L¨owenheim-Skolem , the of Dependent Choice and a categorical extension principle

Christian Esp´ındola∗

Abstract In this paper we prove the equivalence, over ZF , of L¨owenheim-Skolem theorem (LS), the Axiom of Dependent Choice (DC), and a certain functor extension property we introduce. We also consider variations of that property equivalent to other well known choice principles.

1 Introduction

It has been known since 1920 ([6], [3]) that the the played a crucial rˆolein the proof of the following form of L¨owenheim-Skolem theorem: Theorem 1.1. LS. Every model M of a first order theory T with countable signature has an elementary submodel N which is (at most)1countable. As noted by Skolem himself, the condition that the countable model for T be an elemen- tary submodel of M made an apparently essential use of the axiom of choice, by relying on the so called Skolem functions, which can be found in the modern standard proofs of the theorem. A careful examination of the proof shows that DC is sufficient already (see [4]), although the exact strength of LS was not known. On the other hand, a categorical proof of LS was given in [1], following a construction that Joyal had used in his unpublished proof of G¨odel’scompleteness theorem, and which allows to consider a certain functor extension property related to LS. The main theorem of this paper will show that the three statements are in fact equivalent over ZF .

2 The main theorem

To state the functor extension principle we introduce some definitions: Definition 2.1. A Boolean category is a category having the following three properties: 1) It has all finite limits. 2) Every arrow f : A → B can be factored as f : A  C  B, where C, called the image of f, is the least subobject of B through which f can factor. The arrow f : A  C not factoring through any proper subobject of C is called a cover. 3) Images are stable under base change (i.e., pullbacks preserve covers). 4) The poset Sub(X) of subobjects of a given object X has finite unions and these are stable under pullbacks. 5) Every subobject A in the poset Sub(X) has a , i.e., there exists a subobject B such that the intersection A ∧ B is initial in Sub(X) and A ∨ B = X (in particular, Sub(X) is a Boolean algebra, and we denote B = ¬A). Definition 2.2. A functor between Boolean categories is said to be Boolean provided it preserves finite limits and images factorizations, unions and complements.

∗This research was financed by the project “Constructive and category-theoretic foundations of mathematics (dnr 2008-5076)” from the Swedish Research Council (VR). 1From now on, we shall use the word countable to refer to an either finite or countably infinite .

1 Consider the category of small Boolean categories with Boolean functors. Given a small Boolean category C0, [1] describes the construction of succesive conservative embeddings C0 → C1 → ... → Cω, where each Cn+1 has sections for every cover from an object in Cn to the terminal object. It also provides a way to extend Boolean functors with domain C to Boolean functors with domain Cω. Although the construction of Cω can be performed in ZF by using Grothendieck fibrations and the construction of the bicolimit through the category of fractions (see [2]), assuming the axiom of choice it can be proved, in fact, that Cω is the free completion of C by adding sections of covers over the terminal object. In other words, given a Boolean category D such that every cover over the terminal object has a section, any Boolean functor F : C → D has a Boolean extension F : Cω → D which sends sections into sections and that is, moreover, the unique (up to natural isomorphism) Boolean functor with that property. Our functor extension principle concerns the particular case where C is countably small (i.e., it has a of objects and morphisms) and D = Set is the (large) category of sets (note that Boolean functors F : C → Set can still be properly defined in ZF , but we use this notation for convenience). That this extension property is related to some choice principle should not be a surprise, since if we do not impose any restriction on the size of C, the existence of such an extension is trivially equivalent to the axiom of global choice: an extension of the identity functor in Set provides sections for each cover Si → {∗}, i.e., provides a choice for each set. However, the interest of this principle lies in the fact that restricting the to a countable set makes it equivalent to DC. Theorem 2.3. In Zermelo-Fraenkel , the following are equivalent: 1. The Axiom of Dependent Choice.

2. Every Boolean functor F : C0 → Set from a countably small Boolean category C0 has a Boolean extension F : Cω → Set. 3. The L¨owenheim-Skolemtheorem.

Proof. (1 ⇒ 2) This implication is essentially contained in [1], lemma 4.2. The construction of F avoids choice because the pseudofunctor considered can be defined in ZF by using the countability of the set of objects and arrows when considering the pullback functors involved (and one can consider Grothendieck’s construction of the bicolimit, as explained in [2], Ex.6). The first step of the extension from C to C1 only uses countable choice (derivable from DC), as it requires to consider choice functions for a countable . The iteration of this first step uses DC.

(2 ⇒ 3) This implication is also contained in the categorical proof of LS given in [1], but its essential set-theoretic content can be extracted as follows. Given a first order theory T with countable signature, its associated syntactic category CT is a small Boolean category with a countable set of objects. The process of embedding it into the finite coproduct completion C0 and constructing now the free completion of C0 by sections of covers over the terminal object gives a Boolean conservative embedding CT → Cω, where Cω is a small Boolean category with the property that every cover over a subobject of the terminal object has a section. 0 0 Let T be the Henkinization of T . We shall see that Cω is a model for T , by finding suitable interpretations for the added constants. Suppose that φ(x) is a formula with one free variable containing only constants of level at most n − 1; we may also assume that these have already been interpreted in Cω. Then the constant c associated to φ(x) can be interpreted in the following way:

2 m n [[x, φ(x)]] / / [x, >] o o P W W X

q s p c r t

    m0  n0  [[{}, ∃xφ(x)]] / / [{}, >] o o ¬[[{}, ∃xφ(x)]]

