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The Stationarity of the Collection of the Locally Regulars THE STATIONARITY OF THE COLLECTION OF THE LOCALLY REGULARS GUNTER FUCHS Abstract. I analyze various natural assumptions which imply that the set L[x] f!1 j x ⊆ !g is stationary in !1. The focal questions are which implications hold between them, what their consistency strengths are, and which large cardinal assumptions outright imply them. 1. Introduction The motivation for this paper was a question asked by Philipp Schlicht, Pawe l Kawa and Daisuke Ikegami at the Young Set Theory meeting 2008 in Bonn. Recall: L[x] Definition 1.1. !1 is inaccessible to reals iff !1 < !1, for all x 2 R. Fact 1.2. The following are equivalent: (1) !1 is inaccessible to reals. V (2) For every x ⊆ !, !1 is inaccessible in L[x]. (3) !1 is inaccessible in L[b], for every bounded subset b of !1. In order to formulate the initial question concisely, I want to make the following definition. Definition 1.3. Let L[x] Ω = f!1 j x 2 Rg \ !1: I am mostly interested in Ω in the case that !1 is inaccessible to reals. In that L[x] case, Ω = f!1 j x 2 Rg. The original question was: Question 1.4. Is Ω stationary? To exclude trivial counterexamples, one should ask instead: Question 1.5. If !1 is inaccessible to reals, then does it follow that Ω is stationary? It is also natural to ask: Question 1.6. If !1 is inaccessible to reals, then does it follow that Ω contains a club? I will analyze this and a number of related questions and concepts. In section 2, I will show that Ω is a very natural set. It, and its stationarity, can be expressed in many different ways, particularly if !1 is inaccessible to reals. One description is Date: July 20, 2017. 2010 Mathematics Subject Classification. 03E57,03E55,03E45. The final publication is available at Springer via http://dx.doi.org/10.1007/s00153-015-0437-8. 1 2 GUNTER FUCHS that Ω is the set of countable ordinals that are regular in L, and another is that Ω M is the set of countable ordinals that are of the form !1 , for some inner model M. Using this, it is very easy to see that the consistency strength of the stationarity of Ω is exactly a Mahlo cardinal. This is shown in section 3. In section 4, I isolate a property that seems central in this context, which implies that Ω is stationary: the nonexistence of a reshaped set. I show this implication, and that the consistency strength of no reshaped set is again a Mahlo cardinal. To get the consistency, I show that MA!1 (σ-centered), together with the inaccessibility of !1 to reals, implies that there is no reshaped set. On the other hand, I show that the nonexistence of a reshaped set does not imply MA!1 (σ-centered). It is also independent of the projective sets having the regularity properties. In section 5, I introduce relativized versions Ωa of Ω, where, basically, L in the definition of Ω is replaced with L[a] in the definition of Ωa, for a real a. I introduce a strengthening of “Ω is stationary" by defining that Ω is uniformly stationary if Ωa is stationary, for every real a. I show that the nonexistence of a reshaped set implies the uniform stationarity of Ω. Section 6 introduces the concept of local Mahloness. Thus, !1 is Mahlo at a ⊆ ! if the set of countable ordinals that are inaccessible in L is stationary (in V), and !1 is locally Mahlo if this is true for all reals a. I show that uniform stationarity of Ω is the equivalent to local Mahloness of !1. Local Mahloness is a weaker form of the statement that !1 is Mahlo to reals, which means that !1 is Mahlo in L[a], for every real a. In section 7, I show that a variety of large cardinal or forcing axioms imply that there is no reshaped set, and thus, that Ω is uniformly stationary. This is further evidence that these are very natural assumptions. Finally, in section 8, I show that no additional implications between these con- cepts are provable than those that I have shown so far, so figure 1 (on page 10) is complete. I would like to thank the referee for some very valuable comments and suggesti- ons. 2. Characterizations of Ω In this section, I would like to argue that Ω is a very