On the Hanf Number of Souslin Logic

We show it is consistent with ZFC that the Hanf number of Ellentuck's Souslin logic should be exactly D -2' ?1. Survey of Souslin logic. Ellentuck [4] introduces an infinitary language he calls Souslin logic, an extension of SY,. We work throughout with countable vocabularies (sets of relation, function, and constant symbols). For such a vocabulary Y, Souslin logic SeE over S is the smallest class of formulas containing the atomic formulas and closed under (i) negation, (ii) finite and denumerable conjunction and disjunction, (iii) finite strings of universal and existential quantifiers, and (iv) two new operations. The new operations allow us to form, given a family (Dps: s E "w ) of formulas, the following: (*) V A (Pf I n, fE t nEco (**) A V (Pfin. fea. n Ew (Here aw is the set of all functions from a to the set t of natural numbers, Ad = Uni wco, and I indicates restriction.) To identify formulas of STE with set-theoretic objects, follow Keisler's identification [5, Chapter 1] of formulas of S-, . with set-theoretic objects, adding a clause identifying (*) above with, say the ordered pair (1, (,ps: s E ) and similarly for (**). Most facts about the model theory of SE follow from two simple observations. The first is that the class of countable wellorders is a pseudoelementary class (PC) for StE; in other words, the class of models of an existential 2nd-order sentence 3R(p ( c, R) with < E STE,; in yet other words, the class of reducts to vocabulary { } of a sentence of SE for some larger vocabulary S = f ? , R}. For a structure W = (A, ? ') is a countable wellorder if and only if it can be expanded to a model of the sentence of Souslin logic which is the conjunction of the usual axioms for a linear order with: Vx V (x = cn) A A V (Cf(n) ? Cf(n+ 1)). nEw fEaw nEw The second observation is that SE is a fragment of the game-quantifier language S/1i of Vaught [7], and inherits many properties from that language. Received November 5, 1976. AMS (MOS) subject classifications (1970). Primary 02H10. Research supported by NSF MCS 76-10224. 568 ( 1978, Association for Symbolic Logic This content downloaded from 157.55.39.35 on Mon, 29 Aug 2016 06:11:32 UTC All use subject to http://about.jstor.org/terms ON THE HANF NUMBER OF SOUSLIN LOGIC 569 In particular, the satisfaction relation for STE inherits 1~and Hl-definitions, making STE an absolute logic in the sense of Barwise [1]. From these two observations and general theorems of [1] we have (cf. [2, Chapter 3], and [3] for more consequences of the same sort): (1) DOWNWARD LOWENHEIM-SKOLEM THEOREM. Any sentence of STE which has a model has a countable model [1, Proposition 2.1]. (2) KARP PROPERTY. Any two structures which satisfy the same sentences of St< also satisfy the same sentences of StE [1, Proposition 2.5]. (3) ANTI-INTERPOLATION THEOREM. Craig 's Interpolation Theorem fails for STE [1, Proposition 2.13]. Ellentuck [4], however, shows STE has a property sharply distinguishing it from S.,'WG and other languages considered by Barwise [1]. Let S contain the binary predicate ? among other symbols. A sentence up of StE admits an order-type 0 if there is a model A 1= up with ? a linear order of type 0. For (p E SE define: [o, if (p admits nonwellorders, the least ordinal a such that up admits only a(p)= dorinals a, if such exists, l if up admits only wellorders, but there is no l bound a on the ordinals admitted. F0, if (p has arbitararily large models, K((p)= the least K such that p has no model of Lcardinality ? K, if such exists. The ordinal bound of SE is sup {fa(p): p E StE}, and the Hanf number is SUp{K((p): p E SEE}. Ellentuck shows each a (p) is< W2, and each K((p)<<2, so the ordinal bound and the Hanf number of SE are respectively ? w(2 and C :2, with strict inequality holding if CH does (since then card YE = K1). We will show by an example in the next section that it is consistent with ZFC that equality should hold. By contrast, the Hanf number of S.,t1G is larger than the first inaccessible, the first Mahlo, the first weakly compact cardinal, if such things exist. Before presenting our example, let us mention some generalizations of Ellentuck's results. By weak YS,2' we mean the smallest set of formulas containing the atomic and negation-atomic formulas of Y and closed under countable conjunction and disjunction, finite strings of quantifiers, and the formation of disjunctions (but not conjunctions) of sets of formulas of size K1. Ellentuck's adaptation of classical approximations to operation (s) in [4, Theorem 8], in effect shows: (4) Every sentence of StE is equivalent to a sentence of weak St.2W. The "consistency property" argument of [4, Theorems 9, 10], shows: (5) The ordinal bound of weak 'T.2. is ? (02(6) The Hanf number of weak ST,2, is ?W2 Indeed, we have equality in (5) and (6). (For (5), observe that for every a < 02 there is a sentence whose models are precisely the wellorders of type < a; for (6), argue as in Keisler [5, Example 18].) This content downloaded from 157.55.39.35 on Mon, 29 Aug 2016 06:11:32 UTC All use subject to http://about.jstor.org/terms