In this section, we prove that the modal logic of grounds of any generic multiverse is contained inS4.2Top.
LetW be a countable transitive model of ZFCfor the rest of this section. We will rst go
to a ground of W which has the same cardinals and power sets of cardinals up to a certain
height as the mantle. Then, we add generic subsets toℵ0 many of these cardinals and dene downwards buttons, each stating that there is no M-generic subset of one of these cardinals.
After showing that these buttons are independent, theorem 5.3 gives the desired result. We begin by recursively dening a sequence of cardinals similar to the i-sequence which
gives us the cardinals of which we add generic subsets: Denition 7.1. Recursively, we dene
Γ0=ℵ0 ,
Γα+1= (2Γα)+ forα∈Ord,
Γλ=
[
α<λ
Γα for all limit ordinalsλ.
The recursive denition ensures that there is a formulaφ(x, α)deningΓαwith the parameter
α.
Lemma 7.2. There is a groundU ofW such that inU
1. for allκ≤ΓM
ω we have thatκis a cardinal i κis a cardinal inM,
2. P(κ) =P(κ)M for all cardinalsκ≤ΓM
ω, and hence Γn= ΓMn for alln≤ω,
4. P(Fn(Γn,2,Γn)) =P(Fn(Γn,2,Γn))M
Proof. The proof is similar to the proof of lemma 6.5. First, for each cardinal κ≤ΓM
ω of the
mantle we nd a groundWκ ofW in whichκis a cardinal: For each injectionf :κ→αwith
α < κin W, we nd a ground without this injection as there is no such injection inM. In a
common ground of these set-many grounds,κis a cardinal.
Likewise, in a common groundW0 of allWκ the cardinals up toΓMω are the same as in the
mantle.
Further, starting from W0 for eachκ≤ΓM
ω we nd a ground Wκ0 in which P(κ) =P(κ)M:
For each subset of κ which is not inM, there is a ground of W0 which does not contain this
subset. A common groundWκ0 of all these grounds satisesP(κ) =P(κ)M.
Conditions 3. and 4. can be achieved in the same way and so we nd the desired ground
U.
Note that all the conditions for the groundU also hold in any ground ofU. More generally,
this proof technique allows us to nd e.g. a ground of W in whichVκ=VκM, or for any set X
inW withX∩M=∅, to nd a ground W0 withW0∩X =∅.
Letφ(n)be the statement
There is no Fn(Γn+1,2,Γn+1)M-generic lter overM.
In the groundU given by the lemma,
Fn(Γn,2,Γn)M= Fn(Γn,2,Γn)
and
P(Fn(Γn,2,Γn)) =P(Fn(Γn,2,Γn))M.
So in particular, Fn(Γn,2,Γn) has the same dense sets in U and M and therefore a subset
G⊆Fn(Γn,2,Γn)M isFn(Γn,2,Γn)M-generic over Mif and only if it isFn(Γn,2,Γn)U-generic
overU. The same holds for any ground ofU.
Lemma 7.3. LetU be as in lemma 7.2. LetPbe the partial order
Y n∈ω Fn(Γn+1,2,Γn+1)M= Y n∈ω Fn(Γn+1,2,Γn+1)U,
letGbeP-generic overU and letV :=U[G]. Then, the family{φ(n) :n∈ω}is an independent family of downwards buttons inV.
Proof. Clearly, all buttons are unpushed inV. LetV0 be a ground ofV and let A={n∈ω:¬φ(n)V0}.
Further, letU0be a common ground ofV0andU. So,V0is a forcing extension ofU0and for each n∈Ait contains anF n(Γn+1,2,Γn+1)U
0
show that these lters are mutually generic. So, letn < mand letGbeFn(Γn,2,Γn)U
0
-generic lter overU0andH anFn(Γm+1,2,Γm+1)U
0
-generic lter overU0. SinceΓm+1is regular (inU0) we know thatF n(Γm+1,2,Γm+1)U
0
isΓm+1-closed. The partial orderFn(Γn,2,Γn)U
0 has size Γ<Γn+1 n+1 = ((2 Γn)+)2Γn ≤(22Γn)2Γn = 22Γn <Γ n+2≤Γm+1. So, inU0[H]we still have that
P(Fn(Γn+1,2,Γn+1)M) =P(F n(Γn+1,2,Γn+1)M)U 0
by theorem A.19. Thus,Gintersects all dense subsets of the partial order Fn(Γn+1,2,Γn+1)M. So, we get that G is Fn(Γn+1,2,Γn+1)M-generic over U0[H]. Therefore, the lter G×H is
Fn(Γn+1,2,Γn+1)M×Fn(Γm+1,2,Γm+1)M-generic overU0by theorem A.25 and soGandH are mutually generic.
Thus,V0contains aQ
n∈AFn(Γn+1,2,Γn+1)M-generic lterKoverU0. So,U0[K]is a ground ofV0 and there is anQ
n∈BFn(Γn+1,2,Γn+1)M-generic lterK
0 in U0[K]for any setB ⊆Ain U0 andU0[K0]is a ground ofV0. Clearly,¬φ(n)holds inU0[K0]for alln∈B.
It remains to show thatφ(n)holds inU0[K0]forn∈ω−B. Fix such ann. We know that
anF n(Γn+1,2,Γn+1)M-generic lter LoverMand hence overU0 is generic over all projections
of K0 and hence also over the product, i.e. L is F n(Γn+1,2,Γn+1)M-generic over U0[K0] and hence not inU0[K0].
We can summarize those results as follows:
Theorem 7.4. For any modelW ofZFC, there is a modelU in the generic multiverse ofW in
which there is a uniform family of innitely many independent buttons. So,MLG(U)⊆S4.2Top
and henceMLG(MultU)⊆S4.2Top.