ξ 1 , . . ., ξ n ) of terms.Therefore, we shall write In particular, the notation • Foundation scheme.
Theorem 4.5.For every finite sequence F [x, x 1 , . . ., x n ] of formulas, we have : We show, by induction on the rank of a, that : • Power set axiom.Given a set a, let b = P(Cl(a)×Π)×Π.For every set x, we put : / a }, and therefore z ε / y = z ε x → z ε / a .Thus : ).Now, it is obvious that y ∈ P(Cl(a)×Π), and therefore (y, π) ∈ b for every π ∈ Π.Thus, we have y ε / b = Π = ⊥ .It follows that : • Collection scheme.Given a set a, and a formula F Function symbols and equality.
According to our needs, we shall add to the language of ZF ε , some function symbols f, g, . . . of any arity.A k-ary function symbol f will be interpreted, in the realizability model N , by a functional relation, which is defined in the ground model M by a formula F [x 1 , . . ., x k , y] of ZF.Thus, we assume that M |= ∀x 1 . . .
The axiom schemes of ZF ε , written in the extended language, are still realized in the model N , because the above proofs remain valid.
On the other hand, in order to make sure that the axiom schemes of ZF, which use a k-ary function symbol f , are still realized, one must check that this symbol is compatible with ≃, i.e. that the following formula is realized in N : We now add a new rule to build formulas of ZF ε : If t, u are two terms and F is a formula of ZF ε , then The truth value of these new formulas is defined as follows, assuming that t, u, F are closed, with parameters in N : But we have η || − ⊥ → ⊥, and therefore η ⋆ k π ξ .π ∈ ⊥ ⊥.Proposition 4.7 shows that the formulas t = u and ∀x(u ε / x → t ε / x) (Leibniz equality) are interchangeable.
We now show that the axioms of equality are realized.Proof.Trivial, by definition of ֒→.
Conservation of well-foundedness.Theorem 4.9 says that every well founded relation in the ground model M, gives a well founded relation in the realizability model N .
Theorem 4.9.Let f be a binary function such that f (x, y) = 1 is a well founded relation in the ground model M.Then, for every formula Sets in M give type-like sets in N .
We define a unary function symbol ‫ג‬ by putting ‫(ג‬a) = a × Π for every individual a (element of the ground model M).
For each set E of the ground model M, we also introduce the unary function 1 E with values in {0, 1}, defined as follows : ) are also interchangeable.We have : As already said, we shall add to the language of ZF ε , some function symbols of any arity, which will be interpreted in the ground model M by some functional relations.Then every formula of the form ∀ x(t ) which is satisfied in the model M, is realized in the model N (t 1 , u 1 , . . ., t k , u k , t, u are terms of the language).Indeed, we verify immediately that : ) and therefore, we have : ), in other words : Notice, in particular, that the characteristic function 1 E , which takes its values in the set 2 = {0, 1} in the model M, sends ‫ג‬E into ‫2ג‬ in the realizability model N .We shall denote ∧, ∨, ¬ the (trivial) Boolean algebra operations in {0, 1} (they should not be confused with the logical connectives ∧, ∨, ¬).In this way, we have defined three function symbols of the language of ZF ε ; thus, in the realizability model N , they define a Boolean algebra structure on the set ‫.2ג‬ Remarks.i) A set of the form ‫ג‬E behaves somewhat like a type, in the sense of computer science, because any function of the model M with domain (resp.range) ii) The Boolean algebra ‫2ג‬ is, in general, non trivial i.e. it has ε-elements = 0, 1.Notice that they are all empty : indeed, it is easy to check that I || − ∀x ‫2ג‬ ∀y(x = 1 → y ε / x).
The set N of integers in N .
We add to the language of ZF ε a constant symbol 0 and a unary function symbol s.Their interpretation in the model M is as follows : 0 is ∅ ; s(a) is {a}×Π for every set a, in other words s(a) = ‫{(ג‬a}).
