In this section, we construct a model of set theory which is a slight modification of a model from [Rei06]2. Our aim in constructing this model is to find a model such that the modal logic of inner models of this model is exactly S4.2Top.
Definition 103. Let γ be a regular cardinal. The forcing poset Add(γ) which adds aCohen subset of γ is the following:
(i) p∈Add(γ) if p is a function such that dom(p)⊆γ, range(p)⊆ {0,1}and |p|< γ. (ii) Ifp, q∈Add(γ), thenp≤q ifp⊆q.
Note that for the case that γ =ω, Add(ω) is the same as Coh. The next two lemmas will come in handy in our main proof. Their proofs are standard.
Lemma 104. Let M⊆N be models of set theory. Let γ be an infinite regular cardinal in N. InN, let S⊆γ. Then if S is a Cohen subset of γ over M, then for each ordinal α < γ, S∩α∈M.
Lemma 105. Let γ be a regular cardinal. Then |Add(γ)| = 2<γ. Therefore, if 2<γ = γ, then |Add(γ)|=γ.
We now define the class-forcing poset which we shall use to construct the model we want.
2
The modification here is that in both, [Rei06] and [HL13], the class forcing which was used added a Cohen subset to each regular cardinal ofL. However, as we were unable to prove for this model that an analogue of Theorem 108 holds, we modified their construction to one for which we could do so.
Definition 106. Let SuccL denote the class of infinite successor cardinals in L. Define in L the following (class-sized) poset with Easton support:
P=∆
Y
γ∈SuccL
Add(γ).
That is, p∈P if
(i) p is a class function such that dom(p) = SuccL; (ii) For eachγ ∈SuccL,p(γ)∈Add(γ);
(iii) For each such p, for each regular cardinal γ, |{λ ∈ SuccL | p(λ) 6= 0} ∩γ| < γ (the class
{γ ∈SuccL|p(γ)6= 0}is called to be thesupport of p). The ordering is defined by p≤q ifp⊆q.
Also, for each p∈Pand each γ ∈SuccL, we can decomposep into three parts: p<γ =p[0, γ);
pγ =p[γ, γ]; p>γ =p(γ,∞).
Using this decomposition, for each γ ∈SuccL, we can decomposePinto three parts: P<γ ={p<γ |p∈P};
Pγ={pγ|p∈P}; P>γ ={p>γ |p∈P}.
It is clear that P∼=P<γ×Pγ×P>γ.
Proposition 107. (V=L) Let γ ∈SuccL. Then (i) P>γ is≤γ-closed;
(ii) P<γ has size less than γ.
Proof. The first part is trivial. For the second part, letγ=κ+. We use the fact that theGCH is true in L. The result then follows from the following chain of equivalences:
|P<γ|= λ<γ Y λ∈SuccL |Add(λ)|= λ<γ Y λ∈SuccL 2<λ = λ<γ Y λ∈SuccL λ≤ λ<γ Y λ∈SuccL κ≤κ×κ=κ < γ.
Theorem 108. (V=L) Letγ be an infinite successor cardinal. Let Qγ =P<γ×P>γ. Then forcing
Proof. Since Qγ = P<γ ×P>γ and P>γ is ≤ γ-closed, by Proposition 24, we only need to show that forcing with P<γ does not add a Cohen subset of γ. Suppose towards a contradiction that this is not so. Let L[G] be a generic extension by P<γ such thatS ∈L[G] is a Cohen subset of γ. Therefore, for each α < γ, S∩α ∈ L. Now, by the previous proposition, |P<γ|< γ. Hence, by Lemma 98, it follows thatLhas theγ-approximation property inL[G]. But then,S=S
α<γ(S∩α), andhS∩α|α < γiis a⊆-increasing sequence of lengthγ of elements ofL, and hence,S∈L, which is a contradiction. Therefore, forcing with Qγ does not add any Cohen subsets of γ.
Definition 109. Let G be L-generic for P, and let MR =∆ L[G]. Let γ be an infinite successor
cardinal. Then G>γ =G∩P>γ. Let
ϕR =∆∀κ∈SuccL∃G⊆κ(Gis an L-Cohen subset ofκ). Let
ψR=∆ ∃B[(Bis a complete atomless Boolean algebra )∧(kϕRkB= 1B)]. Clearly,MRϕR, and for any modelN,N is a ground ofMR if and only if N ψR.
Hence, ifN is an inner model ofMRsuch thatN ¬ψR, then by Corollary 102, no further inner modelN0 of N can be a model of ψR. That is,MR (¬ψR(¬ψ
R)). Hence, the statement ¬ψR is a pure button.
We now prove an interesting property of this model which we shall use in the next section, namely that every ground of this model itself has a non-trivial ground.
Lemma 110. LetN be a ground of MR. Then there is an infinite successor cardinal γ such that
N ⊇L[G>γ]. In particular, L[G>γ] is a ground (and hence, a definable inner model) ofN.
Proof. Towards a contradiction, suppose this is not so. LetQ∈N be a forcing poset and letH be Q-generic overN such thatMR=N[H]. For some infinite successor cardinalγ large enough, letτ
be a name forG>γ. Letp∈Qbe such that
p“τ isP>γ-generic over L.
By assumption,G>γ is a class-function which is not in N, but in a forcing extension of N byQ
(which is a set). Therefore, for any q ≥ p, q can decide only a set-sized initial segment of G>γ. However, for every β > α, there is ar ≥psuch that rdecides G>γ(γ, β). Therefore, we can form a class-length strictly increasing chain of conditions in Q, thus contadicting that it is a set.
In [HL13], Hamkins and L¨owe studied a model which was a slight modification of MR, and proved a similar property of this model. They used this property to show that.2is valid for the modal logic of grounds ofMR, and hence that the modal logic of grounds of this model is exactly
S4.2.