Given the interpretation m of [x, φ(x)], the resulting interpretation m0 for [{}, ∃xφ(x)] is given by the image factorization of pm. Let s be a section2 of the cover q; consider the complement of the subobject [[{}, ∃xφ(x)]] and take its pullback P along p. Let t be the section of the resulting cover r; then ms and nt determine a section from the terminal object which we can interpret as the constant c. With this interpretation, it is clear that the formula ∃xφ(x) → φ(c) is satisfied, since the interpretation of [{}, φ(c)] is the pullback of m along c, which is precisely the same as [[{}, ∃xφ(x)]] with the projections s and m0. 0 If we consider the syntactic category CT 0 , the fact that Cω is a model for T implies that 3 there is a Boolean functor J : CT 0 → Cω through which the embedding I : CT → Cω factors . By 2, any model M for T , interpreted as a functor F : CT → Set, has a Boolean extension 0 0 F : Cω → Set, and the composition FJ provides a model M for the Henkinized theory T which extends M. Finally, a countable submodel for M can be constructed taking as the underlying universe the set N of the interpretations of all constants of T 0 and restricting to N the interpretations of all symbols of M.

(3 ⇒ 1) Let S be a set with a binary relation R such that for every x ∈ S, the set {y ∈ S / xRy} is nonempty. Consider the theory T over the language L = {R} that contains a binary relation symbol, and whose only non logical axiom is ∀x∃yR(x, y). Since S is a model for T which has a countable signature, by 3 it contains a countable submodel N . Hence, there is a between the underlying set N of N and either the set of natural numbers or some finite ordinal. If f is such a bijection, we can now recursively define a sequence by −1 −1 putting x0 = f (min{f(n) / n ∈ N}) and xn = f (min{f(n) / n ∈ N ∧ xn−1Rn}).

Remark 2.4. The proof above implicitely shows a direct set-theoretic deduction of LS from DC that makes no use of Skolem functions, and instead replaces Skolemization with Henk- inization. Indeed, every model of a theory can be extended to a model of its Henkinization by just interpreting the constants (this requires an application of DC since the constants are added in levels), and then we can restrict the interpretations of the symbols to the countable set of constants of the Henkinized theory.

3 Further variations

By inspecting the proof of theorem 2.3 we see that the only full use of DC in the proof of the implication (1 ⇒ 2) was to pass from the extension F : C1 → Set to the final extension F through a countable iteration. Obtaining the extension F1 only required the use of countable choice. As we shall see now, the existence of such an extension is equivalent to that axiom; moreover, we require instead that the set of objects and morphisms be any aleph κ (i.e., that the category be κ-small), we get an statement equivalent to the axiom of choice for a family of κ sets (ACκ, following [5]). Theorem 3.1. Let κ be any aleph. In Zermelo Fraenkel set theory, the following are equiv- alent:

1. ACκ.

2Note that there is a canonical choice of sections for covers over subobjects of the terminal object. 3This also implies that the Henkinized theory is a conservative extension, since I is a conservative functor.

3 2. Every Boolean functor F : C0 → Set from a κ-small Boolean category C0 has a Boolean extension F1 : C1 → Set. Moreover, the statement obtained from 2 replacing κ-small with small is equivalent to the full Axiom of Choice.

Proof. (1 ⇒ 2) We just need to observe that for defining the Boolean extension it is enough to find elements in a family of sets indexed by κ.

(2 ⇒ 1) Let {Si}i∈κ be a family of disjoint nonempty sets. Consider the theory T whose language has an unary relation Ri for each i ∈ κ and that has each of the ∃xRi(x) for i in κ. Let T1 be the theory resulting from T after the first step of the Henkinization has S been performed. Then i∈κ Si is a model for T under the interpretation Ri(x) ⇐⇒ x ∈ Si. If we consider the corresponding Boolean functor F : CT → Set and extend it to the finite coproduct completion C0, we can use 2 to find a Boolean extension F1 : C1 → Set. This also provides a Boolean functor G : CT1 → Set, since C1 can be seen to be a model of T1 by the same argument we applied in the implication (2 ⇒ 3) of theorem 2.3. Therefore a choice function for the family is given by the image through G of the constants of T1 corresponding to the formulas ∃xRi(x).

Finally, the same proof above can be adapted to the case where there is no restriction on the cardinality, giving the last equivalence.

As a final remark, we note that although the equivalence between LS and DC can be proved without any appeal to categorical formulations (as can been seen from the proof of theorem 2.3 and remark 2.4), their equivalence with the functor extension principle, as well as the equivalence result of theorem 3.1, needs here the use of categorical logic and functorial semantics.

References

[1] Esp´ındola,Christian: A categorical proof of the L¨owenheim-Skolem theorem - http://cms.dm.uba.ar/Members/cespindo/skolem.pdf [2] Verdier, Jean-Louis - Grothendieck, Alexandre - Artin, Michael: SGA 4 (1963/64) - Th´eorie des topos et cohomologie ´etaledes sch´emas,Tome 2 - Springer Lecture Notes in Mathematics, vol. 270 (1972) [3] van Heijenoort, Jean: From Frege to G¨odel: A source book in , 1879-1931 - Harvard University Press (1967) [4] Howard, Paul - Rubin, Jean: Consequences of the axiom of choice - Mathe- matical surveys and monographs (1998) [5] Jech, Thomas: The Axiom of Choice - North Holland Pub. Co. (1973) [6] Skolem, Thoralf: Logisch-kombinatorische Untersuchungen ¨uber die Erf¨ull- barkeit oder Beweisbarkeit mathematischer S¨atze nebst einem Theo- reme ¨uber dichte Mengen - Videnskapsselskapet Skrifter, I. Matematisk- naturvidenskabelig Klasse, vol. 6 (1920), pp. 1-36

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