An introduction to ℙ_{max} forcing

Paul B. Larson^{∗}

October 26, 2009

0.1 Pre-introduction

These notes are an account of a six-hour lecture series I presented at Carnegie Mellon University on September 9, 2006, at the inaugural meeting of the Appalachian Set Theory series. My remarks (occasionally improvised) were faithfully transcribed by Peter LeFanu Lumsdaine and Yimu Yin, who then presented me with a rough draft of this article. Their account aimed to capture the feel of the discussions (including some direct quotations), and I’ve tried to preserve that as much as possible.

1 Introduction

The forcing construction ℙ_{max} was invented by W. Hugh Woodin in the early 1990’s in the wake of his result that the saturation of the nonstationary ideal on ω_{1} (NS_{ω_{1}}) plus the existence of a measurable cardinal implies that there is a definable counterexample to the Continuum Hypothesis (in particular, it implies that δ_{2}^{1} = ω_{2}, which implies ¬CH). These notes outline a proof of the Π_{2} maximality of the ℙ_{max} extension, which we can state as follows.

Theorem 1 ([9]). Suppose that there exist proper class many Woodin cardinals, A ⊆ ℝ, A ∈ L(ℝ), φ is Π_{2} in the extended language containing two additional unary predicates, and in some set forcing extension

⟨H(ω_{2}), ∈, NS_{ω_{1}}, A^{∗}⟩ ⊧ φ

(where A^{∗} is the reinterpretation of A in this extension). Then

L(ℝ)^{ℙ_{max}} ⊧ [⟨H(ω_{2}), ∈, NS_{ω_{1}}, A⟩ ⊧ φ] .

Forcing with ℙ_{max} does not add reals, so there is no need to reinterpret A in the last line of the theorem. The theorem says that any such Π_{2} statement that we can force in any extension must hold in the ℙ_{max} extension of L(ℝ), so H(ω_{2}) of L(ℝ)^{ℙ_{max}} is maximal, or complete, in a certain sense; among other things, it models ZFC, Martin’s Axiom, certain fragments of Martin’s Maximum [1], and the negation of the Continuum Hypothesis. The reinterpretation A^{∗} will be defined later, in terms of tree representations for sets of reals. We will not give the definition of Woodin cardinals (but see [4]).

∗Research supported by NSF grants DMS-0401603 and DMS-0801009; conference expenses supported by NSF grant DMS-0631446

We have reworked the standard proof of Theorem 1 in order to minimize the prerequisites. In particular, the need for (mentioning) sharps has been eliminated. However, they and much more will need to be reintroduced to go any further than the material presented here.

“Woodin’s book on ℙ_{max}, The axiom of determinacy, forcing axioms and the nonstationary ideal [9] runs to around 1000 pages. My article for the Handbook of Set Theory [5], introducing ℙ_{max}, has about 65. The advance notes for these lectures are about 30 pages, and previous lecture courses have taken about 12-15 hours to cover ℙ_{max}; so today will, of course, have to be brief… ”

2 Setup: iterations and the definition of ℙ_{max}

Suppose that M ⊧ ZFC and than I ∈ M is a normal ideal on ω_{1}^{M} in M. Force over M with ((P(ω_{1}) \ I)/I)^{M}. The resulting generic G is now an M-normal ultrafilter on ω_{1}^{M}; so we may form the corresponding ultrapower and elementary embedding j : M → Ult(M, G) := {f : ω_{1}^{M} → M | f ∈ M}/ =_{G}. (“We’ll use this a thousand times today.”) Note that crit(j) = ω_{1}^{M}, j(ω_{1}^{M}) ≥ ω_{2}^{M}, Ord^{Ult(M,G)} = Ord^{M}, and for A ∈ P(ω_{1})^{M}, A ∈ G ↔ ω_{1}^{M} ∈ j(A). There is no need to assume that A is transitive, though it will be in the cases were are interested in. When an ultrapower is well-founded, we identify it with its transitive collapse.

Definition 2. I is precipitous if Ult(M, G) thus constructed is well-founded from the point of view of M[G], for all M-generic G. (N.B. this is definable in M via forcing.)

We need a pair of theories satisfying the following conditions.

• T_{0}, a theory consistent with ZFC and strong enough to make sense of the generic ultrapower construction above and prove that j : M → Ult(M, G) is elementary.

• T_{1}, a theory consistent with ZFC and at least as strong as T_{0} + “every set lies in some H(κ) ⊧ T_{0}.”

In [5], we take T_{0} (which we call ZFC^{∘}) to be ZFC - Replacement - Powerset plus “P(P(ω_{1})) exists” plus the scheme saying that definable trees of height ω_{1} have maximal branches. Then we let T_{1} = T_{0} + Powerset + Choice + Σ_{1}- Replacement (though we don’t express it in these terms). In [9], Woodin has an even weaker fragment of ZFC (which he calls ZFC^{∗}) playing the role of T_{0}. Today we may as well let T_{0} be ZFC and T_{1} be ZFC plus the existence of a proper class of strongly inaccessible cardinals. From now on we will just use the terms T_{0} and T_{1}.

Our basic construction is the generic ultrapower. We now extend to iterated ultrapowers. Suppose we have (M_{0}, I_{0}), G_{0} ⊆ (P(ω_{1})/I_{0})^{M_{0}}, j_{0} : (M_{0}, I_{0}) → Ult(M_{0}, G_{0}), all as before; let M_{1} = Ult(M_{0}, G_{0}), I_{1} = j_{0}(I_{0}). Now we can take the generic ultrapower of M_{1} by I_{1}, and iterate. At limit stages, we have a directed system of elementary embeddings, so can just take the direct limit, so we can keep going up to length ω_{1}. (No further, as if we force again there, we collapse ω_{1}^{V}, so are back to countable length!) Note that the final model of the iteration, M_{ω_{1}}, is an element of H(ω_{2}).

Definition 3. An iteration of (M, I) (as above; M countable) of length γ consists of M_{α}, I_{α} (α ≤ γ), G_{η} (η < γ), and j_{α,β} (α ≤ β ≤ γ), satisfying

• M_{0} = M, I_{0} = I

• G_{η} is M_{η}-generic for (P (ω_{1})/I_{η})^{M_{η}}

• j_{η,η+1} is the canonical embedding of M_{η} into Ult(M_{η}, G_{η}) = M_{η+1}

• j_{α,β} : M_{α} → M_{β} are a commuting family of elementary embeddings

• I_{β} = j_{0,β}(I)

• For limit β, M_{β} is the direct limit of {M_{α} | α < β} under the embeddings j_{α,η} ( α ≤ η < β).

