Our aim in this chapter was to study the modal logic of the relation of being a definable inner model. So far, we have seen that this moda logic is contains the theory S4.2Top. In order to show that there is a model whose modal logic is exactlyS4.2Top, we will first show that the relation of being an inner model has, as an initial segment, a relation that we understand better, the relation of being a forcing ground. Once we have shown this, we will be able to the results from [HL13], where a model was constructed whose modal logic of grounds is exactlyS4.2, to obtain our main result.
We fix some notation. Let ΓIM be the following relation:
(M, N)∈ΓIM iffN is a definable inner model of M,
and let ΓP¯ be the relation:
(M, N)∈ΓP¯ iff N is a forcing ground ofM.
In order to show that ΓP¯ is an initial segment of ΓIM, we split the task into two parts, one where
we show that ΓP¯ is contained in ΓIM, and one where we show that the former is actually an initial
segment of the later.
4.2.1 The Laver-Woodin Theorem
The first task is achieved by the Laver-Woodin Theorem, which shows that if (M, N)∈ΓP¯, then
(M, N)∈ΓIM. Our treatment follows [WDR12]. We note that while we talk about models of NBG
everywhere, all of the proofs go through withZFC itself.
Definition 97. (Hamkins) Let δ be an uncountable regular cardinal. LetM be a transitive class model ofNBG.
(i) Then M is said to have theδ-covering property if for everyσ⊂M with |σ|< δ, there is a τ ∈M such that|τ|< δ andσ ⊆τ.
(ii) The pair M is said to have the δ-approximation property if for every cardinal κ such that cf(κ)≥δ and every⊆-increasing sequence of sets hτα |α < κi from M,∪τα∈M.
The next lemma shows that if V is a model of set theory, andV[G] is a forcing extension of it, thenV has these properties inV[G] for all cardinals which are large enough.
Lemma 98. Let V be a model of NBG. Let δ be an uncountable regular cardinal. Let P∈ V be a poset of size less than δ. Let G be a V-generic filter for P. Then V has the δ-covering and
δ-approximation properties in V[G].
Proof. We first show the δ-covering property. Letσ be aP-name andp∈Pa condition such that
p“σ⊂V and |σ|< δ”. Now, let
S ={λ < δ| ∃q≥p[q |σ|=λ]}.
Now, ifq1, q2 ≥p andλ1, λ2 are such that qi |σ|=λi for λ1 6=λ2, then it follows that q1 ⊥q2.
thatγ < δ (since S ⊂δ). In this case, let ˙f be a name such that p “ ˙f :γ→σ is a surjection”. Then if
τ ={x| ∃q ≥p∃α < γ[qf˙(α) =x]}.,
it is clear thatτ ∈V,σ ⊆τ and|τ|< δ, hence establishing the δ-covering property. Now, for δ-approximation. Letp∈P be a condition such that
p“cf(κ)≥δ and hτα |α < κiis a ⊆-increasing sequence of sets from V”.
For eachα < κ, letpα ≥p be a condition which decides the value ofτα. Since |P|< δ ≤cf(κ), it
follows that there must be some q∈Psuch that for cofinally many α < κ,q=pα. Since p“hτα|α < κi is a ⊆-increasing sequence of sets from V”,
it follows thatq decides the value of∪τα, and hence, ∪τα∈V.
Theorem 99. (Laver [Lav07], Woodin) Letδ be inN a regular uncountable cardinal. Let M, N be transitive class models of NBG such that both satisfy the δ-covering and δ-approximation property. Suppose thatδ+= (δ+)M = (δ+)N, and that N ∩ P(δ) =M∩ P(δ).
(i) Then M =N.
(ii) In particular, M isΣ2-definable from M∩ P(δ).
Proof. (i) We show by induction on ordinalsγ that for allA⊆γ, A∈M ⇐⇒ A∈N.
Ifγ ≤δ, this is clear. Hence, assume thatγ > δ. Then, by the induction hypothesis,M and N have the same cardinals ≤γ. Also, if γ is not a cardinal in these models, then in both the models, there is a bijection between γ and|γ|, which allows us to conclude by applying the induction hypothesis on|γ|that the powerset of γ is the same in both models. Hence, we may assume that γ is a cardinal in bothM and N.
(a) cf(γ) ≥δ. ThenA∈M iff A∩α∈M for eachα < γ. The forward direction is clear, and for the reverse direction, we use the δ-approximation property for the sequence of setshA∩α|α < γi. Therefore, by using the induction hypothesis onA∩αforα < γ, we see that we are done.