canonical set. Particularly under the assumption that !1 is inaccessible to reals, it has many equivalent and very natural definitions. I will use some basic techniques from coding theory. Definition 2.1. A set b ⊆ !1 is reshaped if for all ξ < !1, ξ is countable in L[b\ξ], L[b\ξ] in other words, if !1 > ξ. A proof of the following theorem can be found in [JBW82, Thm. 1.1], see also [JS70]. Theorem 2.2 (Solovay). Assume V = L[b], where b is reshaped. Then there is a 1 <! <! c.c.c. forcing Pb ⊆ [!] × [b] such that Pb adds a real x with b 2 L[x]. Another useful fact on reshaped sets is: Theorem 2.3 (Jensen, Schindler). Let A ⊆ !1 be such that H!2 ⊆ L[A]. Then there is a σ-distributive and stationary set preserving forcing P, called the reshaping 1It's even σ-centered, which will be of importance later. THE STATIONARITY OF THE COLLECTION OF THE LOCALLY REGULARS 3 forcing, consisting of bounded subsets of !1, which adds a set b ⊆ !1 that is reshaped and codes A, in the sense that A 2 L[b].2 Proof. The forcing P consists of bounded subsets s of !1 such that for every ξ, L[s \ ξ] sees that ξ is countable, and such that for every limit ordinal ξ with ξ + 2(n + 1) ≤ sup(s), ξ + 2(n + 1) 2 s iff ξ + n 2 A. The ordering is that of end-extension. The first condition insures that the generic subset of !1 will be reshaped, and the second one insures that A will be coded. A proof that this forcing is countably distributive can be found in [Sch04, Lemma 4.5], and a proof that it preserves stationary subsets of !1 can be found in [Sch04, Lemma 4.8]. Stationarity preservation was originally shown in [Sch01]. Definition 2.4. Let ¯ L[b] Ω = f!1 j L[b] j= \b is reshaped"g \ !1 and set ¯ 0 L[c] Ω = f!1 j c ⊆ Ong \ !1 Observation 2.5. Ω ⊆ Ω¯ ⊆ Ω¯ 0. Proof. The second inclusion is trivial, and so is the first: if x ⊆ !, then L[x] j= L[x] L[x] ¯ x is reshaped, so if !1 < !1, then !1 2 Ω. If !1 is inaccessible to reals, then the converse is also true: 0 Theorem 2.6. If !1 is inaccessible to reals, then Ω¯ ⊆ Ω¯ ⊆ Ω. L[c] ¯ 0 Proof. For the first inclusion, let ρ = !1 2 Ω , for some c ⊆ On. Working inside L[c] α 0 L[c], for every α < !1 (that is, !1 ), let f : ! −! α be a surjection. Let c = 0 α L[c] 0 L[c] L[c ] {≺α; m; f (m) j α < !1 and m < !gg. Clearly then, c ⊆ !1 = !1 = ρ. Now, in L[c0], theorem 2.3 can be applied (with A = c0), so the reshaping forcing 0 P with respect to c is σ-distributive. In V, since !1 is inaccessible to reals, there L[c0] is a b ⊆ !1 which is generic for the reshaping forcing (because there are only countably many subsets of that forcing in L[c0]). Since b codes c0 and the forcing L[b] ¯ preserves !1, it follows that ρ = !1 , and so, ρ 2 Ω. The argument for the second inclusion is similar. Let ρ 2 Ω.¯ Let b ⊆ ρ be as in ¯ L[b] the definition of Ω, so ρ = !1 , where b is reshaped in L[b]. Note that ρ < !1. One L[b] can now force with the Solovay's almost disjoint coding forcing P = (Pb) over L[b]. That forcing is contained in [!]<! × [b]<!. Note that P(P) \ L[b] is countable V in V, as !1 is inaccessible in L[b]. So let x be a generic real over L[b]. Then since L[x] L[b] b 2 L[x], and P is c.c.c. in L[b], !1 = !1 = ρ 2 Ω. This provides the following very nice and canonical characterization of Ω, assu- ming !1 is inaccessible to reals. Theorem 2.7. If !1 is inaccessible to reals, then Ω is the collection of all countable ordinals α with the property that for some inner model M j= ZFC, M j= α = !1. M Proof. For the substantial direction, suppose M is an inner model, and α = !1 < V L[c] !1 . Working inside M, construct a set c ⊆ α such that α = !1 , as in the proof ¯ 0 of theorem 2.6. This shows that α 2 Ω = Ω. 2The reshaping forcing was introduced and shown to be !-distributive by Jensen (see [JBW82, Thm. 1.4]). Schindler showed that it preserves stationary sets, see [Sch01, Claim 3']). 4 GUNTER FUCHS Theorem 2.8. If !1 is inaccessible to reals, then Ω is the collection of all countable ordinals α > ! that are regular in some inner model M j= ZFC, and this is the same as the collection of all countable ordinals greater than ! that are regular in L.
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