In practice, we almost always have γ = ω_{1}^{N} for some larger N ⊇ M. We will generally write ⟨M_{α}, I_{α}, G_{η}, j_{α,β} | α ≤ β ≤ γ, η < γ⟩ for the iteration, or just “j is an iteration” to mean that j is the j_{0,γ} of an iteration, with j(M) for M_{γ}. (As we will see, in some circumstances, if we know M_{0}, M_{γ}, j_{0,γ}, we can (with slight assumptions) recover the full iteration.) We say that the M_{α}’s are iterates of (M, I); (M, I) is iterable if all iterates are well-founded; and (M, I) is an iterable pair if M is a countable transitive model of T_{0}, I a normal ideal on P(ω_{1}) in M, and (M, I) is iterable.

If M is well-founded and M ⊧ “I is precipitous,” then certainly (M, I) is finitely iterable (i.e., its finite-length iterations produce wellfounded models); and in fact, we will show that in this case (M, I) is iterable to any α ∈ Ord^{M}.

The proof of the following lemma is left an exercise (the proof is by induction on the length of the iteration). In a typical application, M is H(κ)^{N} for some suitable κ.

Lemma 4. Suppose that M ∈ N are models of T_{0}, M is closed under ω_{1}- sequences from N, and P(P(ω_{1}))^{M} = P(P(ω_{1}))^{N}. Let I ∈ M be an M-normal ideal on ω_{1}^{M}. Then the following hold.

• for each iteration ⟨M_{α}, I_{α}, G_{η}, j_{α,β} | α ≤ β ≤ γ, η < γ⟩ of (M, I) there is a unique iteration ⟨N_{α}, I_{α}, G_{η}, j_{α,β}^{∗} | α ≤ β ≤ γ, η < γ⟩ of (N, I) such that ∀β ≤ γ, j_{0,β}^{∗}(M) = M_{β}, M_{β} is closed under ω_{1}-sequences from N_{β}, P(P(ω_{1}))^{M_{β}} = P(P(ω_{1}))^{N_{β}}, and j_{α,β}^{∗} ↾_{M_{α}}= j_{α,β}.

• for each iteration ⟨N_{α}, I_{α}, G_{η}, j_{α,β}^{∗} | α ≤ β ≤ γ, η < γ⟩ of (N, I) there is a unique iteration ⟨M_{α}, I_{α}, G_{η}, j_{α,β} | α ≤ β ≤ γ, η < γ⟩ of (M, I) such that ∀β ≤ γ, j_{0,β}^{∗}(M) = M_{β}, M_{β} is closed under ω_{1}-sequences from N_{β}, P(P(ω_{1}))^{M_{β}} = P(P(ω_{1}))^{N_{β}}, and j_{α,β}^{∗} ↾_{M_{α}}= j_{α,β}.

Corollary 5. In the context of Lemma 4, if (M, I) has an ill-founded iterate by an iteration of length α, then so does (N, I).

Lemma 6 below then shows that (M, I) is iterable if N contains ω_{1} (recall that iterations can have length at most ω_{1}, and note that an illfounded iteration of length ω_{1} must be illfounded at some countable stage).

First we fix a coding of elements of H(ω_{1}) by reals. Fix a recursive bijection π : ω × ω → ω, and say X ⊆ ω codes a ∈ H(ω_{1}) if

(tc({a}), ∈) ≅ (ω, {(i, j) | π(i, j) ∈ X}),

where tc(b) denotes the transitive closure of b. Then ∈ and = are Σ_{1}^{1} (as permutations of ω induce different codes for the same object).

Lemma 6. Suppose that N is a transitive model of T_{1}, γ ∈ Ord^{N}, and I is a normal precipitous ideal on ω_{1}^{N} in N. Then any iterate of (N, I) by an iteration of length γ is well-founded.

Proof. It suffices to prove that iterations of the form (H(κ)^{N}, I) produce wellfounded models for all κ ∈ N such that H(κ)^{N} ⊧ T_{0}; for if any iterate of N is ill-founded, then some ordinal in N is large enough to witness this (i.e. sup(rge(f)), where f witnesses ill-foundedness) and by assumption (as N ⊧ T_{1}) this is contained in some H(κ)^{N} that models T_{0}.

Let (γ̄, κ̄, η̄) be the lexicographically minimal triple (γ, κ, η) satisfying (with N) the formula φ(N, γ, κ, η) defined by “H(κ)^{N} ⊧ T_{0} and there exists an iteration of (H(κ)^{N}, I) of length γ which is ill-founded below the image of η”.

Using our fixed coding of elements of H(ω_{1}) by reals there is a Σ_{1}^{1} formula φ′(x, y, z) saying “x codes a model of T_{0} and a normal ideal in the model on the ω_{1} of the model and there exists an iteration of this pair whose length is coded by y and which is illfounded below the image of the element of this model coded by z.”

For all cardinals κ, ρ ∈ N and all ordinals γ, η ∈ N, if ρ ∈ N is larger than |H(κ)|^{N}, |η|^{N} and |γ|^{N}, then there exist reals coding H(κ)^{N}, η, and γ in any forcing extension of N by Coll(ω, ρ). Such an extension is correct about whether these reals satisfy φ′. However, this is a homogeneous forcing extension of N; so there is a formula ψ(γ, κ, η) saying that in every forcing extension in which H(κ) (of the ground model), η and γ are all countable there exist reals coding H(κ), η and γ which satisfy φ′. It follows that that N ⊧ ψ(γ, κ, η) if and only if φ(N, γ, κ, η) holds, and furthermore, for all well-founded iterates N^{∗} of N, and all γ, κ, η ∈ N^{∗}, N^{∗} ⊧ ψ(γ, κ, η) if and only if φ(N^{∗}, γ, κ, η) holds.

Since I is precipitous in N, γ̄ is a limit ordinal, and clearly η̄ is a limit ordinal as well. Fix an iteration ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ̄⟩ of (H(κ̄)^{N}, I) such that j_{0γ̄}(η̄) is not wellfounded, and let ⟨N_{α}, G_{β}, j′_{αδ} : β < α ≤ δ ≤ γ̄⟩ be the corresponding iteration of N as in Lemma 4. By the minimality of γ̄ we have that N_{α} is wellfounded for all α < γ̄. Since Mγ̄ is the direct limit of the iteration leading up to it, we may fix γ^{∗} < γ̄ and η^{∗} < j_{0γ^{∗}}(η̄) such that j_{γ^{∗}γ̄}(η^{∗}) is not wellfounded. By Lemma 4, j′_{γ^{∗},γ̄}(η^{∗}) = j_{γ^{∗},γ̄}(η^{∗}) and j′_{γ^{∗},γ̄}(η̄) = j_{γ^{∗},γ̄}(η̄).