In the realizability model N , s(a) is the singleton of a. Indeed, we have trivially :  This shows, in particular, that the function s is compatible with the extensional equivalence ≃.
written with / ∈, ⊆, 0, s).Then, indeed, the formula F is compatible with the extensional equivalence ≃.Since the function s is compatible with ≃, we deduce from lemma 4.14 that the formula : ∀y(y ∈ N → sy ∈ N) is realized in N ; the formula 0 ∈ N is also obviously realized.From the recurrence scheme just proved, we deduce that : N is the set of integers of the model N , considered as a model of ZF .
Theorem 4.16.i) Let f : N k → N be a recursive function.Then, the formula : The proof is done in [18,15].ii) We have N || − (∀x 1 ε ‫ג‬N) . . .(∀x k ε ‫ג‬N) g(x 1 , . . ., x k ) ε ‫.2ג‬ Now, since g is recursive, we have, by (i) : Hence the result, by lemma 4.17.Remarks.i) In the present paper, theorem 4.16 is used only in trivial particular cases.ii) Let us recall the difference between ‫ג‬N and N (the set of integers in the model N ) ; we have : Notice that we have K || − ∀x(x ε / ‫ג‬N → x ε / N), in other words K || − N ⊂ ‫ג‬N.This means that, in N , the set N of integers is strongly included in ‫ג‬N.In the particular realizability model considered below (and, in fact, in every non trivial realizability model), the formula ‫ג‬N ⊆ N is realized.

Non extensional and dependent choice.
For each formula F (x, y 1 , . . ., y m ) of ZF ε , we add a function symbol f F of arity m + 1, with the axiom : It is the axiom scheme of non extensional choice, in abbreviated form NEAC.
Remarks.i) The axiom scheme NEAC does not imply the axiom of choice in ZF, because we do not suppose that the symbol f F is compatible with the extensional equivalence ≃.It is the reason why we speak about non extensional axiom of choice.On the other hand, as we show below, it implies DC (the axiom of dependent choice).
ii) It seems that we could take for f F a m-ary function symbol and use the following simpler (and logically equivalent) axiom scheme NEAC' : ∀ y(F [f F ( y), y] → ∀x F [x, y]).But this axiom scheme cannot be realized, even though the axiom scheme NEAC is realized by a very simple proof-like term (theorem 4.18), provided the instruction ς is present.More precisely, we can define a function f F in M, such that NEAC is realized in N , but this is impossible for NEAC'.
and therefore, by hypothesis on ξ, we have : ξ ⋆ j .π ′ ∈ ⊥ ⊥.This is in contradiction with π ′ ∈ P j .NEAC implies DC.Let us call DCS (dependent choice scheme) the following axiom scheme : where F is a formula of ZF ε with free variables x, y, z ; the formula S F is written below.In the following, we omit the variables z (the parameters), for sake of simplicity.The usual axiom of dependent choice DC is obtained by taking for F [x, y, z 0 , z 1 ] the formula y ε z 0 ∧ (x ε z 0 → <x, y> ε z 1 ).We now show how to define the formula S F , so that ZF ε , NEAC ⊢ DCS ; we shall conclude that DC is realized.So, let us assume ∀x∃y F [x, y].By NEAC, there is a function symbol f such that : This means : "y = f (k, x) for the first integer k such that F [x, f (k, x)] ".Therefore, R F is functional, i.e. we have ∀x∃!y R F (x, y).S F is defined so as to represent a sequence obtained by iteration of the function given by R F , beginning (arbitrarily) at 0 : It should be clear that, with this definition of S F , we obtain : ∀n ent ∃!y S F [n, y] and ∀n ent ∃y∃y ′ {S F [n, y], S F [sn, y ′ ], F [y, y ′ ]}.Thus, DCS is provable from ZF ε and NEAC.
