Forcing axioms and inner models

Outline Andrés E. Department of Mathematics California Institute of Technology Boise State University, February 27 2008

Outline Outline 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with ω2 M = ω 2 3 Elementary embeddings Discontinuities Consistency strength 4 Elementary embeddings

Outline Introduction 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with ω2 M = ω 2 3 Elementary embeddings Discontinuities Consistency strength 4 Elementary embeddings

The questions that we study here grew out of recent results in the theory of forcing axioms and some interesting conjectures that these results suggest. We seek to study the relation between the universe of sets V and an inner model M under assumptions that imply some form of closeness between M and V. Throughout this talk, M always denotes an inner model. We measure closeness in terms of agreement of cardinals.

The usual picture of the universe of sets consists of a skeleton provided by the ordinals, around which the universe grows by iterating the power set operation. We denote the ordinals by α, β,... and the corresponding stages of the iteration by V α, V β,... We identify cardinals with initial ordinals. The full universe is denoted V. Inner models M have the same ordinals, but their notion of power set may be more restrictive. This thinness typically implies that ordinals interpreted in M to be cardinals really are not cardinals in V.

The smallest possible inner model is called L and was first studied by Gödel. Large cardinal axioms formalize the idea that many stages V α must closely resemble V. In the presence of large cardinals, the cardinals of L are much different from the cardinals of V. Another important inner model is K. It shares many of the nice properties of L but closely resembles the large cardinal structure of V. The stages in the construction of K are called mice. There are a few legends explaining this unfortunate name.

Large cardinals are important because they provide us with a scale to measure the consistency strength of diverse assumptions. A typical large cardinal states that there are elementary embeddings j : M N between some inner models M and N (perhaps M = N = L, or M = V, or... ) Any such embedding restricts to a map j : ORD ORD from the ordinals to themselves. This map is increasing (j(α) α) and non trivial (j id). The first α such that j(α) > α is called the critical point of j. It is typically rather large.

Forcing axioms state that the universe is saturated in a technical sense. Just like V may be fatter than an inner model M, in the method of forcing we obtain a way of fattening V itself as if it were an inner model of a larger universe V. Of course, this is only an ideal extension, since V is the collection of all sets. This universe V would have been obtained from V by adding to it a new set G and any additional sets that this entails (like P(G) or G N). A forcing axiom states that close approximations to these ideal sets G already exist. The forcing axiom that concerns us in this talk is PFA, the proper forcing axiom. It has very high consistency strength.

I will mostly use the forcing axioms to motivate the results I discuss. In particular, no understanding in the technicalities of proper forcing is required. I proceed to state two results that motivated much of what follows.

Theorem (, Veličković) Assume ω M 2 = ω 2 and PFA holds in M and in V. Then R M. Question Does the conclusion of the theorem hold under the assumption that ω M 2 = ω 2 and PFA holds in V?

Theorem (Viale) Assume M and V have the same cardinals and reals, PFA holds in V and stationary sets in M are stationary. Then ORD ω M. Question Can we remove the assumption on preservation of stationary sets from the statement of the theorem?

Outline Introduction Same ω 1 Same ω 2 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with ω2 M = ω 2 3 Elementary embeddings Discontinuities Consistency strength 4 Elementary embeddings

Same ω 1 Same ω 2 Inner models that compute cardinals correctly Recall: Theorem (C. Veličković) Assume ω M 2 = ω 2 and PFA holds in M and in V. Then R M. What can we say about the reals of M under the sole assumption that M and V agree on some cardinals?

Same ω 1 Same ω 2 In general, very little can be said when ω1 M = ω 1. For example, if 0 exists, it does not follow that 0 M. In fact, there is in this case (in V ) a set forcing extension of L that computes ω 1 correctly. This observation is folklore: Use the L-indiscernibles (ι α : α ω V 1 ) to guide an inductive construction of a Col(ω, < ω V 1 )-generic over L.

Same ω 1 Same ω 2 Inner models that compute ω 2 correctly The results in this section are joint work with Ralf Schindler and Martin Zeman. At this stage, our results are only preliminary. In contrast with the situation for ω 1, if M computes ω 2 correctly and 0 exists, then 0 M. This fact has been known for some time and is probably due to Friedman. Our results extend the situation from 0 to larger mice, although it is not clear how far they can go.

Same ω 1 Same ω 2 Theorem Assume ω2 M = ω 2. If 0 does not exist, then every sound mouse projecting to ω belongs to M. In other words, K V and K M have the same reals. If 0 exists, then every sound mouse projecting to ω and below 0 is in M. We expect that the result holds with 0 replaced by larger mice. Question Assume that AD L(R), M V and CAR M = CAR. Does M = AD L(R)?

Outline Introduction Discontinuities Consistency strength 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with ω2 M = ω 2 3 Elementary embeddings Discontinuities Consistency strength 4 Elementary embeddings

Discontinuities Consistency strength Cardinal preserving elementary embeddings Recall: Theorem (Viale) Assume M and V have the same cardinals and reals, PFA holds in V and stationary sets in M are stationary. Then ORD ω M. We expect the assumption on stationary sets is superfluous.

Discontinuities Consistency strength There was a scenario that would prevent Viale s theorem to generalize as expected: Assume that there is an elementary embedding. These embeddings can be obtained using the stationary tower forcing under mild assumptions. (But they cannot be created by set forcing.) Foreman has shown that for any such embedding, ORD ω M.

Discontinuities Consistency strength Theorem If PFA holds, then no embedding is cardinal preserving. This is a consequence of recent work on forcing axioms and the following result. Theorem Suppose that is elementary, and CAR M = CAR. Let κ = cp(j). Then for all λ > κ, j(λ) > λ. In particular, if j λ λ, then cf M (λ) κ.

Discontinuities Consistency strength It follows from this result that the critical point of such an embedding is Π 1 -indescribable in a very strong sense. For example: Corollary If there is a cardinal preserving then there is a proper class of weakly inaccessible cardinals. Proof. Let κ = cp(j). Any weakly inaccessible cardinal λ in V is also weakly inaccessible in M and therefore j(λ) is (another) weakly inaccessible cardinal. κ is weakly inaccessible in M, so there are (in V, thus in M) weakly inaccessible cardinals above κ. If there are only set many of them, their supremum would be a fixed point of j.

Discontinuities Consistency strength It is expected that there are no cardinal preserving embeddings. Since such an M computes incorrectly many cofinalities, it is to be expected that such an embedding would require considerable consistency strength. For example: Theorem Assume that there is a cardinal preserving embedding. Then there are inner models with strong cardinals.

Outline Introduction 1 Introduction 2 Inner models M with ω1 M = ω 1 Inner models M with ω2 M = ω 2 3 Elementary embeddings Discontinuities Consistency strength 4 Elementary embeddings

How about cardinal preserving embeddings? The first observation should not be too surprising. Theorem If is the embedding generated by a set extender, then j is not cardinal preserving.

Even though M may be correct about cofinalities, the existence of cardinal preserving embeddings is of significant large cardinal strength. Theorem If is cardinal preserving, then cp(j) is cp(j) + -strongly compact (and significantly more). This indicates these embeddings cannot be analyzed in terms of current inner model theory.

Theorem (, Woodin) Assume is such that if κ = cp(j), then cf(λ) = cf M (λ) for any λ sup j[j(κ)]. Then 1 j(κ) is strongly inaccessible and 2 V j(κ) = κ is strongly compact.