[figure: M_{0}; M_{γ^{∗}}; Mγ̄; η̄; η^{∗}]

But now, N_{γ^{∗}} ⊧ ψ(γ̄ − γ^{∗}, j_{0,γ^{∗}}(κ̄), η^{∗}), γ̄ − γ^{∗} ≤ γ̄, and η^{∗} < j_{0,γ^{∗}}(η̄), contradicting minimality of (j_{0,γ^{∗}}(γ̄), j_{0,γ^{∗}}((κ̄), j_{0,γ^{∗}}(η̄)) in N_{γ^{∗}}.

We note that ZFC does not imply the existence of iterable pairs. However, by Lemma 6, if there exist a normal, precipitous ideal J on ω_{1}, and a measurable cardinal κ with a κ-complete ultrafilter μ, then there exist iterable pairs. The main point here is that if θ > κ is a regular cardinal and X is a countable elementary submodel of H(θ) with κ, J ∈ X, then X can be end-extended below κ by taking γ to be any member of ⋂(X ∩ μ), and letting X[γ] be the set of values f(γ) for all functions f in X with domain κ. Applying this fact ω_{1} many times, we get that the transitive collapse M of X ∩ V_{κ} is a countable model which is a rank initial segment of a model containing ω_{1}. Letting I be the image of J under the transitive collapse, then, (M, I) is an iterable pair. This is a special case of the proof of Lemma 22, and a key point in Woodin’s proof (which appears in Chapter 3 of [9]) that if there exists a measurable cardinal and the nonstationary ideal on ω_{1} is saturated, then CH fails.

If there is a precipitous ideal on ω_{1}, then sharps exist for subsets of ω_{1}, and a countable iterable model will be correct about these sharps. We will work around this today to avoid having to talk about sharps.

Lemma 7. If (M, I) is an iterable pair and A is an element of P(ω_{1})^{M}, then (ω_{1}^{L[A]})^{M} = ω_{1}^{L[A]}.

Proof. Let ⟨M_{α}, I_{α}, G_{η}, j_{α,β} | α ≤ β ≤ ω_{1}, η < ω_{1}⟩ be an iteration of (M, I). The ordinals of M and M_{1} are the same, so L[A]^{M} = L[A]^{M_{1}}. The critical point of j_{1ω_{1}} is greater than ω_{1}^{M}, and thus greater than the ω_{1} of L[A]^{M}. The restriction of j_{1ω_{1}} to L[A]^{M} embeds L[A]^{M} elementarily into L[A]^{M_{ω_{1}}} , which means that L[A]^{M} and L[A]^{M_{ω_{1}}} have the same ω_{1}. Since ω_{1} ⊂ M_{ω_{1}}, (ω_{1}^{L[A]})^{M_{ω_{1}}} = ω_{1}^{L[A]}.

Now we can define ℙ_{max}.

Definition 8. The partial order ℙ_{max} is the set of pairs ⟨(M, I), a⟩ such that

1. M is a countable transitive model of T_{0} + MA_{ℵ_{1}}

2. (M, I) is an iterable pair

3. a ∈ P(ω_{1})^{M} and ∃x ∈ P(ω)^{M} such that ω_{1}^{L[x,a]} = ω_{1}^{M}

ordered by: p < q (where p = ⟨(M, I), a⟩, q = ⟨(N, J), b⟩) if there is some iteration j : (N, J) → (N^{∗}, J^{∗}) such that

1. j ∈ M

2. j(b) = a

3. J^{∗} = N^{∗} ∩ I ( and hence j(ω_{1}^{N}) = ω_{1}^{M})

4. q ∈ H(ω_{1})^{M}

Note that since j ∈ M in definition of the ℙ_{max} order above, N and N^{∗} are both in M as well.

Definition 9. We say that (M, I) is a ℙ_{max} precondition if there exists an a such that ⟨(M, I), a⟩ ∈ ℙ_{max}, or equivalently just if (M, I) satisfies conditions 1 and 2 in the definition of ℙ_{max} conditions above.

Suppose that (M, I) is an iterable pair, and j: (M, I) → (M′, I′) is an iteration. Then for any A ∈ P(ω_{1})^{M} which is bounded in ω_{1}^{M}, j(A) = A. By Lemma 7, it follows then that ω_{1}^{L[A]} < ω_{1}^{M}, since j(ω_{1}^{M}) > ω_{1}^{M} if j is nontrivial. Therefore, the set a from a ℙ_{max} condition ⟨(M, I), a⟩ must always be unbounded in ω_{1}^{M} to make ω_{1}^{L[x,a]} = ω_{1}^{M} possible.

If p_{0} < p_{1} < p_{2} (p_{i} = ⟨(M_{i}, I_{i}), a_{i}⟩), and these are witnessed by j_{1,0}, j_{2,1}, then p_{0} < p_{2} is witnessed by j_{1,0}(j_{2,1}): j_{1,0} ∈ H(ω_{2})^{M_{0}}, j_{2,1} ∈ H(ω_{2})^{M_{1}}; j_{2,1} is an iteration of (M_{2}, I_{2}), and j_{1,0}((M_{2}, I_{2})) = (M_{2}, I_{2}).

Under our fixed coding, “(M, I) is iterable” is Π_{2}^{1} in a code for (M, I) : roughly, “for anything satisfying the first-order properties of being an iteration, either there is no infinite descending sequence in the ordinals of the final model, or there is an infinite descending sequence in the indices of the iteration.” Since iterable models embed elementarily into models containing ω_{1}, they are Π_{2}^{1}-correct. It follows that “(M, I) is iterable” is absolute to iterable models containing a code for (M, I).

So now we see that ℙ_{max} ∈ L(ℝ) — all constructions involved are nicely codable.

3 First properties of ℙ_{max}

The requirement that the models in ℙ_{max} conditions satisfy MA_{ℵ_{1}} is used for a particular consequence of MA_{ℵ_{1}} known as almost disjoint coding [2]. That is, it follows from MA_{ℵ_{1}} that if Z = {z_{α} : α < ω_{1}} is a collection of infinite subsets of ω whose pairwise intersections are finite (i.e., Z is an almost disjoint family), then for each B ⊆ ω_{1} there exists a y ⊆ ω such that for all α < ω_{1}, α ∈ B if and only if y ∩ z_{α} is infinite. This is used to show that if ⟨(M, I), a⟩ is a ℙ_{max} condition, then any iteration of (M, I) is uniquely determined by the image of a (Lemma 10 below), so the order on each comparable pair of conditions is witnessed by a unique iteration.