For each integer n ≥ 2, the set n = {0, 1, . . ., n−1} is a ring : the ring of integers modulo n ; the Boolean algebra {0, 1} is a set of idempotents in this ring.These ring operations extend to the realizability model, giving a ring structure on ‫ג‬n, and ‫2ג‬ is a set of idempotents in ‫ג‬n.For each a ε ‫,2ג‬ the equation ax = x defines an ideal in ‫ג‬n, which we denote as a‫ג‬n.The application x −→ ax is a retraction from ‫ג‬n onto a‫ג‬n.
Remark.This formula means that, if n > 2m, a ε ‫,2ג‬ a = 0, then there is no surjection from ‫ג‬m onto a‫ג‬n : it suffices to take F (x, y, z) ≡ <x, y> ε z. i) ∀n N ∀a ‫2ג‬ (a = 0 → there is no surjection from ‫ג‬n onto a‫(ג‬n + 1)).
ii) ∀n N ∀a ‫2ג‬ ∀b ‫2ג‬ (a∧b = 0, b = 0 → there is no surjection from a‫ג‬n onto b‫.)2ג‬ iii) ∀n N ∀a ‫2ג‬ ∀b ‫2ג‬ (a∧b = a, a = b → there is no surjection from a‫ג‬n onto b‫.)2ג‬ Proof.i) Suppose that there is a surjection from ‫ג‬n onto a‫(ג‬n + 1).Then, by the recurrence scheme (theorem 4.12(ii)), we see that, for each integer k ∈ N, there exists a surjection from (‫ג‬n) k onto (a‫(ג‬n + 1)) k ; and, by proposition 5.6(ii) and the recurrence scheme, it follows that there is a surjection from ‫(ג‬n k ) onto a‫((ג‬n + 1) k ).But, for k > n, we have (n + 1) k > 2n k and this contradicts theorem 5.7.ii) Since a∧b = 0, the rings (a + b)‫ג‬n and a‫ג‬n × b‫ג‬n are isomorphic.Reasoning by contradiction, there would exist a surjection from (a + b)‫ג‬n onto b‫×2ג‬b‫ג‬n, thus also onto b ‫2(ג‬n) (proposition 5.6(ii)), thus a surjection from ‫ג‬n onto b‫2(ג‬n), which contradicts (i).iii) Otherwise, there would exist a surjection from a‫ג‬n onto (b− a)‫,2ג‬ which contradicts (ii).
Applications.i) By DC, since ‫2ג‬ is atomless, there exists in ‫2ג‬ a strictly decreasing sequence.Hence, by

F
is now defined by recurrence on the length of F : • F is atomic ; then F has one of the forms ⊤, ⊥, a ε / b, a ⊆ b, a / ∈ b where a, b are parameters in M. We set : ⊤ = ∅ ; ⊥ = Π ; a ε / b = {π ∈ Π; (a, π) ∈ b}. a ⊆ b , a / ∈ b are defined simultaneously by induction on (rk(a)∪ rk(b),rk(a)∩ rk(b)) (rk(a) being the rank of a).
By the induction hypothesis, we have t[ ξ/ x] || − ∀yB[y, a/ z] ; therefore t[ ξ/ x] || − B[b/y, a/ z] for every parameter b.We get the desired result by taking b = τ [ a]. 6.The result follows from the following : Theorem 4.3.For every formulas A, B, we have cc || −
b ε / s(a) = b = a (i.e.∅ if a = b and Π if a = b) and it follows that : I || − ∀x∀y(y ε / sx → x = y) ; I || − ∀x∀y(x = y → y ε / sx).For each n ∈ N, the term s n 0 will also be written n.Remark.In the definition of the set of integers in the realizability model N , we prefer to use the singleton as the successor function s, instead of the usual one x −→ x ∪ {x}, which is more complicated to define.It would give : s(a) = {(a, K. π); π ∈ Π} ∪ {(x, 0 .π); (x, π) ∈ a}.