We are now in a position to calculate the modal logic ofMR.
Lemma 111. InMR, there are arbitrarily large finite families of mutually independent¬ψR-switches and ¬ψR-buttons. Consequently, the modal logic of inner models ofMR is exactly S4.2Top.
Proof. For each natural number n, let bn be the statement “there is no L-generic subset of ℵLn”. Clearly, each of thebn is a¬ψR-button. Also, partition the regular cardinals aboveℵω intoℵ0-many
classes, hΓnin∈ω such that each class contains unboundedly many cardinals. Enumerate each class as Γn ={γαn | α∈ ORD}. Let sn be the statement “the least α such that there is an L-generic subset of γαn is even”. To see that each of the sn are a ¬ψR-switch, notice that any inner model of MR such thatMRψR is in fact a ground model ofMR. Hence, it has a ground of the form L[Gα] forα large enough. One can then choose a ground where any specific cardinals do not have an L-generic subset, hence allowing us to obtain any possible combination of the switchboard. To see that all of these switches and buttons are mutually independent, we appeal to Theorem 108.
Also, it is clear that ¬ψR is a fatal button for all of these statements. We then appeal to Theorem 95 and Theorem 93 to see that we are done.
And finally, our main theorem:
Theorem 112. If NBG is consistent, then the NBG-provable modal logic of inner models is exactly S4.2Top.
Chapter 5
The Modal Logic of ccc Forcing
In this chapter, we improve upon the upper bounds forMLccc that were obtained in Theorem 67 by providing labellings for some Kripke frames. The labellings are a natural generalisation of the labelling of the two element fork that was used to show that S4.2 is not valid for MLccc in Theorem 67. In particular, we useω1-trees for these labellings as well.
In Section 5.2 we study two classes of frames, spiked pre-Boolean algebras, Definition 113, and
topless pre-Boolean algebras, Definition 116, each of which contain the two element fork frame. Of these, the first one is a notion that has not been considered in the literature before, whereas the second has been. We give upper and lower bounds with respect to some standard modal theories for both the modal theories that are characterised by these frames in the hierarchy of normal modal logics upto containment. In particular, both these theories occupy a similar place in this hierarchy with respect to many familiar logics. We then show that these two theories are not the same by showing that one of them is not contained in the other.
In Section 5.3 we describe the labelling of one of these classes of frames. This requires the existence of an arbitrarily large finite number ofω1-trees with some specific properties. In Section 5.3
we appeal to a theorem of Abraham and Shelah to obtain these trees in a model of ♦ω1. We do not,
however, prove this result or its corollary that we use.
The existence of these labellings depends on being able to understand when a tree can be specialised or made non-Aronszajn without adding a branch or specialising another. We study these properties in Section 5.4. Using the results from this section, we show in Section 5.5 that these labellings are complete. Finally, in the last section, Section 5.7, we discuss generalisations of this method and some questions related to the modal logic of ccc forcing.
5.1
Preliminaries
Recall that a partial order (T,≤) is called atree if for each elementt∈T, the set {s∈T |s < t}is well-ordered and if there is a unique node r∈T such that for every t∈T, r≤t. In this case,r is called the root of T. For an element t∈T, the level or height of t, levT(t), is defined to be the ordertype of the set {s∈ T |s < t}. We may omit mentioning T here if it is clear from context. Given an ordinalα, the αth level ofT, Tα is defined to be the set {t∈T |levT(t) =α}. The height of the tree T is defined to be the least ordinalλ such thatTλ=∅. In such a case, we callT a λ-tree.
In this report we will only talk about ω1-trees. An ω1-tree is normal if the following hold:
(ii) Ifα < ω1 is a limit ordinal, and if s, t∈Tα, then {p∈T |p < s} 6={p∈T |p < t}. Given p∈T, the tree Tp is defined as follows:
Tp
∆
={q ∈T |p≤q}.
While this terminology might seem to cause conflict with the definition of the αth level ofT, this will in practice not be so since elements of the tree will usually be denoted by letters of the English alphabet and ordinals by letters of the Greek alphabet.
A branch of T is a downwards-closed subsetb of T such that for anys, t∈b,skt. That is, a branch is a downwards-closed linearly ordered subset ofT. If T is an ω1-tree, then a branch b is
cofinal if its ordertype isω1. By default, a branch is assumed to be cofinal.
Henceforth, all (ω1-)trees that will be mentioned will be normal unless specified otherwise. We
will also assume that their levels are countable. This justifies the choice of wording of the next definition. An ω1-treeT isAronszajn if it has no cofinal branches. It isSuslin if all of its antichains
are countable. Hence, a Suslin tree is a ccc (forcing) poset. A special (ω1-)tree will be an Aronszajn
tree to which no ccc poset can add a branch to by forcing. Note that this is not the standard definition, which asserts that anω1-tree is special if it is the union of countably many antichains.
However, this definition is weaker, in the sense that every tree which is special in the ‘traditional’ sense is special in this sense as well. It follows that standard results about special trees such as the fact that any Aronszajn tree can be specialised by a ccc poset, and the standard method of constructing a special ω1-tree can be used when speaking about special trees in this sense as well.
The matter is discussed in some more detail in Section 5.3. Let S and T be two trees of the same height. Then
T ⊗S =∆{(x, y)∈T×S|levT(x) = levS(y)}.
This set with the co-ordinatewise ordering is clearly also a tree (we call this tree thetensor product
of the composite trees). Further, the poset we obtain from this tree can be embedded as a dense subset of the poset obtained from the product T ×S in a canonical way. This implies in particular that the largest size of an antichain in T×S is the same as the largest size of an antichain inT ⊗S (and vice versa). Also, this operation is clearly abelian, in the sense that S⊗T is isomorphic to T ⊗S. It is also associative, in the sense that ifT1, T2, T3 are Aronszajn trees, then (T1⊗T2)⊗T3 is
isomorphic toT1⊗(T2⊗T3). From this, we can, without any ambiguity, define the tensor product
of a finite set of trees as follows: ifX ={T1, T2· · ·Tn} is a set of trees, then O
X=∆ T1⊗T2⊗ · · · ⊗Tn.