(b) γ > δ, cf(γ) < δ, and|A|< δ. We will use the δ-approximation property to find a set S ⊇A such that S ∈M ∩N. We do this by using the δ-covering property as follows: define increasing sequences hEα |α < δi and hFα |α < δi of subsets of γ such that (1) |Eα|,|Fα|< δ; (2) A⊆E0; (3) Eα ⊆Fα; (4) S α<βFα ⊆Eβ; (5) Eα ∈M and Fα ∈N.
Then S= [ α<γ Eα= [ α<γ Fα
clearly satisfies the requirements thatA⊆S andS∈M∩N. Now, letθbe the ordertype of S, and letπ :S→θ be the Mostowski collapse ofS. As subsets of ordinals can only be collapsed in one way, it follows thatπ∈M ∩N. Now, it is clear that |S| ≤δ, and hence,θ < δ+. But now, the hypothesis of the theorem tell us that
δ+= (δ+)M = (δ+)N.
Applying now the induction hypothesis to δ+ and π(A)⊂δ+, it follows that A∈M ⇐⇒ π[A]∈M ⇐⇒ π[A]∈N ⇐⇒ A∈N. (c) γ > δ, cf(γ)< δ, and |A| ≥δ. We claim thatA∈M iff
(1)M A∩α∈M for allα < γ;
(2)M For everyσ ⊆γ such that|σ|< δ and σ∈M,A∩σ∈M,
and analogously,A∈N iff (1)N and (2)N. If we could show this, then by the induction hypothesis and Case (b), we would be done.
The forward direction is obvious, so assume (1)M and (2)M. Fix first a large θ with cf(θ)> γ and a formula defining M which is absolute inVθ. Now, define an increasing chainhXα |α < δi of elementary substructures ofVθ and an increasing chainhYα|α < δi of subsets of Vθ∩M such that:
(1) |Xα|,|Yα|< δ; (2) A∈X0; (3) sup(X0∩γ) =γ; (4) Xα∩M ⊆Yα; (5) Yα ∈M; (6) S α<β(Yα∪Xα)⊆Xβ.
To do this, use the Downward L¨owenheim-Skolem Theorem to get the Xα, and the δ-covering property on M to get Yα. Let X = S
α<δXα and let Y =Sα<δYα. Then X≺Vθ andY =X∩M ≺Vθ∩M.
SinceYα∈M and|Yα|< δ, it follows by assumption (2)M that for eachα < δ,A∩Yα∈M. Then, by theδ-approximation property, it follows that A∩Y ∈M.
Now, for any α∈Y ∩γ, notice that A∩α∈Y because A∈ X and α∈X, and since X≺Vθ,A∩α∈X. Also, by (1)M,A∩α∈M. Hence,A∩α∈X∩M =Y. Also, for every b∈Y, if b∩Y = (A∩Y)∩α, then Y b=A∩α, and therefore, b=A∩α. This is because Y ∩α∈M, and Y ≺Vθ∩M.
Therefore, the sequencehA∩α|α∈Y ∩γiis definable inM with parametersγ, Y, A∩Y. In particular, this sequence belongs to M, and it follows then thatA=S
α<γAγ is inM. (ii) This part follows: A∈M if there is a large regular cardinalθ, and anN ⊂Vθ which is a model
ofNBG−Powersetsatisfyingδ-covering andδ-approximation and such thatM∩P(δ) =N∩P(δ) and A∈N. This is a Σ2 statement.
4.2.2 Grigorieff ’s Theorem
Now, we shall appeal to a theorem of Grigorieff to accomplish the second task, namely, show that ΓP¯ is aninitial segment of ΓIM.
Theorem 101. (Grigorieff ) Let V be a model of NBG. LetB∈V be a complete atomless Boolean algebra. LetGbe V-generic forB and V[G] the corresponding generic extension. LetM be an inner model of V[G]such that V ⊆M ⊆V[G]. Then there is a complete atomless Boolean subalgebra C of B in V such that M =V[C∩G].
Corollary 102. Let V be a model of NBG. Let M ⊆V be an inner model. If M is not a ground of
V, then there is no inner model of M which is a ground of V. That is, ΓP¯ is an initial segment of
ΓIM.
Proof. LetN ⊆M be an inner model ofM and a ground ofV. Let B∈N be a complete atomless
Boolean algebra and G an N-generic for this Boolean algebra such that V = N[G]. Then by Grigorieff’s theorem, there isC, a complete subalgebra ofB, inN and H anN-generic forC such
thatM =N[H]. Then by Corollary 35, it follows that V is a generic extension of M as well. The second part follows.