Lemma 10. Let ⟨(M, I), a⟩ be a ℙ_{max} condition and let A be a subset of ω_{1}. Then there is at most one iteration of (M, I) for which A is the image of a.

Proof. Fix a real x in M such that ω_{1}^{M} = ω_{1}^{L[a,x]}, and let Z = ⟨z_{α} : α < ω_{1}^{M}⟩ be the almost disjoint family defined recursively from the constructibility order in L[a, x] on P(ω)^{L[a,x]} (using a and x as parameters) by letting ⟨z_{i} : i < ω⟩ be the constructibly least partition of ω into infinite pieces, and, for each α ∈ [ω, ω_{1}^{M}), letting z_{α} be the constructibly least infinite z ⊂ ω almost disjoint from each z_{β} (β < α). Suppose that

I = ⟨M_{α}, G_{β}, j_{αδ} : β < α ≤ δ ≤ γ⟩

and

I′ = ⟨M′_{α}, G′_{β}, j′_{αδ} : β < α ≤ δ ≤ γ′⟩

are two iterations of (M, I) such that j_{0γ}(a) = A = j′_{0γ′}(a). Then

j_{0γ}(ω_{1}^{M}) = ω_{1}^{L[x,A]} = j_{0γ}(ω_{1}^{M′})

and j_{0γ}(Z) = j′_{0γ′}(Z) (this uses Lemma 7 to see that the constructibility order on reals in L[A, x] is computed correctly in M_{γ} and M′_{γ′}). Let ⟨z_{α} : α < j_{0γ}(ω_{1}^{M})⟩ enumerate j_{0γ}(Z).

Without loss of generality, γ ≤ γ′. We show by induction on α < γ that, for each such α, G_{α} = G′_{α}. This will suffice. Fix α and suppose that

{G_{β} : β < α} = {G′_{β} : β < α}.

Then M_{α} = M′_{α}. For each B ∈ P(ω_{1})^{M_{α}}, B ∈ G_{α} if and only if ω_{1}^{M_{α}} ∈ j_{α(α+1)}(B), and B ∈ G′_{α} if and only if ω_{1}^{M_{α}} ∈ j′_{α(α+1)}(B). Applying almost disjoint coding, fix x ∈ P(ω)^{M_{α}} such that for all η < ω_{1}^{M_{α}}, η ∈ B if and only if x ∩ z_{η} is infinite. Then B ∈ G_{α} if and only if x ∩ z_{ω_{1}^{Mα}} is infinite if and only if B ∈ G′_{α}.

Lemma 11. (T_{0}) Suppose that (M, I) is an iterable pair, and J is a normal ideal on ω_{1}. Then there exists an iteration j: (M, I) → (M^{∗}, I^{∗}) of length ω_{1} such that I^{∗} = M^{∗} ∩ J.

Proof. Note that I^{∗} ⊆ M^{∗} ∩ J holds for any such ω_{1}-length iteration. To see this, first note that the critical sequence of an iteration of length ω_{1} is a club. Every element B of I^{∗} is j_{α,ω_{1}}(b) for some α < ω_{1} and b ∈ I_{α}. Then for all β ∈ [α, ω_{1}), j_{α,β}(b) ∉ G_{β}, so ω_{1}^{M_{β}} ∉ j_{α,ω_{1}}(b) = B; thus B ∈ NS_{ω_{1}}, but J is normal, so NS_{ω_{1}} ⊆ J.

Conversely, for ⊇: as J is normal, we may let ⟨E_{i}^{α} | α < ω_{1}, i < ω⟩ be a partition of ω_{1} into J-positive pieces. Now, as we construct an iteration, let {e_{i}^{α} | i < ω} enumerate P(ω_{1})^{M_{α}} \ I_{α}, and build each G_{β} in such a way that if ω_{1}^{N_{β}} ∈ E_{i}^{α} for some α ≤ β and i < ω, then j_{α,β}(e_{i}^{α}) is in G_{β}.

Now, for all B ∈ P(ω_{1})^{M_{α}} \ I_{ω_{1}}, ∃α < ω_{1}, i < ω such that B = j_{α,ω_{1}}(e_{i}^{α}) and for all β ∈ [α, ω_{1}), ω_{1}^{M_{β}} ∈ E_{i}^{α} ⇒ ω_{1}^{M_{β}} ∈ j_{0,β+1}(e_{i}^{α}) ⇒ ω_{1}^{M_{β}} ∈ B.

So in particular, we have a club C ⊆ ω_{1} such that C ∩E_{i}^{α} ⊆ B, so B ∉ J.

We may consider this as an iteration game G((M, I), J, B): two players collaborate on building an iteration of (M, I), and play is as follows at each round α:

• if α ∉ B, player I does nothing, and player II chooses G_{α};

• if α ∈ B, player I specifies some element for G_{α}, and player II must choose some G_{α} containing it.

Player I wins if I_{ω_{1}} = M_{ω_{1}} ∩ J. The above proof shows that player I has a winning strategy iff B ∉ J. (More precisely, it shows ⇐; ⇒ is because if B ∈ J then II may choose some I-positive set and keep its images out of every G_{α}.)

The following lemma shows that ℙ_{max} satisfies a homogeneity property strong enough to imply that the theory of the generic extension can be computed in the ground model. Since the existence of a proper class of Woodin cardinals implies that the theory of L(ℝ) is generically absolute, it implies that the theory of the ℙ_{max} extension of L(ℝ) is generically absolute as well.

Lemma 12. Suppose for each x ∈ H(ω_{1}) there exists a ℙ_{max} precondition (M, I) such that x ∈ M. Then ∀p_{0}, p_{1} ∈ ℙ_{max}, ∃q_{0}, q_{1} ∈ ℙ_{max} such that each q_{i} ≤ p_{i}, and ℙ_{max} ↾ q_{0} ≅ ℙ_{max} ↾ q_{1}.

Proof. Take p_{i} = ⟨(M_{i}, I_{i}), a_{i}⟩. Then let (N, J) be a ℙ_{max} precondition such that p_{0}, p_{1} ∈ H(ω_{1})^{N}. Now take j_{i}: (M_{i}, I_{i}) → (M_{i}^{∗}, I_{i}^{∗}) to be iterations in N such that I_{i}^{∗} = M_{i}^{∗} ∩ J (we may do so, by the previous theorem applied in N).

