The rôle of absoluteness and correctness

Ralf Schindler

http://www.logic.univie.ac.at/^{∼}rds/

Barcelona, Sept 03, 16 thru 19

Outline.

1^{st} and 2^{nd} talk: We’ll consider a natural context in which the study of correctness and absoluteness arises.

3^{rd} talk: Absoluteness w.r.t. certain classes of forcings.

4^{th} talk: Bounded forcing axioms.

You’ll see 26 theorems and a couple of lemmas. Many important theorems will not be stated.

You’ll be asked to solve 7 exercises.

Let M be a transitive model of ZFC. We say that M is Σ_{n}^{1} correct (in V) iff

M ≺_{Σ_{n}^{1}} V ,

i.e., for any Σ_{n}^{1} formula φ and for all reals x⃗ ∈ M,

M ⊧ φ(x⃗) ⇔ V ⊧ φ(x⃗).

Examples: Every M is Σ_{1}^{1} correct (by the absoluteness of well-foundedness). If ω_{1} ⊂ M, then M is Σ_{2}^{1} correct (Shoenfield). In fact: If for any real x ∈ M, x^{#} (from the point of view of V!) is in M, then M is Σ_{2}^{1} correct.

Again let M be a transitive model of ZFC. We say that M is (2-step) Σ_{n}^{1} absolute iff for all posets ℙ ∈ M and for all posets ℚ ∈ M^{ℙ},

M^{ℙ} ≺_{Σ_{n}^{1}} M^{ℙ∗ℚ},

i.e., for any Σ_{n}^{1} formula φ and for all reals x⃗ ∈ M^{ℙ},

M^{ℙ} ⊧ φ(x⃗) ⇔ M^{ℙ∗ℚ} ⊧ φ(x⃗).

Example: By Shoenfield, any M is Σ_{2}^{1} absolute.

At the third level, absoluteness may be characterized in terms of a large cardinal assumption.

Theorem 1 (Martin-Solovay, Woodin) The following are equivalent.

(1) V is Σ_{3}^{1} absolute.

(2) V is closed under sharps, i.e., X^{#} exists for all X ∈ V.

In what follows I’ll always think of X^{#} as an Xmouse, i.e., as an iterable X-premouse which is a(n amenable) structure of the form

⟨J_{α}[X]; ∈, X, U⟩.

Here, X should be a set of ordinals. J_{α}[X] is supposed to have a largest cardinal, κ, say. U is supposed to be a κ-complete normal measure on κ from the point of view of J_{α}[X].

An X-premouse is an X-mouse iff it is iterable.

Proof of Theorem 1.

(2) ⇒ (1): We aim to avoid the “Martin-Solovay tree.” We’ll consider a different tree instead. Let φ be Π_{2}^{1}, and let x be a real in V^{ℙ} such that V^{ℙ} ⊧ φ(x). Let X be such that ℙ ∈ X^{#} . Consider the tree T searching for a 5-tuple ⟨N, ℚ, g, y, π⟩ such that

(a) N is a ctble. transitive premouse,

(b) ℚ ∈ N is a poset,

(c) g is ℚ-generic over N,

(d) y is a real in N[g] such that N[g] ⊧ φ(y),

(e) π: N → X^{#} is elementary, where π(ℚ) = ℙ.

T is illfounded in V^{ℙ} (!).

Therefore, T is ill-founded in V as well. Let ⟨N, ℚ, g, y, π⟩ ∈ V be given by a branch thru T.

Then N[g] ∈ V is a (ctble.) Y -premouse for some Y, and y is a real in N[g] with N[g] ⊧ φ(y).

The point is that N is actually iterable, due to the existence of π. This gives that N[g] is also iterable (i.e., is a Y -mouse), as the iteration of N “lifts” to an iteration of N[g]. But then N[g] ⊧ φ(y) implies V ⊧ φ(y).

We have shown that if φ(−) has a solution in V^{ℙ} then φ(−) already has a solution in V. As generic extensions of V are closed under sharps (if V is closed under sharps) this suffices.

(1) ⇒ (2): This exploits Jensen’s Covering Lemma.

Theorem 2 (Jensen). Suppose that 0^{#} does not exist. Then for each X ⊂ OR there is some Y ∈ L such that Y ⊃ X and Card^{V} (Y) ≤ Card^{V} (X) + ℵ_{1}. In particular, κ^{+L} = κ^{+V} whenever κ is a singular cardinal (in V).

Now suppose that X^{#} does not exist.

Let κ be such that κ^{+L[X]} = κ^{+} .

In V^{Col(ω,κ)}, there is a real x (coding X) such that:

(*) for all ctble. ordinals ξ, L[x] ⊧ ξ is ctble.

(*) is no longer true in V^{Col(ω,κ^{+} )}, of course. However, (*) is a Π_{3}^{1} statement about x. Contradiction!

For higher levels, we have the following.

Theorem 3 (Woodin, Hauser). The following theories are equiconsistent.

(1) ZFC+ V is Σ_{n+4}^{1} absolute.

(2) ZFC + there are n + 1 strong cardinals.

Proof of Theorem 3.

(2) ⇒ (1): This is shown by exploiting a theorem of Woodin on propagating universal Baireness.

Definition (Feng-Magidor-Woodin). A ⊂ ℝ is κ universally Baire if there are trees T, U such that

(a) A = p[T], and

(b) p[U] = ℝ \ p[T] in any V^{ℙ}, where ℙ ∈ H_{κ}.

A is universally Baire iff A is κ universally Baire for all κ.

Theorem 4 (Woodin). Suppose that A ⊂ ℝ × ℝ is κ universally Baire as witnessed by T, U. Suppose that λ is a strong cardinal, where 2^{2^{λ}} < κ. Then in V^{Col(ω,2^{2λ})} , the projection of A is κ universally Baire.

Now let κ_{1} < ⋯ κ_{n+1} be strong cardinals. It is easy to show via an induction, using Theorem 4, that in

V^{Col(ω,2^{2κn+1} )} ,

Σ_{n+4}^{1} absoluteness holds.

(1) ⇒ (2):

Theorem 5 (Steel; 90). Suppose that V is closed under sharps, but there is no inner model with a Woodin cardinal. Then the core model exists.

