Mathematical Logic

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ : For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ . With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to < κ . By a well-known theorem of Lindström [4], first order logicLωω is the strongest logic which satisifies the compactness theorem and the downward LöwenheimSkolem theorem. For weakly compact κ , the infinitary logic Lκω satisfies both the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ . In [1] Jon Barwise pointed out that Lκω is not maximal with respect to these properties, and asked what is the strongest logic based on a weakly compact cardinal κ which still satisfies the (κ, κ)-compactness property and some other natural conditions suggested by κ . We prove (Corollary 5) that for weakly compact κ there is no strongest extension of Lκω with the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ . This shows that there is no extension of Lκω which would satisify the most obvious generalization of Lindström’s Theorem. A stronger result (Theorem 11) is proved under an additional assumption. We use the notation and terminology of [2, Chapter II] as much as possible. We will work with concrete logics such as first order logic Lωω, infinitary logic Lκλ and their extensions Lωω({Qi : i ∈ I }) and Lκλ({Qi : i ∈ I }) by generalized quantifiers. Therefore it is not at all critical which definition of a logic one uses as long as these logics are included and some basic closure properties are respected. We use L ≤ L′ to denote the sublogic relation. Let P be a property of logics. A logic L∗ is strongest extension of L with P , if ∗ We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions. S. Shelah: Institute of Mathematics, Hebrew University, Jerusalem, Israel. Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726] J. Väänänen: Department of Mathematics, University of Helsinki, Helsinki, Finland. e-mail: [email protected] Research partially supported by grant 40734 of the Academy of Finland. 64 S. Shelah, J. Väänänen 1. L ≤ L∗, 2. L∗ has property P , and whenever L is a sublogic L′ and L′ has property P , then L′ ≤ L∗. Let L be a logic and κ and λ infinite cardinals. L is (κ, λ)-compact if for all ⊆ L of power κ , if each subset of of cardinality < λ has a model, then has a model. L is κ-compact if it is (κ, ω)-compact. L is weakly κ-compact if L is (κ, κ)-compact. L is fully compact if it is κ-compact for all κ . L has the Löwenheim-Skolem property down to κ , denoted by LS(κ) if every φ ∈ L which has a model, has a model of cardinality ≤ κ . If every sentence φ ∈ L which has a model, has a model of cardinality < κ , we say that L satisfies LS(< κ). The order-type of the well-ordering R is denoted by otp(R). Theorem 1. [4] The logic Lωω is strongest extension of Lωω with א0-compactness and LS(א0). Let C be a class of cardinals. Let A |= Q Cxyφ(x, y, z) ⇐⇒ {〈a, b〉 : A |= φ(a, b, c)} is a linear order with cofinality in C. By [9], Lωω(Q C) is always fully compact. For C an interval we use the notation Q [κ,λ) and Q cf [κ,λ]. Proposition 2. There is no strongest κ-compact extension of Lωω. In fact: 1. there are fully compact logics Ln, n < ω, such that Ln ≤ Ln+1 for all n < ω, but no א0-compact logic extends each Ln. 2. There is an א0-compact logic L1 and a fully compact logic L2 such that no א0-compact logic extends both L1 and L2. Proof. LetLn = Lωω({Q [אω,∞]}∪{Q אl : l < n}). By [9], eachLn is fully compact. Clearly, no א0-compact logic can extend each Ln. For the second claim, let L1 be the logic Lωω(Q1), where Q1 is the quantifier “there exists uncountable many” introduced by Mostowski [8]. This logic is א0-compact [3], see [2, Chapter IV] for more