Now set q_{i} = ⟨(N, J), j_{i}(a_{i})⟩ ∈ ℙ_{max}. Certainly these satisfy q_{i} < p_{i} as desired. (To see that these q_{i} are indeed conditions, note that the witnessing x_{i} for p_{i} (i.e., the x ∈ P(ω)^{M_{i}} such that ω_{1}^{L[x,a_{i}]} = ω_{1}^{M_{i}}) still works for q_{i}.) But now ℙ_{max} ↾ q_{0} ≅ ℙ_{max} ↾ q_{1}, for given any r_{0} = ⟨(N′, I′), b⟩ < q_{0}, there is unique j : (N, J) → (N^{∗}, J^{∗}) witnessing this (and we have b = j(j_{0}(a_{0}))); now take r_{0} to r_{1} := ⟨(N′, J′), j(j_{1}(a_{1}))⟩ < q_{1}, also witnessed by j.

Given γ ∈ [ω_{1}, ω_{2}), a canonical function for γ is a function f: ω_{1} → ω_{1} such that for some (equivalently, every) bijection π : ω_{1} → γ, {α < ω_{1} | ot(π[α]) = f(α)} contains a club. In a normal ultrapower context, we then have: f: ω_{1} → Ord, [f]_{Ult} = j(f)(ω_{1}) = ot(π[ω_{1}]) = γ.

Suppose that ⟨M_{α}, I_{α}, G_{η}, j_{α,β} | α ≤ β ≤ ω_{1}, η < ω_{1}⟩ is an iteration of length ω_{1} of some (M, I), and let π : ω_{1} → Ord^{M_{ω_{1}}} be a bijection; then π induces a canonical function g. For club-many α < ω_{1}, ω_{1}^{M_{α}} = α, and also π[α] = j_{α,ω_{1}}[Ord^{M_{α}}] (recall that we take direct limits at limit stages of iterations); so there is a club C ⊂ ω_{1} such that for all α ∈ C, ot(π[α]) = Ord^{M_{α}}. If f ∈ (ω_{1}^{ω_{1}})^{M_{β}}, for some β < α, then for any α ∈ [β, ω_{1}),

j_{β,α+1}(f)(ω_{1}^{M_{α}}) < ω_{1}^{M_{α+1}} < Ord^{M_{α+1}} = Ord^{M_{α}},

which equals g(α) if α ∈ C. It follows that for any f ∈ (ω_{1}^{ω_{1}})^{M_{ω_{1}}} , g(α) > f(α) for club-many α. We will use this fact to show that ℙ_{max} σ-closed (this is an alternate proof avoiding sharps; some of what follows can be done more easily and in more generality with sharps).

Lemma 13. Suppose that for each x ∈ H(ω_{1}) there exists a ℙ_{max} precondition (M, I) such that x ∈ M. If p_{i} ∈ ℙ_{max} (i < ω) are such that ∀i p_{i+1} < p_{i}, then ∃q ∈ ℙ_{max} such that ∀i q < p_{i}.

The proof of Lemma 13 involves some new notions. Say that p_{i} = ⟨(M_{i}, I_{i}), a_{i}⟩, and for each j < i < ω, let k_{i,j} : M_{j} → M_{j}^{∗} ∈ M_{i} be the unique witness for p_{i} < p_{j}. By the uniqueness of witnesses, the k′_{i,j}s commute, so let N_{i}, J_{i} be the images of M_{i}, I_{i} in the limit of the directed system given by the embeddings k_{i,j}. Each (N_{i}, J_{i}) is an iterate of the corresponding (M_{i}, I_{i}) by an iteration of length sup{ω_{1}^{M_{i}} : i < ω}, so each N_{i} is wellfounded. Let b = ⋃_{i<ω} a_{i}. Then:

1. Each (N_{i}, J_{i}) is iterable;

2. for all i, ω_{1}^{N_{i}} = sup_{j<ω} ω_{1}^{M_{j}};

3. i < j ⇒ N_{i} ∈ H(ω_{2})^{N_{j}};

4. for all i, J_{i} = J_{i+1} ∩ N_{i};

5. for each i there exists some iteration j_{i} : (M_{i}, I_{i}) → (N_{i}, J_{i}) in N_{i+1} such that j_{1}(a_{i}) = b (and so in N_{i+1}, there is a canonical function for Ord^{N_{i}} that dominates on a club every member of (ω_{1}^{ω_{1}})^{N_{i}}).

We call a sequence ⟨(N_{i}, J_{i}) : i < ω⟩ satisfying (1) to (5) above a ℙ_{max} limit sequence. An ⟨(N_{i}, J_{i}) | i < ω⟩-normal ultrafilter is a filter G ⊆ ⋃_{i}(P(ω_{1})^{N_{i}} \J_{i}) such that for all i < ω, and for all regressive f ∈ (ω_{1}^{ω_{1}})^{N_{i}} ∃e ∈ G such that f is constant on e. Then we have Ult(⟨(N_{i}, J_{i}) : i < ω⟩, G), i.e. a sequence of models whose jth model [Ult(⟨(N_{i}, J_{i}) : i < ω⟩, G)]_{j} is

{f : ω_{1}^{N_{0}} → N_{j} | f ∈ ⋃{N_{i} | i < ω}}/ =_{G} .

Now we will iterate this operation.

Definition 14. An iteration of ⟨(N_{i}, J_{i}) : i < ω⟩ of length γ is some

⟨⟨(N_{i}^{α}, J_{i}^{α}) : i < ω⟩, G_{η}, j_{α,β} | α ≤ β ≤ γ, η < γ⟩

in which:

• each G_{η} is a langle(N_{i}^{η}, J_{i}^{η}) | i < ω⟩-normal ultrafilter contained in ⋃_{i}(P(ω_{1})^{N_{i}^{η}} \ J_{i}^{η}) such that for each i < ω, j_{η,η+1} ↾ N_{i}^{η} → N_{i}^{η+1} is the induced ultrapower

• the j_{α,β} commute, and for limit β, N_{i}^{β} is the direct limit of N_{i}^{α} (α < β) under j_{α,ρ} ↾ N_{i}^{α} (α ≤ ρ < β).

In an iteration of this form, for each pair i, α, there is in N_{i+1}^{α} an iteration of (M_{i}, I_{i}) of length ω_{1}^{N_{0}^{α}}, with final model N_{i}^{α}. Since each (M_{i}, I_{i}) is iterable, the wellfoundedness of each N_{i}^{α} will follow from the wellfoundedness of ω_{1}^{N_{0}^{α}}.