The core model, K, is an inner model of the form L[E], where E codes a sequence ⟨E_{α}; α⟩ of (partial) extenders.

E_{α} codes an embedding

J_{α}[E ↾ α] → N,

the ultrapower of J_{α}[E ↾ α] by E_{α}.

Theorem 6 (Mitchell, Schimmerling, Steel). Suppose that V is closed under sharps, but there is no inner model with a Woodin cardinal. If β ≥ ℵ_{2} then cf^{V} (β^{+K} ) ≥ Card^{V} (β). In particular, κ^{+K} = κ^{+V} whenever κ is a singular cardinal (in V).

Theorem 7 (Hauser). Suppose that β < ω_{1}^{V} is either a limit cardinal or a double successor cardinal in K, and there are at most n strong cardinals in K||β. Then K||β^{+K} is Σ_{n+3}^{1} (x) in the codes for any real x coding K||β.

That K||β^{+K} is Σ_{m}^{1}(y) in the codes means that

{z ∈ ℝ; z codes some M ◁ K||β^{+K} }

is Σ_{m}^{1}(y).

Now suppose V is Σ_{n+4}^{1} absolute, but K has at most n strong cardinals.

Let κ be a singular cardinal such that there are at most n strong cardinals in K||κ.

In V^{Col(ω,κ)}, there is then a real x (coding K||κ), such that

(*) for all ctble. ordinals ξ, K||κ^{+} [x] ⊧ ξ is ctble.

By Theorem 7, this statement is Π_{n+4}^{1} in x, and it can no longer be true in V^{Col(ω,κ^{+} )}. Contradiction!

The following theorem says that the study of models which are correct and/or absolute arises in natural contexts.

Theorem 8 (Woodin, Steel, Sch). The following theories are equiconsistent.

(1) ZFC + every projective set of reals is Lebesgue measurable and has the property of Baire, and every projective binary relation can be uniformized by a projective function.

(2) ZFC + for every integer n, there is a transitive model M_{n} of ZFC of height ω_{1} such that M_{n} is Σ_{n}^{1} correct and Σ_{n}^{1} absolute.

(3) ZFC + there is an inner model W and there is a strictly increasing sequence ⟨κ_{n}; n < ω⟩ of ctble. ordinals such that for each n, κ_{n} is < ω_{1}^{V} - strong in W.

We actually have (1) ⇒ (2) (Woodin) and (2) ⇒ (3) (Sch). The implication from (3) to (1) is only consistency-wise (Steel).

Proof of Theorem 8.

(1) ⇒ (2): Let A ⊂ ℝ × ℝ. A can be ∗∗- uniformized iff there is a Borel function f: ℝ → ℝ such that for comeager many x,

∃y⟨x, y⟩ ∈ A ⇒ ⟨x, f(x)⟩ ∈ A.

By (1), every projective A can be ∗∗-uniformized.

Lemma. Suppose (1). In fact, suppose that every projective A ⊂ ℝ×ℝ can be ∗∗-uniformized. If x is a Cohen real then for every n, V is Σ_{n}^{1} correct in V [x].

Proof by induction on n. Suppose V [x] ⊧ ∃z∀yφ(z, y), where φ is Σ_{n−2}^{1}. Let

p | |− ∀yφ(τ, y).

We can think of τ as a Borel function such that V [x] ⊧ ∀yφ(τ(x), y).

Assume ∃z∀yφ(z, y) is false in V. Then by ∗∗- uniformizability there’s a Borel function f such that

A = {z; ¬ φ(τ(z), f(z))}

is comeager. A contains a comeager Borel subset B by (1).

Have, in V,

∀z ∈ B ¬ φ(τ(z), f(z)),

which is Π_{n−2}^{1}, and hence true in V [x] by inductive hypothesis. I.e., in V [x],

∀z ∈ B ¬ φ(τ(z), f(z)).

As x ∈ B (as x is a Cohen real),

V [x] ⊧ ¬ φ(τ(x), f(x)).

Contradiction!

Similarily:

Lemma. Suppose (1). If x is a random real then for every n, V is Σ_{n}^{1} correct in V [x].

Theorem 9 (Shelah;81). If every projective set of reals is Lebesgue measurable then there is no unctble. projective sequence of distinct reals.

We can now construct the models witnessing (2). Fix n.

M_{n} will be of the form

L(F) ∩ H_{ω_{1}},

where F: H_{ω_{1}} → H_{ω_{1}} is a partial “Skolem function.”

Let U ⊂ ℝ × ℝ be a universal Π_{n−1}^{1} set, and let f be a projective function uniformizing U. We may and shall assume that for each φ in Π_{n−1}^{1}, f induces a Skolem function f_{φ}, i.e.,

∃xφ(x, y⃗) ⇒ φ(f_{φ}(y⃗), y⃗),

where x,y⃗ range over reals. This is a projective statement and therefore also holds in V [x] where x is a Cohen real.

Let ℙ ∈ H_{ω_{1}} be a nontrivial poset. Basically, ℙ is Cohen forcing. Let x be ℙ generic over V, let φ be Π_{n−1}^{1}, and suppose

V [x] ⊧ ∃xφ(x, y⃗).

Therefore,

V [x] ⊧ φ(f_{φ}(y⃗), y⃗).

Let τ be a canonical term for f_{φ}(y⃗), depending on terms σ⃗ for y⃗. Set

F (φ, Q, σ⃗) = τ.

We have that F is projective in the codes.

Set M_{n} = L(F) ∩ H_{ω_{1}}.

Claim 1. M_{n} ⊧ ZF C.

Proof: otherwise there’d be an unctble. projective sequence of distinct reals.

Claim 2. M_{n} is Σ_{n}^{1} correct. Proof: By the construction of M_{n}. Claim 3. M_{n} is Σ_{n}^{1} absolute. Proof: Let ℙ∗ℚ̇ ∈ M_{n}. M_{n}^{ℙ} is Σ_{n}^{1} correct in V^{ℙ}, and M_{n}^{ℙ∗ℚ̇} is Σ_{n}^{1} correct in V^{ℙ∗ℚ̇} (by the proof of Claim 2). Also, V^{ℙ} is Σ_{n}^{1} correct in V^{ℙ∗ℚ̇}. Hence M_{n}^{ℙ} is Σ_{n}^{1} correct in M_{n}^{ℙ∗ℚ̇}.

