Order with successors is not interpretable in RCF

Using the monotonicity theorem of L. van den Dries for RCF-definable real functions, and a further result of that author about RCF-definable equivalence relations on R, we show that the theory of order with successors is not interpretable in the theory RCF. This confirms a conjecture by J. Mycielski, P. Pudlák and A. Stern. 1. Let RCF be the theory of real closed fields. We may view RCF as the first order theory of the structure 〈R; +, ·,≤, 0, 1〉, where R is the set of real numbers and +, ,≤, 0, 1 have the usual meaning. It is conjectured in [6], (P23), that the order theory of ω = {0, 1, 2, . . .}, i.e., Th(〈ω;≤〉), cannot be interpreted in RCF: (1.1) |Th(〈ω;≤〉)| 6≤ |RCF| . Here |T | denotes the chapter of mathematics containing a given theory T (i.e., the class of all theories which both interpret and are interpretable in T ). |T1| ≤ |T2| means that T1 is interpretable in T2. Our aim in this note is to prove (1.1). Familiarity with the definition of interpretability, as proposed by Jan Mycielski in [5], will be assumed. For further information, the reader is referred to the survey [6]. It is known that the order theories of ω and Q are not interpretable in each other: Both non-interpretability results (1.2) |Th(〈ω;≤〉)| 6≤ |Th(〈Q;≤〉)| and |Th(〈Q;≤〉)| 6≤ |Th(〈ω;≤〉)| were obtained by J. Kraj́ıček [4]. (The second part of (1.2) was proved independently by A. Stern; a proof which combines the methods of both authors is given in [8].) Obviously (1.3) |Th(〈Q;≤〉)| ≤ |RCF| , so, using the second part of (1.2), we conclude that RCF is not interpretable in Th(〈ω;≤〉). Hence, by (1.1), the chapters |RCF| and |Th(〈ω;≤〉)| are incomparable. The first part of (1.2), easier to establish, clearly also follows from (1.1) and (1.3). 282 S. Świerczkowski Since |Th(〈Z;≤〉)| = |Th(〈ω;≤〉)|, we may replace ω by Z in the above considerations. Another consequence of (1.1) is the well known fact that ω is not an RCF-definable subset of R (stemming from the undecidability of arithmetic). To state our result quite exactly, let us specify first the axioms of a preordering with successors. We shall call a binary relation a pre-ordering if is reflexive, transitive and there is universal comparability: ∀x∀y(x y ∨ y x) . For any such pre-ordering we abbreviate (x y)∧(y x) as x ≈ y. Clearly, ≈ is an equivalence relation; we call it the equivalence relation associated with . We denote by Succ(x, y) the formula x y ∧ x 6≈ y ∧ ∀z(x z y → z ≈ x ∨ z ≈ y) , reading it: “y is an immediate successor of x”. We say that is a pre-ordering with successors if every x has an immediate successor, i.e., if ∀x∃y Succ(x, y). It is clear that (1.1) will result if we prove: Theorem 1.1. Any sentence which proves the existence of a pre-ordering with successors is not interpretable in RCF. We take this opportunity to mention another recent result about RCF, and to pose a problem. A theory T is called connected if T interprets in the union T1∪̇T2 of two theories with disjoint languages only if T interprets in T1 or in T2. The problem if RCF is connected was raised in [6], (P2). A positive answer was found recently by A. Stern and the author [9]. A theory T is called compact if there is a finitely axiomatizable theory T ′ such that |T | = |T ′|, i.e., T and T ′ interpret each other. Problem. Is RCF compact? 2. A subset S of R, d ≥ 1, will be called definable if S is definable in the language {+, ·,≤, 0, 1} of RCF, i.e., if there is a formula φ in that language and there are some a1, . . . , ak ∈ R such that, for all x1, . . . , xd ∈ R, (x1, . . . , xd) ∈ S ↔ φ(x1, . . . , xd, a1, . . . , ak) . Functions and relations on Cartesian powers of R will be called definable if they have definable graphs. We shall show that for every d ≥ 1, there does not exist on R a definable relation of pre-ordering with successors. Clearly, this will be enough to establish Theorem 1.1. The following known facts will be needed in the proof. Theorem 2.1 ([1]). Given a definable equivalence relation ≈ on R (d ≥ 1), there is a definable function f : R → R which selects one representative Order with successors 283 from each (≈)-equivalence class, i.e., such that for any d-tuples x, y ∈ R, x ≈ y ↔ f(x) = f(y) and x ∈ f(x) . The next theorem is proved, in the general setting of O-minimal structures, in [7]: Theorem 2.2 (Monotonicity Theorem in [2], [3]). For every definable function f : R → R there is a partition of R into finitely many points and open intervals such that on each of the intervals f is either constant or strictly monotone and continuous. So, for a definable f : R → R there are only finitely many y ∈ R such that the pre-image f−1(y) is infinite. Combining this observation with the case d = 1 of Theorem 2.1, we get: Lemma 2.3. Every definable equivalence relation on R has only finitely many infinite equivalence classes. 3. We need two more lemmas: Lemma 3.1. Let L0 ⊂ L1 ⊂ L2 ⊂ . . . be a strictly increasing sequence of subsets of R such that each boundary ∂Li is a finite set. Suppose further that the number of elements #(∂Li) of any of these boundaries does not exceed a fixed constant K <∞. Then the difference Li+1 \Li is infinite for infinitely many i. P r o o f. Suppose there is an i such that Lj+1 \ Lj is finite for all j ≥ i. Let j > i and consider any x ∈ Lj \Li. If x is in the interior of Lj , then, as x 6∈ Li, we must have x ∈ ∂Li, for otherwise a neighbourhood of x would be disjoint from Li, and Lj \ Li would be infinite. So Lj \ Li ⊆ ∂Li ∪ ∂Lj . Since there are at least j − i elements in Lj \ Li, we get j − i ≤ #(∂Li) + #(∂Lj) ≤ 2K , which is obviously impossible for all j ≥ i. Lemma 3.2. There does not exist a definable pre-ordering of R for which there is an infinite sequence x0, x1, x2, . . . such that every xi+1 is an immediate successor of xi. P r o o f. Suppose that is a definable pre-ordering of R and (xi)i<ω is a sequence such that Succ(xi, xi+1) holds for each i. By the Tarski– Seidenberg quantifier elimination theorem for RCF, the formula x y is RCF-equivalent to a Boolean combination of atomic formulas: