Mathematical Logic

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K , where K is the core model below a μ-measurable cardinal. 1. Homogeneously Souslin sets In this paper we shall deal with homogeneously Souslin sets of reals, or rather with sets of reals which admit an ω-closed embedding normal form. Definition 1.1. (Cf. [4, p. 92].) Let A ⊂ ω. Let α ∈ OR. We say that A has an α-closed embedding normal form if and only if the following holds true. There is a commutative system ((Ms : s ∈ ω), (πst : s, t ∈ ω, s ⊂ t)) such that M0 = V , each Ms is an inner model of ZFC with Ms ⊂ Ms , each πst : Ms → Mt is an elementary embedding, and if x ∈ ω and (Mx, (πx n,x : n < ω)) is the direct limit of ((Mx n : n < ω), (πx n,x m : n ≤ m < ω)) then x ∈ A ⇔ Mx is wellfounded. As we shall not need it here, we do not repeat the definition of the concept of being homogeneously Souslin in this paper (cf. [4, p. 87]). We just remind the reader of the following facts. Lemma 1.2. Let A ⊂ ω. (1) If A is coanalytic and if κ is a measurable cardinal then A is κ-homogeneously Souslin (cf. [3], [4, Theorem 2.2]). (2) If A is κ-homogeneously Souslin, where κ is a (measurable) cardinal, then A is determined (cf. [4, Theorem 2.3]) and has a κ-closed embedding normal form (cf. [4, p. 92]). (3) If A has a 2א0 -closed embedding normal form then A is homogeneously Souslin (cf. [7, Lemma 2.5], [2, Theorem 5.2]). P. Koepke: Mathematisches Institut, Universität Bonn, Beringstraße 1, Bonn, Germany. e-mail: [email protected] R. Schindler: Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstraße 62, Münster, Germany. e-mail: [email protected] 54 P. Koepke, R. Schindler Our aim is to prove a converse to Lemma 1.2 (1) under appropriate anti-large cardinal hypotheses. Definition 1.3. A cardinal κ is called μ-measurable if there is an embedding π : V → M such thatM is transitive, κ = crit(π), and {X ⊂ κ|κ ∈ π(X)} ∈ M (cf. [5]). We say that 0¶ does not exist if for every iterable premouse M, if EM ν = ∅ then M||crit(EM ν ) is a model of “there is no strong cardinal (as being witnessed by the extenders from the M-sequence)” (cf. [8, p. 272]). Suppose that 0¶ does not exist, and letK denote the core model (cf. [8, Chap. 8]). We say that K does not have a μ-measurable cardinal if K |= “there is no μ-measurable cardinal.” If K does not have a μ-measurable cardinal then every total extender on the K-sequence has exactly one generator (i.e., can be construed as a measure in the usual sense). Definition 1.4. We say that 0long does not exist if for every iterable premouse M, if we let A be the set of critical points of the total measures from the M-sequence then A = ∅ or else otp(A) < min(A) (cf. [1]). We can now state the main results of our paper. Theorem 1.5. Suppose that 0¶ does not exist, andK does not have aμ-measurable cardinal. Suppose that V = K . Let A ⊂ ω have an ω-closed embedding normal form. Then A is coanalytic. Theorem 1.6. Suppose that 0long does not exist. Let A ⊂ ω have an ω-closed embedding normal form. Then A is coanalytic. Our main technical tool will be the concept of a “shift map.” Shift maps will be defined in the next section, where we shall also show that if K does not have a μ-measurable cardinal then any elementary embedding from one universal weasel into another one is a shift map. The final section will prove Theorems 1.5 and 1.6. As to prerequisites, we shall assume familiarity with the core model theory as presented in [8, Chap. 8].