(2) ⇒ (3):

The model W will be a “core model.”

Definition (Sch;00). 0^{|^{•}} (“zero hand grenade”) is the “sharp” for an inner model with a proper class of strong cardinals.

We may assume that 0^{|^{•}} does not exist.

The following result partially strengthens Theorem 5.

Theorem 10 (Sch;00). If 0^{|^{•}} does not exist then the core model exists.

(2) ⇒ (3) will be shown by comparing K of V with K of M_{n} for n < ω.

Theorem 11 (Sch;97). Suppose that there are n strong cardinals in K ∩ H_{ω_{1}^{V}}. Then:

(a) If ω_{1}^{V} is inaccessible in K then K||ω_{1}^{V} is Δ_{n+5}^{1} in the codes.

(b) If ω_{1}^{V} is the cardinal successor of δ in K then there is some δ̃, δ < δ̃ < ω_{1}^{V} such that K||ω_{1}^{V} is Δ_{4}^{1}(x) in the codes for any real x coding K||δ̃.

That K||ω_{1}^{V} is Δ_{m}^{1}(y) in the codes means that

{z ∈ ℝ; z codes some M ◁ K||ω_{1}^{V} }

is Δ_{m}^{1}(y).

Now suppose that K ∩ H_{ω_{1}^{V}} has only finitely many strong cardinals.

Then ω_{1}^{V} must be inaccessible in K, because otherwise, by Theorem 11 (b), there would be an unctble. projective sequence of distinct reals.

Theorem 11 readily implies

Corollary. There is a Π_{n+5}^{1} formula φ such that inside any transitive model of ZFC + ¬0^{|^{•}}, for all reals x, φ(x) if and only if x codes an ordinal α such that α is the n^{th} strong cardinal of K ∩ H_{ω_{1}^{V}}.

Let n − 1 = # of strong cardinals in K ∩ H_{ω_{1}^{V}}.

Look at M_{m}, m ≥ n + 4. It is a model of Σ_{n+4}^{1} absoluteness. By Theorem 7, K^{M_{m}} therefore has (at least) n many strong cardinals. Let α be the n^{th} strong cardinal of K^{M_{m}}. W.l.o.g., α^{++K} < ω_{1} in M_{m}, as otherwise we may replace M_{m} by M_{m}^{Col(ω,α^{++} )} .

Let φ be as in the previous Corollary. Let x ∈ ℝ ∩ M_{m} code α. Then φ(x) holds in M_{m}, and hence in V. Contradiction!

CON(3) ⇒ CON(1): W.l.o.g., 0^{|^{•}} does not exist. First build K inside W. One can verify that K||ω_{1}^{V} then has infinitely many strong cardinals. Let κ_{1} < κ_{2} < ⋯ be the first ω many of them. Set κ = sup_{n}κ_{n}, and

M = K^{Col(ω,κ)}.

The claim is that M is a model of (1).

Claim 1 (Woodin). In M, every projective set of reals is Lebesgue measurable and has the Baire property.

Proof. Let A ⊂ ℝ be Σ_{n+3}^{1} . By Theorem 4, there are

T, U ∈ K^{Col(ω,κ_{n+1}^{++} )} =_{def} M̃_{n+4}

such that in M (!), A = p[T] and ℝ \ A = p[U]. From the point of view of M, M̃_{n+4} has ctbly. many reals, so that in particular the set of reals that are not random over M̃_{n+4} is null and the set of reals that are not Cohen generic over M̃_{n+4} is meager.

By Solovay, A is hence Lebesgue measurable and has the Baire property.

Remark. Cutting off M̃_{n+4} at κ^{+} = ω_{1}^{M} gives a model M_{n+4} as in (2).

Claim 2 (Steel). In M, every projective A ⊂ ℝ × ℝ can be uniformized by a projective function F.

Proof. Let B ⊂ ℝ^{3} be a universal Π_{n}^{1} set.

Let g ∈ M be Col(ω, κ_{n−2}^{++}) generic over K. g is basically a real, and it will be an inevitable parameter in the definition of F.

Let T, U ∈ K[g] witness p[T] is κ^{+} universally Baire in K[g] and B = p[T] in M.

Let A ⊂ ℝ×ℝ be Π_{n}^{1}(z_{0}), A = {(x, y); (x, y, z_{0}) ∈ p[T]}, say.

Definition. A premouse M is (z_{0}, x) good iff it is least in the mouse order such that

(a) M ▷ K||κ_{n−1}^{+},

(b) the phalanx ⟨⟨K, M⟩, κ_{n−1}^{+}⟩ is iterable,

(c) M has a top extender, Ḟ^{M}, with critical point κ_{n−1}, and

(d) if π: M →Ḟ^{M} M̃ and π̃: M[g] → M̃[g] is the “lifting” of π then there is some real y such that

(x, y, z_{0}) ∈ p[π̃(T ↾ κ_{n−1})].

The set of (z_{0}, x) good premice is Π_{n+1}^{1} (z_{0}, x, g) in the codes (uniformly in z_{0}, x).

The key fact is that there is some y with (x, y) ∈ A iff there is some y with (x, y, z_{0}) ∈ p[T] iff there is a (z_{0}, x) good premouse.

Definition. Let M be (z_{0}, x) good. We write (y, α⃗)^{M} for the leftmost witness to

∃y (x, y, z_{0}) ∈ p[π̃(T ↾ κ_{n−1})].

M is stable iff for no iterate M^{∗} of M strictly above κ_{n−1},

(y, α⃗)^{M^{∗}} <_{lex} (y, α⃗)^{M}.

The set of all stable (z_{0}, x) good premice M is Π_{n+2}^{1} (z_{0}, x, g) in the codes (uniformly in z_{0}, x).

An iteration argument shows that if there is a (z_{0}, x) good premouse then there is a stable (z_{0}, x) good premouse.

We may now uniformize A as follows.

Set F (x) = the y such that there is some stable (z_{0}, x) good premouse M and there is α⃗ such that

(y, α⃗) = (y, α⃗)^{M}.