For each α < γ,

ω_{1}^{N_{0}^{α+1}} = sup{Ord^{N_{i}^{α}} | i < ω}.

To see this, fix for each i ∈ ω a canonical function g_{i} ∈ N_{i+1}^{α} for Ord^{N_{i}}. Then each g_{i} dominates on a club every member of (ω_{1}^{ω_{1}})^{N_{i}^{α}}. The g_{i} are cofinal under mod-NS_{ω_{1}} domination in ⋃_{i}(ω_{1}^{ω_{1}})^{N_{i}^{α}}, and each g_{i} represents an ordinal in N_{i+1}^{α}

in this ultrapower, which shows that ω_{1}^{N_{0}^{α+1}} = sup{Ord^{N_{i}^{α}} | i < ω}. For limit β,

ω_{1}^{N_{0}^{β}} = sup{ω_{1}^{N_{0}^{α}} | α < β}.

It follows that each N_{0}^{α} is wellfounded.

We have shown the following.

Fact 15. All iterations of ℙ_{max} pre-limit sequences give well-founded models.

Again this can be rephrased in terms of games. Let G_{ω}(⟨(N_{i}, J_{i}) | i < ω⟩, I, B) be the game of length ω_{1}, in which I and II collaborate to build an iteration of ⟨(N_{i}, J_{i}) : i < ω⟩ of length ω_{1}, in which at stage α:

• if α ∈ B, player I chooses e ∈ ⋃_{i}(P(ω_{1})^{N_{i}^{α}} \ J_{i}^{α}), and player II chooses G_{α}, a ⟨(N_{i}^{α}, J_{α}^{i}) | i < ω⟩-normal ultrafilter contained in ⋃_{i}(P(ω_{1})^{N_{i}^{α}} \J_{i}^{α}), with e ∈ G_{α};

• if α ∉ B, player I does nothing, and player II chooses any suitable G_{α}.

Player I wins if ∀i j_{0,ω_{1}}(J_{i}) = I ∩ M_{i}^{ω_{1}}. The argument just given (along with the argument for Lemma 11) shows the following.

Lemma 16. Suppose ⟨(N_{i}, J_{i}) | i < ω⟩ is a ℙ_{max} pre-limit-sequence, I is a maximal ideal on ω_{1} and B ⊆ ω_{1}. Then Player I has a winning strategy in G_{ω}(⟨(N_{i}, J_{i}) | i < ω⟩, I, B) if and only if B is not in I.

We now return to the proof of Lemma 13. We have that the limit sequence ⟨(N_{i}, J_{i}) : i < ω⟩ induced by the descending sequence p_{i} (i ∈ ω) is iterable. Fix a ℙ_{max} precondition (M′, I′) such that this sequence is in H(ω_{1})^{M′}. Apply a winning strategy for player I in M′ for G_{ω_{1}}(⟨(N_{i}, J_{i}) | i < ω⟩, I′, ω_{1}) to get an iteration j of ⟨(N_{i}, J_{i}) : i < ω⟩ of length ω_{1}^{M′}. Then for all i < ω, j(j_{i}) witnesses that p_{i} > ⟨(M′, I′), j(b)⟩.

Thus ℙ_{max} forcing is σ-closed, so it does not add any reals; so L(ℝ)^{V^{ℙmax}} = L(ℝ)^{V}.

4 Existence of ℙ_{max} conditions

Definition 17. Given A ⊆ ℝ, and an iterable pair (M, I), we say (N, I) is Aiterable if A ∩ M ∈ M and for any iteration j : (M, I) → (M^{∗}, I^{∗}), j(A ∩ M) = A ∩ M^{∗}.

In this section we will work through a proof of the following existence theorem for ℙ_{max} conditions.

Lemma 18 (Main existence lemma). Suppose there are infinitely many Woodin cardinals below some measurable cardinal, and let A ∈ P(ℝ) ∩ L(ℝ). Then there exists an A-iterable ℙ_{max} precondition (M, I) such that for every set forcing extension M^{+} of M and every precipitous ideal I^{+} ∈ M^{+} on ω_{1}^{M^{+}}, (M^{+}, I^{+}) is A-iterable.

We need to introduce towers of measures and homogeneous tree. See [8] for a detailed discussion of this material

Definition 19. Given Z ≠ ϕ, a tower of measures on Z is a sequence ⟨μ_{i} | i < ω⟩ such that each μ_{i} ⊆ P(Z^{i}) is an ultrafilter, and for all k < i < j and all A ∈ μ_{i}, we have {b ∈ Z^{j} | b↾ i ∈ A} ∈ μ_{j} and {b↾ k | b ∈ A} ∈ μ_{k}.

Such a tower is countably complete if whenever ⟨A_{i} | i < ω⟩ is such that eaach A_{i} ∈ μ_{i}, there is a ∈ Z^{ω} such that ∀i a↾ i ∈ A_{i}.

We note briefly that countable completeness is equivalent to: the direct limit of Ult(V, μ_{i}) is well-founded.

Definition 20. A tree on ω × Z is a set T ⊆ (ω × Z)^{<ω} such that for all i < ω, t ∈ T we have t ↾ i ∈ T. The projection of T is p[T] := {y ∈ ω^{ω} | ∃c ∈ Z^{ω} ∀i < ω (y ↾ i, c↾ i) ∈ T }.

Such a tree is weakly κ-homogeneous (for κ a cardinal) if there exist κcomplete ultrafilters μ_{a,b} ⊆ P(Z^{|a|}) such that ∀a, b ∈ ω^{<ω} with |a| = |b|,

{c ∈ Z^{|a|} | (a, c) ∈ T } ∈ μ_{a,b},

and such that for each x ∈ p[T] there exists a b ∈ ω^{ω} such that ⟨μ_{x↾i,b↾i} | i < ω} is a countably complete tower.

Weakly homogeneous trees originated from work of Kechris, Martin and Solovay. The following fact is due to Woodin. A proof appears in [6].

Fact 21. If δ is a limit of Woodin cardinals and there is a measurable cardinal above δ, then for each A ∈ P(ℝ) ∩ L(ℝ) and γ < δ, there exists a γ-weakly-homogeneous tree T such that p[T] = A.