recent results. Let L2 be the logic Lωω(QB), where QB is the quantifier “there is a branch” introduced by Shelah [10]. More exactly, QBxytuM(x)T (y)(t ≤ u) if and only if ≤T is a partial order of T ⊆ M and there are D,≤D , f and B such that: 1. ≤D is a total order of D ⊆ M 2. f : 〈T ,≤T 〉 → 〈D,≤D〉 is strictly increasing 3. ∀s ∈ D∃p ∈ T (f (p) = s) 4. B ⊆ T is totally ordered by ≤T 5. ∀b ∈ B((p ∈ T &p ≤T b) → p ∈ B) 6. ∀s ∈ D∃b ∈ B(s ≤D f (b)). The reader is referred to [10] for a proof of the full compactness of L2. A note on extensions of infinitary logic 65 Suppose there were an א0-compact logic L containing both L1 and L2 as a sublogic. It is easy to see that the class of countable well-orders can be expressed as a relativized pseudoelementary class in L. This contradicts א0-compactness of L. Lauri Hella pointed out that by elaborating the proof of claim (2) of the above proposition, we can make L1 fully compact. It was proved in [11] that, assuming GCH, there is no strongest extension of Lωω which is א0-compact. Our proof of (1) of the above proposition essentially occurs in a note, based on a suggestion of Paolo Lipparini, added after Theorem 8 of [11]. Proposition 3. Suppose κ > א0. There is no strongest extension of Lκ+ω with LS(κ). Proof. Let L1 = Lκ+ω(Qcf א0) and L2 = Lκ+ω(Qcf [א1,κ]). By using standard arguments with elementary chains of submodels, it is easy to see that both L1 and L2 have LS(κ), but the consistent sentence R is a linear order with no last element ∧ ¬Qcf א0xyR(x, y) ∧ ¬Qcf [א1,κ]xyR(x, y) has no models of size ≤ κ . It was proved in [11] that there is no strongest extension of Lωω with LS(ω). Lemma 4. Suppose κ is weakly compact. Then Lκω(Q cf {א0}) and Lκω(Q cf [א1,κ]) are weakly κ-compact. Moreover, if κ > ω, these logics satisfy LS(κ). Proof. The claim concerning LS(κ) is proved with a standard elementary chain argument. We prove the weak compactness ofLκω(Q [א1,κ]). The case ofLκω(Q {א0}) is similar, but easier. For this end, suppose T is a set of sentences of Lκω(Q [א1,κ]) and |T | = κ . We may assume T ⊆ κ . If α < κ , then we assume that there is a model Mα |= T ∩ α. In view of LS(κ), it is not a loss of generality to assume that Mα = 〈κ,Rα〉, where Rα ⊆ κ × κ . Let R(α, β, γ ) ⇐⇒ Rα(β, γ ). By weak compactness there is a transitive M of cardinality κ such that 〈H(κ), , T , R〉 ≺Lκκ 〈M, , T ∗, R∗〉 and κ ∈ M . Let M = 〈M,S〉, where S(x, y) ⇐⇒ R∗(κ, x, y). We claim that M |= T . We need only worry about the cofinality-quantifier. Cofinalities < κ can be expressed in Lκκ , so they are preserved both ways. Therefore also cofinality κ is preserved, and no other cofinalities can occur as the models have cardinality κ . Since the logics Lκω(Q א0)) and Lκω(Q [א1,κ]) cannot both be sublogics of a logic with LS(κ), we get from the above lemma: Corollary 5. Suppose κ > ω is weakly compact. Then there is no strongest weakly κ-compact extension of Lκω with LS(κ). 66 S. Shelah, J. Väänänen The logic Lκω actually satisfies the property LS(< κ) which is stronger than LS(κ). To prove a result like the above corollary for the property LS(< κ) we have to work a little harder. At the same time we extend the proof to extensions of Lκκ . Here the cofinality quantifiers Q C will not help as Q cf {λ} is definable in Lκκ for λ < κ . Therefore we use more refined order-type quantifiers. Definition 6. Let Lκλ(Q) denote the formal extension of Lκλ by the generalized quantifier symbol Qxyφ(x, y, z). If Y is a class of ordinals, we get a logic Lκλ(Q,Y) from Lκλ(Q) by defining the semantics by A |= Qxyφ(x, y, c) ⇐⇒ otp({〈a, b〉 : A |= φ(a, b, c)}) ∈ Y . If φ ∈ Lκλ(Q,Y) and A |= φ, we say that A |= φ holds in the Y-interpretation. If A is a model, then