F is Δ_{n+3}^{1} (z_{0}, g).

I.e., any Π_{n}^{1}(z_{0}) relation can be uniformized by a Δ_{n+3}^{1} (z_{0}, g) function.

Steel’s Con(3) ⇒ Con(1) implies:

Corollary. The statement that every projective set of reals is Lebesgue measurable and has the property of Baire and every projective subset of ℝ^{2} can be uniformized by a projective function is not equivalent with projective determinacy.

In fact, said statement, call it φ, is not equivalent with Γ determinacy for any Γ which is a “level of Π_{2}^{1}.” (If so, Γ determinacy holds in M_{4}, by correctness. But M_{4} need not satisfy φ. Contradiction.)

In contrast:

Theorem 12 (Woodin). Suppose V = L(ℝ). Suppose that every set of reals is Lebesgue measurable and has the property of Baire and that every Σ_{1}^{2} subset of ℝ^{2} can be uniformized by a Σ_{1}^{2} function. Then the Axiom of Determinacy holds.

The proof uses Woodin’s “core model induction.”

Theorem 13 (Steel; 93) Suppose that κ < Ω are measurable cardinals. Suppose further that there is no inner model with a Woodin cardinal, and let K denote the core model of height Ω. Then K is Σ_{3}^{1} correct.

Proof of Theorem 13. We give a proof which is simpler than Steel’s original one. Again, we’ll exploit a tree argument.

Let φ be Π_{2}^{1}, and let x be a real in V such that V ⊧ φ(x).

Consider the tree T searching for a 6-tuple ⟨N, κ̄, Ω̄ , y, π, K̄⟩ such that

(a) N is a ctble. transitive model of ZF C^{−}+κ̄ < Ω̄ are measurable cardinals,

(b) y is a real in N such that N ⊧ φ(y),

(c) K̄ is the core model of N of height Ω̄

(c) π: K̄ → K is elementary. T is illfounded in V (!). Therefore, T is ill-founded in K as well. (Basically, T ∈ K.)

Let ⟨N, κ̄, Ω̄ , y, π, K̄⟩ ∈ K be given by a branch thru T.

Then K̄ is iterable, due to the existence of π. The point is that this yields that N is (linearily) iterable by any measure on κ̄ and its images. (If σ: N → ult(N; μ), where μ is such a measure, then K^{ult(N;μ)} is an iterate of K^{N} with iteration map σ ↾ K^{N}.) But then N ⊧ φ(y) implies that V ⊧ φ(y).

Corollary. Suppose that for any real x, x^{†} exists. Suppose further that there is no inner model with a Woodin cardinal. Then every binary Π_{2}^{1} (and hence Σ_{3}^{1}) relation can be uniformized by a Σ_{3}^{1} function.

Proof. Let φ be Π_{2}^{1}.

Let x, y be reals such that φ(x, y). Set z = x ⊕ y.

By the proof of the preceding theorem,

K_{x}^{z^{†}} ≺_{Σ_{3}^{1}} z^{†}.

(I’m thinking here of z^{†} as a z-mouse in much the same way as in the case of sharps.) Therefore,

K_{x}^{z^{†}} ⊧ ∃ȳφ(x, ȳ).

By an absolutess argument, there is hence an iterable x-premouse M_{x} containing a solution to φ(x, −).

We may now define F (x) to be the “least” solution to φ(x, −) which exists in an iterable x-premouse (if some solution exists at all). As “iterability” is Π_{2}^{1} (at this level), F is Σ_{3}^{1}.

At most one of the pointclasses Σ_{3}^{1}, Π_{3}^{1} can have the uniformization property.

A slight refinement (+ the determinacy proof of Martin-Steel) then shows:

Corollary. Suppose that every projective set of reals is Lebesgue measurable and has the property of Baire. Suppose further that every binary Π_{3}^{1} relation can be uniformized by a Π_{3}^{1} function. Then Π_{2}^{1} determinacy holds.

Exercise 1. Suppose that ¬ 0^{|^{•}}. Suppose further that in K there is a strictly increasing sequence of cardinals ⟨κ_{n}; n < ω⟩ with supremum κ such that each κ_{n} is κ^{+} strong. Find a forcing extension of K in which every projective set of reals is Lebesgue measurable and has the Baire property and every lightface projective subset of ℝ × ℝ can be uniformized by a lightface projective function.

Exercise 2. Suppose that every projective set of reals is Lebesgue measurable and has the Baire property and every Π_{2n+1}^{1} subset of ℝ×ℝ can be uniformized by a Π_{2n+1}^{1} function. Show that Projective Determinacy holds.

The following techniques are probably relevant for solving Exercise 2.

Suppose that the family E ⊂ V_{δ} of extenders witnesses that δ is a Woodin cardinal.

Consider a language L which has ∈, a constant ẋ, and constants ň for every integer n. Atomic formulae are ň ∈ ẋ. Arbitrary conjunctions and disjunctions are allowed.

If x is a real then x ⊧ φ for φ in L is defined in the obvious way.

The theory T in the language L is supposed to have the axioms

_{i<κ}^{⋁} φ_{i} ⟷ _{i<λ}^{⋁} i_{E}(⟨φ_{j}; j < κ⟩)_{i},

whenever E ∈ E, crit(E) = κ ≤ λ, and i_{E}(⟨φ_{j}; j < κ⟩) ↾ λ ∈ ult(V; E).

We let ℙ_{δ} be the Lindenbaum algebra of T. I.e., φ ∼ ψ iff T ⊢ φ ↔ ψ and [φ] ≤ [ψ] iff T ⊢ φ → ψ.

Claim 1. ℙ_{δ} has the δ-c.c. and is in fact a cba.

The proof of Claim 1 uses δ is Woodin.

Claim 2. If x ⊧ T then, setting G_{x} = {[φ]; x ⊧ φ}, G_{x} is ℙ_{δ} generic over V and x ∈ V [G_{x}].

Lemma (Woodin). Let M be sufficiently iterable, M ⊧ δ is Woodin. Let x be any real. There is then a (ctble.) iteration π: M → M^{∗} such that x (or rather G_{x}) is π(ℙ_{δ}^{M}) generic over M.

Proof. Keep hitting the “least” extender E for which x does not satisfy an axiom mentioning E.