Lemma 22. Suppose that T ⊂ (ω × Z)^{<ω} is a γ^{+}-weakly-homogeneous tree, θ > (2^{|T |})^{+} is regular, X ≺ H(θ), T, γ ∈ X, |X| < γ, and ā ∈ p[T]. Then there exists Y ≺ H(θ) with X ⊆ Y, X ∩ γ = Y ∩ γ, |X| = |Y |, and ā ∈ p[T ∩ Y].

Proof. Fix ⟨μ_{a,b} | a, b ∈ ω^{<ω}⟩ in X witnessing the γ^{+}-weak-homogeneity of T. Since ā ∈ p[T], ∃b̄ such that ⟨μ_{ā↾i,b̄↾i} | i < ω⟩ is a countably complete tower. Now let A_{i} = ⋂(X ∩ μ_{ā↾i,b̄↾i}). Each A_{i} is in μ_{ā↾i,b̄↾i}; so there is c̄ ∈ Z^{ω} such that ∀i c̄↾ i ∈ A_{i}. Take Y = X[{c̄↾ i | i < ω}] := {f(c̄↾ i) | f ∈ X, dom(f) = Z^{<ω}, i < ω}.

Elementarity of Y follows from an argument similar to the proof of Łós’s Theorem (see Theorem 1.1.13 of [6]). To see that Y ∩ γ = X ∩ γ, note that if α ∈ Y ∩ γ, then α = f(c ↾ i) for some f ∈ X, domf = Z^{i}; but μ_{a↾i,b↾i} is γ^{+}-complete, so f is constant on a set in μ_{a↾i,b↾i}. But then this constant value is in X, and f(c↾ i) is this value, since c↾ i = ⋂(μ_{a↾i,b↾i} ∩ X).

Note that Y in the proof above is in some sense a limit ultrapower of the transitive collapse of X.

The following was first proved by Foreman, Magidor and Shelah from a supercompact cardinal, and later improved by Woodin.

Fact 23. If δ is Woodin, then Coll(ω_{1}, < δ) forces that NS_{ω_{1}} is presaturated, and hence precipitous.

Recall that an ideal I on ω_{1} is presaturated if for every sequence of maximal antichains {Q_{i} | i < ω} ⊂ P(ω_{1}) \ I, ∀A ∈ I^{+}, ∃B ⊆ A, B ∈ I^{+} such that for all i < ω,

|{E ∈ Q_{i} | E ∩ B ∈ I^{+}}| ≤ ℵ_{1}.

The following was proved by Kakuda and Magidor independently [3, 7].

Fact 24. Any c.c.c. forcing preserves that NS_{ω_{1}} is precipitous.

Recall the hypotheses of main existence lemma: δ is a limit of Woodin cardinals, there exists a measurable cardinal greater than δ, and A is in P(ℝ) ∩ L(ℝ). To prove the lemma, let κ be the least Woodin cardinal, and γ the least strong inaccessible above κ. Fix γ^{+}-weakly-homogeneous trees S, T, with p[S] = A, p[T] = ℝ\A. Fix a regular θ > (2^{|S|})^{+}, (2^{|T |})^{+}. Let X be a countable elementary submodel of H(θ), with S, T, γ, κ ∈ X. Repeatedly apply Lemma 22 above to obtain Y ≺ H(θ) such that X ⊆ Y, X ∩ γ = Y ∩ γ, A = p[S ∩ Y], ℝ ∩ A = p[T ∩ Y]. (Then |Y ∩ Ord| = 2^{ω}.) Now let N be the transitive collapse of Y, and let S̄, T̄, γ̄, κ̄ be the images of S, T, γ, κ therein. Let h be N-generic for Coll(ω_{1}, < κ̄) followed by a c.c.c. poset of size 2^{ω_{1}} to make MA_{ℵ_{1}} hold. Then N[h] ⊧ MA_{ℵ_{1}} + “NS_{ω_{1}} is precipitous”. Then N[h] is iterable, by Lemma 6. Let M be (Vγ̄)^{N[h]}, and let j : (M, NS_{ω_{1}}^{M}) → (M^{∗}, NS_{ω_{1}}^{∗}) be an iteration. By Lemma 4, this induces an iteration of (N[h], NS_{ω_{1}}^{N[h]}) with final model (N^{∗}, I^{∗}) (which we’ll also call j). Now, N^{∗} is well-founded, and p[S̄] ⊆ p[j(S̄)] and p[T̄] ⊆ p[j(T̄)]. But by elementarity, N^{∗} ⊧ p[j(S̄)] ∩ p[j(T̄)] = ∅, and since N^{∗} is well-founded it is correct about this. Then p[S̄] = p[j(S̄)] and p[T̄] = p[j(T̄)], so j(A ∩ M) = p[j(S̄)] ∩ M^{∗} = A ∩ M^{∗}.

Remark 1. Instead of Coll(ω_{1}, < κ̄), we could have taken h to be N-generic for any poset in V N ; γ̄ such that N[h] ⊧ “∃ precipitous I on ω_{1}”, and the rest of the proof would have still gone through.

The main existence lemma gives not only A-iterable preconditions for any A ∈ P(ℝ) ∩ L(ℝ), but also A-iterable preconditions containing any given real x, for any A ∈ P(ℝ) ∩ L(ℝ), applying the lemma to the set {y ⊕ x | y ∈ A}. Thus we have shown: if there exist infinitely many Woodin cardinals below a measurable, then ∀x ∈ ℝ, ∀A ∈ P(ℝ) ∩ L(ℝ), there exists some ℙ_{max} condition ⟨(M, I), a⟩, with x ∈ M, and (M, I) A-iterable.

We didn’t quite show ⟨H(ω_{1})^{M}, ∈, A ∩ M⟩ ≺ ⟨H(ω_{1}), ∈, A⟩. We can do this using A^{♯}, or by using not just S, T as above but similar trees for all sets projective in A. We omit this for now.

Given a filter G ⊂ ℙ_{max}, A_{G} denotes the set ⋃{e | ∃⟨(N, J), e⟩ ∈ G}. We also omit a proof of the following:

Fact 25 (“The combinatorial heart of the ℙ_{max} analysis”). Suppose that for each A ∈ P(ℝ) ∩ L(ℝ) there exists an A-iterable ℙ_{max} precondition (N, I) such that

⟨H(ω_{1})^{M}, ∈, A ∩ M⟩ ≺ ⟨H(ω_{1}), ∈, A⟩,

and suppose that G ⊆ ℙ_{max} is an L(ℝ)-generic filter. Then ∀B ∈ P(ω_{1})^{L(ℝ)[G]}, ∃⟨(M, I), a⟩ ∈ G such that B = j(b) for some b ∈ P(ω_{1})^{M}, where j is the unique iteration of (M, I) satisfying j(a) = A_{G}.

