Canonical functions, non-regular ultrafilters and Ulam's problem on ω1

Our main results are:Theorem 1. Con(ZFC + “every function f: ω1 → ω1 is dominated by a canonical function”) implies Con(ZFC + “there exists an inaccessible limit of measurable cardinals”). [In fact equiconsistency holds.]Theorem 3. Con(ZFC + “there exists a non-regular uniform ultrafilter on ω1”) implies Con(ZFC + “there exists an inaccessible stationary limit of measurable cardinals”).Theorem 5. Con (ZFC + “there exists an ω1-sequence of ω1-complete uniform filters on ω1 s.t. every A ⊆ ω1 is measurable w.r.t. a filter in (Ulam property)”) implies Con(ZFC + “there exists an inaccessible stationary limit of measurable cardinals”).We start with a discussion of the canonical functions and look at some combinatorial principles. Assuming the domination property of Theorem 1, we use the Ketonen diagram to show that ω2V is a limit of measurable cardinals in Jensen's core model KMO for measures of order zero. Using related arguments we show that ω2V is a stationary limit of measurable cardinals in KMO, if there exists a weakly normal ultrafilter on ω1. The proof yields some other results, e.g., on the consistency strength of weak*-saturated filters on ω1, which are of interest in view of the classical Ulam problem.

The well-foundedness of the Mitchell order

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ

Suppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

The ⊲-ordering on normal ultrafilters

If κ is a measurable cardinal, then it is a well-known fact that there is at least one normal ultrafilter over κ. In [K-1], Kunen showed that one cannot say more without further assumptions, for if U is a normal ultrafilter over κ, then L[U] is an inner model of ZFC in which κ has exactly one normal measure. On the other hand, Kunen and Paris showed [K-P] that if κ is measurable in the ground model, then there is a forcing extension in which κ has normal ultrafilters, so it is consistent that κ has the maximum possible number of normal ultrafilters. Starting with assumptions stronger than measurability, Mitchell [Mi-1] filled in the gap by constructing models of ZFC + GCH satisfying “there are exactly λ normal ultrafilters over κ”, where λ could be κ+ or κ++ (measured in the model), or anything ≤ κ. Whether or not Mitchell's results can be obtained by starting only with a measurable cardinal in the ground model and defining a forcing extension is unknown.There are substantial differences between the Mitchell models and the Kunen-Paris models. In the Kunen-Paris models κ can be the only measurable cardinal. However, in the Mitchell model in which κ has exactly 2 normal ultrafilters, one of them contains the set {α < κ: α is measurable} while the other does not. Thus it is natural to ask if it is possible to get a model M of ZFC in which κ is the only measurable cardinal and κ has exactly 2 normal ultrafilters. In this paper we will show that, using appropriate large cardinal assumptions, the answer is yes.

Strongly compact cardinals, elementary embeddings and fixed points

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

Many-times huge and superhuge cardinals

In this paper we consider various generalizations of the notion of hugeness. We remind the reader that a cardinal κ is huge if there exist a cardinal λ > κ, an inner model M which is closed under λ-sequences, and an elementary embedding i: V → M with critical point κ such that i(κ) = λ. We shall call λ a target for κ and shall write κ → (λ) to express this fact. Equivalently, κ is huge with target λ if and only if there exists a normal ultrafilter on P=κ(λ) = {X ⊆ λ:X has order type κ}. For the proof and additional facts on hugeness, see [3].We assume that the reader is familiar with the notions of measurability and supercompactness. If κ is γ-supercompact for each γ < λ, we shall say that κ is < λ-supercompact. We note that if κ → (λ), it follows immediately that κ is < λ-supercompact.Throughout the paper, n shall be used to denote a positive integer, the letters α, β, and δ shall denote ordinals, while κ, λ, γ, and η shall be reserved for cardinals. All addition is ordinal addition. V denotes the universe of all sets.All results except for Theorems 6b and 6c and Lemma 6d can be formalized in ZFC.This paper was written while the first named author was at Rochester Institute of Technology, Rochester, New York. We wish to thank the department of mathematics at R.I.T. for secretarial time and facilities.

Supercompact cardinals and trees of normal ultrafilters

Supercompact cardinals are usually defined in terms of the existence of certain normal ultrafilters. It is well known that there is a natural partial ordering on the collection of all normal ultrafilters associated with a super-compact cardinal, that of normal ultrafilter restriction. Using this notion, we define a tree structure T on the collection of normal ultrafilters associated with a fixed supercompact cardinal. Many results already appearing in the literature can be conveniently phrased in terms of structural properties of T (see, e.g. [4] or [6]). In this paper, we establish additional structural facts concerning T.In §1 we standardize our notation and review some of the basic facts and methods that will be used throughout. §2 begins with a presentation of an important technique, due to Solovay, which will be an important tool for us. Also in §2, we begin a detailed study of the structure of T in terms of branching and the existence of many successors to branches at limit levels. §3 contains results proving the existence of many nodes of T which do not have successors above certain levels of T. This complements work of Magidor [6] who established the existence of many nodes which have successors at all higher levels of T.