Corollary (Woodin). Suppose that M is a model of ZF C^{−} plus there is a measurable cardinal above n Woodin cardinals. If M is sufficiently iterable then M is Σ_{n+2}^{1} correct.

Proof. By induction. n = 0: Shoenfield.

Let n > 0. Let δ be the least Woodin cardinal of M. Let φ(x), where φ is Π_{n+1}^{1} and x is a real (in V). Let π: M → M^{∗} be an iteration such that x is π(ℙ_{δ}^{M}) generic over M^{∗}.

By inductive hypothesis, M^{∗}[x] ⊧ φ(x).

Therefore, there is a condition p in M^{∗} such that

p | |−_{ℙ_{π(δ)}^{M∗}}^{M^{∗}} ∃yφ(y).

By elementarity, there is some p in M such that

p | |−_{ℙ_{δ}^{M}}^{M} ∃yφ(y).

Working in M, let σ: M̄ → M be sufficiently elementary such that M̄ is countable. Let z ∈ M be σ^{−1}(ℙ_{δ}^{M}) generic over M̄. We have

M̄[z] ⊧ φ(z).

The inductive hypothesis, applied inside M, yields

M ⊧ φ(z).

Let Γ be a class of posets.

We say that L(ℝ) is (2-step) absolute w.r.t. Γ iff for all posets ℙ ∈ Γ and for all posets ℚ ∈ V^{ℙ} such that ℙ ∗ ℚ ∈ Γ,

V^{ℙ} ≺_{L(ℝ)} V^{ℙ∗ℚ},

by which we mean that for any formula φ and for all reals x⃗ ∈ V^{ℙ},

L(ℝ^{V^{ℙ}}) ⊧ φ(x⃗) ⇔ L(ℝ^{V^{ℙ∗ℚ}}) ⊧ φ(x⃗).

We shall be interested in L(ℝ) absoluteness w.r.t. Γ, where Γ = c.c.c., proper, semiproper, stationary set preserving, all forcings.

The following provides a key technique for showing L(ℝ) absoluteness w.r.t. Γ in models of set theory, provided Γ is not too large.

Let κ be an inaccessible cardinal. Let x be a real in V^{Col(ω,<κ)∗ℙ}. Then x is “small generic” (over V) iff there is some ℚ ∈ V_{κ} such that x is ℚ-generic over V.

Lemma. Let κ be an inaccessible cardinal. Let ℙ be a poset in V^{Col(ω,<κ)} such that every real in V^{Col(ω,<κ)∗ℙ} is “small generic.” Then V^{Col(ω,<κ)} is L(ℝ) absolute w.r.t. {ℙ}.

Proof. Let G be Col(ω, < κ)-generic over V, and let H be ℙ^{G}-generic over V [G].

Let ⟨e_{n}; n < ω⟩ be an enumeration of the reals of V [G][H] in a further forcing extension. We construct ⟨α_{n}, g_{n}; n < ω⟩ as follows.

Given ⟨α_{n}, g_{n}; n < m⟩ ∈ V [G][H], we pick α_{m} < κ (with α_{m} > α_{m−1} if m > 0) such that e_{m} is Col(ω, < α_{n})-generic over V. Pick g_{m} (⊃ g_{m−1} if m > 0) Col(ω, < α_{n})-generic over V such that g_{m} ∈ V [G][H] and e_{m} ∈ V [g_{m}].

Finally, set G^{∗} = ⋃_{n<ω} g_{n}. As Col(ω, < κ) has the κ-c.c., G^{∗} is Col(ω, < κ)-generic over V. By the same reason and the construction of G^{∗},

ℝ^{V [G^{∗}]} = ℝ^{V [G][H]}.

Now we use the homogeneity of Col(ω, < κ). For any formula φ and reals x⃗ ∈ V [G][H],

L(ℝ^{V [G^{∗}]}) ⊧ φ(x⃗) ⇔
| |−_{V [x⃗]}^{Col(ω,<κ)} L(ℝ̇) ⊧ φ(ˇx⃗) ⇔
L(ℝ^{V [G]}) ⊧ φ(x⃗).

Here ℝ̇ is a name for the set of reals which is mapped to itself by the relevant automorphism of Col(ω, < κ).

Theorem 14 (Harrington, Shelah). The following theories are equiconsistent.

(1) ZFC + L(ℝ) is absolute w.r.t. c.c.c. forcings.

(2) ZFC + there is a weakly compact cardinal.

Theorem 15 (Sch; 99) The following theories are equiconsistent.

(1) ZFC + L(ℝ) is absolute w.r.t. proper forcings.

(2) ZFC + there is a remarkable cardinal.

Proof of Theorem 15. “⇐”: If (1) holds then ω_{1} is weakly compact in L.

“⇒”: Let κ be weakly compact. Let x be a real in V^{Col(ω,<κ)∗ℙ}, where ℙ is c.c.c. in V^{Col(ω,<κ)}. A reflection argument shows that x is “small generic.”

Proof of Theorem. “‘⇐”: If (1) holds then ω_{1} is remarkable in L.

What is a remarkable cardinal, by the way?

Definition (Sch; 99). A cardinal κ is remarkable iff for all regular θ > κ there is a 6-tuple ⟨π, M, κ̄, σ, N, θ̄⟩ such that

(1) π: M → H_{θ} is elementary,

(2) M is ctble. and transitive,

(3) π(κ̄) = κ,

(4) σ: M → N is elementary with crit(σ) = κ̄,

(5) N is ctble. and transitive,

(6) θ̄ = M ∩ OR is regular in N,

(7) σ(κ̄) > θ̄, and

(8) M = HN ; θ̄.

For our purposes, the more useful characterization of “remarkability” is the following one.

Definition. Let κ be a cardinal. Let G be Col(ω, < κ)-generic over V, let θ > κ be a regular cardinal, and let X ∈ [H_{θ}^{V [G]}]^{ω}. We say that X condenses remarkably iff X = ran(π) for some elementary

π: (H_{β}^{V [G∩H_{α}^{V}]}; ∈, H_{β}^{V} , G∩H_{α}^{V}) → (H_{θ}^{V [G]}; ∈, H_{θ}^{V} , G)

where α = crit(π) < β < κ and β is a regular cardinal (in V).

Lemma. A cardinal κ is remarkable iff for all regular cardinals θ > κ we have that

| |−_{Col(ω,<κ)}^{V} “{X ∈ [HV [Ġ] ; θ̌]^{ω}: X condenses

remarkably } is stationary.”

Proof of Theorem 15. “⇐”: Let κ be remarkable, and let x be a real in V^{Col(ω,<κ)∗ℙ}, where ℙ is proper in V^{Col(ω,<κ)}. Let θ be large enough.

In V^{Col(ω,<κ)}, let S be the set of all ctble. X ≺ H_{θ} which condense remarkably.

The set S is stationary in V^{Col(ω,<κ)}, and hence in V^{Col(ω,<κ)∗ℙ}.

We may hence, in V^{Col(ω,<κ)∗ℙ}, pick some ctble.

X ≺ ⟨H_{θ}^{V^{Col(ω,<κ)}} ; ∈, ℙ, x⟩

such that, setting α = X ∩ ω_{1} and β = otp (X ∩ θ),

(i) α < β < κ,

(ii) α and β are cardinals in V, and

(iii) if τ: N ≅ X, where N is transitive, then N = H_{β}^{V^{Col(ω,<α)}} .

By elementarity, x is τ^{−1}(ℙ)-generic over N. But then x is Col(ω, < α) ∗ τ^{−1}(ℙ)-generic over V, i.e., “small generic.”

The technique of showing L(ℝ) absoluteness by verifying that all reals are “small generic” no longer works at the level of semiproper forcings.

Theorem 16 (Shelah) Suppose that δ is a Woodin cardinal. There is then a semiproper ℙ such that in V^{ℙ}, δ = ω_{2} and I_{NS} is ω_{2} saturated.

Theorem 17 (Woodin, ≤ 99). Suppose that I_{NS} is ω_{2} saturated and there is a measurable cardinal. Then u_{2} = ω_{2}.

Now suppose that δ is Woodin and κ is measurable, where κ > δ. Let ℙ be semiproper such that in V^{ℙ}, δ = ω_{2} and I_{NS} is ω_{2} saturated, and hence u_{2} = ω_{2} = δ.

I.e. in V^{ℙ}, for any ξ < ω_{2} there is a real x_{ξ} such that the least x_{ξ}-indiscernible strictly above ω_{1} is actually above ξ.

Hence if ξ is large enough then x_{ξ} cannot be “small generic” over an inner model of V.

Exercise 3. What is the consistency strength of L(ℝ) being absolute w.r.t. semiproper forcings?

Theorem 18 (Sch). If L(ℝ) is absolute w.r.t. stationary set preserving forcings then for every set x there is an inner model with a strong cardinal containing x.

Exercise 4. Suppose that L(ℝ) is absolute w.r.t. stationary set preserving forcings. Does AD hold in L(ℝ)?

Theorem 19 (Woodin). Suppose that there is an iterable inner model M with infinitely many Woodin cardinals which are all ctble. in V. Then L(ℝ) is absolute w.r.t. any forcing.

Proof. Enumerate ℝ^{V} in order type ω (in a forcing extension). Make one real after the other generic over an iterate of M. Done carefully, we obtain a final iterate M^{∗} of M such that

ℝ^{V} = _{i<κ}^{⋃} ℝ^{M^{∗}[G∩V_{i}]},

where κ is the sup of the ω many Woodins of M^{∗} (in fact, κ = ω_{1}^{V}) and G is Col(ω, < κ) generic over M^{∗}.

But then

L(ℝ^{V}) ⊧ φ(x⃗) ⇔
| |−_{M^{∗}[x⃗]}^{Col(ω,<κ)} L(ℝ̇) ⊧ φ(x⃗),

where ℝ̇ is a name for the reals of the symmetric collapse of the ω Woodins of M^{∗}. By elementarity, this holds if and only if

| |−_{M′[x⃗]}^{Col(ω,<κ̄)} L(ℝ̇) ⊧ φ(x⃗),

where M′ is an appropriate iterate of M in V and κ̄ is the sup of the ω Woodins in M′.

However, the same representation of L(ℝ) ⊧ φ(x⃗) has to hold in any generic extension of V.

Let Γ be a class of posets. The Bounded forcing axiom for Γ is the following statement.

For all ℙ ∈ Γ,

H_{ω_{2}} ≺_{Σ_{1}} H_{ω_{2}}^{V^{ℙ}}.

More precisely, for all ℙ ∈ Γ, for any Σ_{1} formula φ, and for all A ∈ H_{ω_{2}}, if V^{ℙ} thinks that H_{ω_{2}} ⊧ φ(A) then so does V.

If ℙ ∈ Γ allows one to shoot a club thru a costationary (and stationary) set S ⊂ ω_{1} then the bounded forcing axiom for Γ is simply false, because “ω_{1} \ S is nonstationary” is Σ_{1} over H_{ω_{2}}.

(Also, Σ_{1} cannot be replaced by Σ_{2}, as CH is Σ_{2} over H_{ω_{2}}.)

We shall be interested in:

BMM (Bounded Martin’s Maximum): Γ = the class of all stationary set preserving posets.

BSPFA (Bounded Semiproper Forcing Axiom): Γ = the class of all semiproper posets.

BPFA (Bounded Proper Forcing Axiom): Γ = the class of all proper posets.

Definition. κ is reflecting iff κ is inaccessible and V_{κ} ≺_{Σ_{2}} V. Theorem 20 (Goldstern-Shelah). Equiconsistent are:

(1) ZFC + BPFA.

(2) ZFC + BSPFA.

(3) ZFC + there is a reflecting cardinal.

Theorem 21 (Todorčević; 02). If BMM holds then the continuum has size ℵ_{2}.

In fact:

Theorem 22 (J. Moore; 03). If BPFA holds then the continuum has size ℵ_{2}.

We shall now be interested in the relation between BMM and Σ_{n}^{1} absoluteness (of V).

Theorem 23 (Sch; 03). If BMM holds then V is Σ_{3}^{1} absolute.

By Theorem 1, this is tantamount to saying that under BMM X^{#} exists for all X ∈ V.

Proof of Theorem 23. Let A ⊂ ω_{1}. We may define f_{A}: ω_{1} → ω_{1} ∪{∞} by letting f_{A}(ξ) = the least α such that

J_{α+1} [A ∩ ξ] ⊧ ξ is ctble,

if some such α exists, and f_{A}(ξ) = ∞ if not.

We say that A is reshaped iff there is a club C ⊂ ω_{1} such that f_{A}(ξ) < ω_{1} for all ξ ∈ C.

If A, B ⊂ ω_{1} then we write A < B iff there is a club C ⊂ ω_{1} such that f_{A}(ξ) < f_{B}(ξ) for all ξ ∈ C (in particular, A is then reshaped).

Obviously, < is well-founded.

Let us now assume that X^{#} does not exist for some X. Suppose w.l.o.g. X = ∅. We shall prove, assuming BMM:

Claim. For all A, there is some B with B < A.

Proof of the Claim: Let A ⊂ ω_{1} be given.

Using Jensen’s Covering Lemma and almost disjoint forcing, it is not hard to obtain an ωclosed forcing extension of V in which H_{ω_{2}} = J_{ω_{2}}[Ā] for some Ā ⊂ ω_{1}.

It obviously suffices to find B ⊂ ω_{1} such that B < A⊕Ā in a further stationary set preserving forcing extension. This is because then B < A in this forcing extension. However, the existence of such a subset B of ω_{1} is easily seen to be Σ_{1} over H_{ω_{2}}. Hence, by BMM there must be such a B in V.

Set D = A ⊕ Ā. In order to find B < D in a stationary set preserving forcing extension we shall use a variant of Jensen’s reshaping forcing. Let (p, c) ∈ ℙ iff

(i) c is a ctble. closed subset of ω_{1}, (ii) p: max(c) → 2, (iii) for all ξ ∈ c,

J_{f_{D}(ξ)}[D ∩ ξ, p ↾ ξ] ⊧ ξ is ctble.

The ordering is by end-extension.

It suffices to prove that ℙ is stationary set preserving.

Let S ⊂ ω_{1} be stationary, and let

(p, c) | |− τ ⊂ ω̌_{1} is club.

Let ⟨X_{i}; i < ω_{1}⟩ be a strictly increasing continuous chain of ctble. elementary submodels of ⟨H_{ω_{2}}; ∈, ℙ, τ⟩ with i ∈ X_{i+1} , all i. Then C = {X_{i} ∩ ω_{1}; i < ω_{1}} ⊂ ω_{1} is club.

Let

X ≺ ⟨H_{ω_{2}}; ∈, ℙ, (p, c), C⟩

such that ξ_{0} = X ∩ ω_{1} ∈ S.

We aim to construct (q, d) < (p, c) such that (q, d) | |− ξ̌_{0} ∈ τ.

Pick E = {e_{0} < e_{1} < e_{2} < ⋯} ⊂ C ∩ ξ_{0} cofinal in ξ_{0}, e_{0} > max(c). Let e′_{n} be the least element of C above e_{n}.

Starting with (p_{0}, c_{0}) = (p, c), construct a sequence ⟨(p_{n}, c_{n}); n < ω⟩ as follows.

Given (p_{n}, c_{n}) ∈ X_{e′_{n−1}} (for n > 0), first pick an

extension (p′_{n}, c′_{n}) such that

(a) max(c′_{n}) > e_{n},

(b) (p′_{n}, c′_{n}) ∈ X_{e′_{n}},

(c) for all ξ ∈ C ∩[e′_{n−1}, e_{n}], p′_{n}(ξ) = 1 iff ξ = e_{n}.

Then let (p_{n+1} , c_{n+1} ) ∈ X_{e′_{n}} be a further extension such that for some ordinal γ_{n} ∈ (e_{n}, e′_{n}),

(p_{n+1} , c_{n+1} ) | |− γ̌_{n} ∈ τ.

In the end, set q = ⋃_{n<ω} q_{n}, and d = ⋃_{n<ω} c_{n} ∪ {ξ_{0}}.

One happily verifies:

E can be read off from C∩ξ_{0} ∈ X together with q, simply as ξ ∈ E iff ξ ∈ C ∩ ξ_{0} and q(ξ) = 1. As dom(q) = ξ_{0}, (q, d) is thus a condition.

But we have

(q, d) | |− γ̌_{n} ∈ τ

for each n, and hence ξ_{0} = sup_{n<ω}γ_{n} is forced by (q, d) to be in τ.

Theorem 24 (Sch;03). If BMM holds then for any X ∈ V there is an inner model with a strong cardinal containing X.

Proof of Theorem 24. The proof uses “Kreshaping” rather than “L-reshaping.”

Suppose that there is some X such that there is no inner model with a strong cardinal containg X. Suppose w.l.o.g. X = ∅.

For A ⊂ ω_{1}, we define f_{A}: ω_{1} → ω_{1} ∪ {∞} by letting f_{A}(ξ) = the least α such that

K(A ∩ ξ)||α ⊧ ξ is ctble.,

if some such α exists, and f_{A}(ξ) = ∞ if not.

Define A < B as before.

Assuming BMM:

Claim. For all A, there is some B with B < A.

The key point is that for all A ⊂ ω_{1} there is some A′ ⊂ ω_{1} such that

ω_{2}^{V} = ω_{2}^{K(A⊕A′)}.

This exploits the fact that K does not have a strong cardinal!

One can then run an argument basically as in the L-case.

What about V being Σ_{4}^{1} absolute?

Theorem 25 (Bagaria;00). If BMM holds and u_{2} < ω_{2} then V is 1-step Σ_{4}^{1} absolute w.r.t. stationary set preserving forcings.

Proof of Theorem 25. If u_{2} < ω_{2} then the Martin-Solovay tree is an element of H_{ω_{2}}.

Theorem 26 (Woodin). BMM does not imply that V is 1-step Σ_{4}^{1} absolute w.r.t. stationary set preserving forcings.

Proof of Theorem 26. Let L[E] denote the minimum model with ω + 1 many Woodin cardinals. Let κ be the supremum of the first ω many Woodins. Let G be Col(ω, < κ) generic over L[E], and set

ℝ^{∗} = _{α<κ}^{⋃} ℝ^{L[E][G∩J_{α}[E]]}.

(I.e., ℝ^{∗} is the set of reals of the symmetric collapse of the first ω many Woodins.)

AD (in fact AD^{+} ) holds in L[E](ℝ^{∗}). Also, L[E](ℝ^{∗}) has (exactly) one Woodin cardinal.

Let K(ℝ^{∗}) denote the core model built over ℝ as computed in L[E](ℝ^{∗}) cut off at the Woodin cardinal.

Claim 1. BMM holds in K(ℝ^{∗})^{ℙ_{max}} .

Proof. Write V = K(ℝ^{∗})^{ℙ_{max}} .

Let φ be Σ_{1}, and let A ⊂ ω_{1}. Let ℙ be stationary set preserving, and assume that in V^{ℙ},

H_{ω_{2}} ⊧ φ(A).

We have to show this also holds in V. Let’s assume this is wrong.

W.l.o.g., A ∉ L(ℝ). Then A is ℙ_{max} generic over L(ℝ). There is hence a condition ⟨(M, I), a⟩ in the generic filter given by A such that the following holds true.

(∗) Let π: (M, I) → (M^{∗}, I^{∗}) be a ctble. iteration. Set a^{∗} = π(a). Let N be a ctble. transitive model of ZFC with

• P(ω_{1}^{M^{∗}}) ∩ M^{∗} ⊂ N,

• ω_{1}^{N} = ω_{1}^{M^{∗}},

• Q_{3}(x) ⊂ N for all x ∈ P(ω_{1}^{N}) ∩ N, and

• if S ∈ P(ω_{1}^{M^{∗}}) ∩ M^{∗} is stationary in M^{∗} (i.e., if S ∉ I^{∗}) then S is stationary in N as well.

Then in N,

H_{ω_{2}} ⊧ ¬φ(a^{∗}).

Recall that for reals x, y, y ∈ Q_{3}(x) means that y is Σ_{3}^{1}(x) in a ctble. ordinal.

In terms of inner model theory, y ∈ Q_{3}(x) means that y ⊂ x is in the “least” inner model with a Woodin cardinal (which is ctble. in V) and which contains x. For reals x, y, “y ∈ Q_{3}(x)” is Σ_{3}^{1} (uniformly).

Therefore, the statement (∗) is Π_{3}^{1} in any code for the condition ⟨(M, I), a⟩.

V is Σ_{3}^{1} absolute, and hence (∗) holds in all generic extensions.

Let θ be very large. In V^{ℙ∗Col(ω,θ)}, the following holds true.

There is an iteration π: (M, I) → (M^{∗}, I^{∗}) with π(a) = A.

Let N = J_{α}[V_{θ}^{V^{ℙ}}], where α is the least V_{θ}^{V^{ℙ}}- indiscernible.

Then π and N witness that (∗) is false in V^{ℙ∗Col(ω,θ)}. Contradiction!

Claim 2. K(ℝ^{∗})^{ℙ_{max}} is not a model of 1-step Σ_{4}^{1} absoluteness w.r.t. stationary set preserving forcings.

Proof. K(ℝ^{∗}) is 1-small above ℝ^{∗}.

In a Col(ω, θ) generic extension of K(ℝ^{∗})^{ℙ_{max}} (where θ is large enough) there is hence a real x such that there is no sharp for an inner model with a Woodin cardinal (ctble. in V) containing x. This is a Σ_{4}^{1} statement which is false in K(ℝ^{∗})^{ℙ_{max}} (the reals of which are ℝ^{∗}).

Such a real x can actually be added by a stationary set preserving forcing.

By Theorems 23 and 25, the strength of BMM is (not necessarily strictly) between one strong cardinal and ω + 1 Woodin cardinals.

Exercise 5. Construct the “canonical” model of BMM ! What is the consistency strength of BMM ?

If A is reshaped then f_{A} can be verified to be above all canonical functions (w.r.t. <).

Let α < ω_{2}. Let g: ω_{1} → α be onto. Let f_{α}: ω_{1} → ω_{1} be defined by

f_{α}(ξ) = otp(g”ξ).

Given α, f_{α} is unique mod the club filter.

By club bounding we mean the statement that for every f: ω_{1} → ω_{1} there is some α < ω_{2} such that f < f_{α}, i.e., there is some club C such that f(ξ) < f_{α}(ξ) for all ξ ∈ C.

Definition (Aspero-Welch). We say that P_{2}(ω_{1})^{+} holds iff for all clubs C in [ω_{2}]^{ω} there are club many ctble. ξ s.t.

{otp(X); X ∈ C ∧ X ∩ ω_{1} = ξ}

is stationary.

Lemma (P. Larson). Club bounding implies P_{2}(ω_{1})^{+} .

Lemma (Sch). Suppose that BMM holds. Then P_{2}(ω_{1})^{+} implies club bounding.

Proof of Lemma. Let f: ω_{1} → ω_{1}. Let (p, c) ∈ ℙ iff

(a) c ⊂ ω_{1} is ctble. and closed,

(b) p: max(c) → ω_{2},

(c) for all ξ ∈ c, f(ξ) < otp(p”ξ). The ordering is by end-extension. It suffices to verify that ℙ is stationary set preserving.

Let S ⊂ ω_{1} be stationary. Let (p, c) | |− τ ⊂ ω̌_{1} is club.

Let X ≺ ⟨H_{ω_{2}}; ∈, ℙ, (p, c)⟩ be ctble. and such that α = X∩ω_{1} ∈ S and β = otp(X∩ω_{2}) > f(α). Such X exists by P_{2}(ω_{1})^{+} (we in fact only need “unbounded” rather than “stationary.”)

We may then construct some (q, d) < (p, c) such that

(q, d) | |− α̌ ∈ τ,

and q: X ∩ ω_{1} → β is onto.

Exercise 6. Does BMM alone imply that club bounding holds?

Lemma (Woodin). Suppose that δ is Woodin and κ is measurable, κ > δ. If BMM holds then u_{2} = ω_{2}.

Theorem 23 above shows that BMM yields that every real has a sharp. It thus makes sense now to ask:

Exercise 7. Does BMM alone imply that u_{2} = ω_{2}?