In other words, all subsets of ω_{1} in extensions come from models in the conditions, and L(ℝ)[G] = L(ℝ)[A_{G}].

Corollary 26. Suppose that for each A ∈ P(ℝ)∩L(ℝ) there exists an A-iterable ℙ_{max} precondition (N, I) such that

⟨H(ω_{1})^{M}, ∈, A ∩ M⟩ ≺ ⟨H(ω_{1}), ∈, A⟩,

and suppose that G ⊆ ℙ_{max} is an L(ℝ)-generic filter. Then NS_{ω_{1}}^{L(ℝ)[G]} is the collection of all sets of the form j(e), where for some ⟨(M, I), a⟩ ∈ G, e ∈ I, and j is the iteration of (M, I) sending a to A_{G}.

Woodin has shown that the hypotheses of Fact 25 are equivalent to the assertion that AD holds in L(ℝ).

5 Π_{2} maximality

Proof of Goal 1 So now fix some Π_{2} sentence φ = ∀x∃yψ(x, y) (in the extended language with two new unary predicates), and some A ∈ P(ℝ) ∩ L(ℝ). To show that

⟨H(ω_{2}), ∈, A, NS_{ω_{1}}⟩^{L(ℝ)^{ℙmax}} ⊧ φ,

it is sufficient to show that for each ⟨(M, I), a⟩ ∈ ℙ_{max} and each b ∈ H(ω_{2})^{M}, there exist ⟨(N, NS_{ω_{1}}^{N}), e⟩ ∈ ℙ_{max} and j : (M, I) → (M^{∗}, I^{∗}) in N such that j(a) = e, I^{∗} = M^{∗} ∩ NS_{ω_{1}}^{N}, and

⟨H(ω_{2})^{N}, ∈, A ∩ N, NS_{ω_{1}}^{N}⟩ ⊧ ∃d ψ(j(b), d).

The argument is like the one for existence of conditions.

So suppose ⟨(M, I), a⟩ is given. Fix P forcing φ. Let δ be the least Woodin cardinal with p ∈ V_{δ}; let κ be the least strong inaccessible above δ. Let S, T be κ^{+}-weakly-homogeneous trees projecting to A, ℝ \ A. Let θ > (2^{|S|})^{+}, (2^{|T |})^{+} be regular. Fix Y ≺ H(θ) with Y ∩ κ countable, p[S ∩ Y] = A, p[T ∩ Y] = ℝ \ A and ⟨(M, I), a⟩ ∈ Y.

Let N be the transitive collapse of Y, and let P̄,S̄,δ̄,κ̄ be the respective images of P,S,δ,κ under this collapse. Let h_{0} be P̄-generic for N. Note that since P ∈ V_{δ}, δ̄ remains Woodin in N[h_{0}]. The reinterpretation of A is the projection of S̄ in the extension. Thus

⟨H(ω_{2})^{N[h_{0}]}, ∈, (p[S̄])^{N[h_{0}]}, NS_{ω_{1}}^{N[h_{0}]}⟩ ⊧ φ.

Pick an iteration j of (M, I) in N such that j(I) = j(M) ∩ NS_{ω_{1}}^{N[h_{0}]}. Then there exists a d ∈ H(ω_{2})^{N[h_{0}]} such that

⟨H(ω_{2})^{N[h_{0}]}, ∈, (p[S̄])^{N[h_{0}]}, NS_{ω_{1}}^{N[h_{0}]}⟩ ⊧ ψ(j(b), d).

Let h_{1} be N[h_{0}]-generic for Coll(ω_{1}, < δ̄)^{N[h_{0}]} followed by some c.c.c. forcing making MA_{ℵ_{1}} hold. Now ⟨((Vκ̄)^{N[h_{0}][h_{1}]}, NS_{ω_{1}}^{N[h_{0}][h_{1}]}), j(a)⟩ is the desired condition. ■

6 Discussion

Question 1. You’ve shown that under these conditions, any forceable Π_{2} statement must hold in the ℙ_{max} extension. Can you give us some cool examples?

Answer. One example is φ_{AC}: “For every stationary, costationary A, B ⊆ ω_{1}, there is some γ ∈ [ω_{1}, ω_{2}), some bijection π : ω_{1} → γ such that

{α < ω_{1} | α ∈ A ↔ (π[α]) ∈ B}

contains a club.” This can be used to get an injection P(ω_{1}) ↪ ω_{2}, which shows that the Axiom of Choice holds in the ℙ_{max} extension of L(ℝ).

Also, in some cases one can use ℙ_{max} to get Π_{2} maximality relative to a given Σ_{2} statement; that is, for a given Σ_{2} statement for H(ω_{2}), you can simultaneously get all Π_{2} statements forceably consistent with it.

Another useful aspect: often, the combinatorics of forcing to kill off one thing while preserving another are not clear; the combinatorics of doing the same by an iteration may be much clearer. For instance, an analysis of iterations may help answer the question of whether there exists a Dowker space on ω_{1}.

References

[1] Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s Maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988.

[2] Ronald B. Jensen and Robert M. Solovay. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pages 84–104. North-Holland, Amsterdam, 1970.

[3] Yuzuru Kakuda. On a condition for Cohen extensions which preserve precipitous ideals. J. Symbolic Logic, 46(2):296–300, 1981.

[4] Akihiro Kanamori. The higher infinite. Springer Monographs in Mathematics. Springer-Verlag, Berlin, second edition, 2003. Large cardinals in set theory from their beginnings.

[5] Paul B. Larson. Forcing over models of determinacy. In Hanbook of Set Theory. Foreman, Magidor, eds., to appear.

[6] Paul B. Larson. The stationary tower, volume 32 of University Lecture Series. American Mathematical Society, Providence, RI, 2004. Notes on a course by W. Hugh Woodin.

[7] Menachem Magidor. Precipitous ideals and Σ_{4}^{1} sets. Israel J. Math., 35(1-2):109–134, 1980.

[8] Donald A. Martin and John R. Steel. A proof of projective determinacy. J. Amer. Math. Soc., 2(1):71–125, 1989.

[9] W. Hugh Woodin. The axiom of determinacy, forcing axioms, and the nonstationary ideal, volume 1 of de Gruyter Series in Logic and its Applications. Walter de Gruyter & Co., Berlin, 1999.

Department of Mathematics Miami University Oxford, Ohio 45056 United States larsonpb@muohio.edu