- J. D. Hamkins and B. Löwe, “Moving up and down in the generic multiverse,” Logic and its Applications, ICLA 2013 LNCS, vol. 7750, pp. 139-147, 2013.
`@ARTICLE{HamkinsLoewe2013:MovingUpAndDownInTheGenericMultiverse, AUTHOR = {Joel David Hamkins and Benedikt L\"owe}, title = {Moving up and down in the generic multiverse}, journal = {Logic and its Applications, ICLA 2013 LNCS}, publisher= {Springer Berlin Heidelberg}, editor= {Lodaya, Kamal}, isbn= {978-3-642-36038-1}, year = {2013}, volume = {7750}, number = {}, pages = {139--147}, doi= {10.1007/978-3-642-36039-8_13}, month = {}, note = {}, url = {http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse}, url = {http://arxiv.org/abs/1208.5061}, eprint = {1208.5061}, archivePrefix = {arXiv}, primaryClass = {math.LO}, abstract = {}, keywords = {}, source = {}, }`

In this extended abstract we investigate the *modal logic of the generic multiverse,* which is a bimodal logic with operators corresponding to the relations “is a forcing extension of”‘ and “is a ground model of”. The fragment of the first relation is the *modal logic of forcing* and was studied by us in earlier work. The fragment of the second relation is the *modal logic of grounds* and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.

The main theorems are as follows:

**Theorem.** If ZFC is consistent, then there is a model of ZFC whose modal logic of forcing and modal logic of grounds are both S4.2.

**Theorem**. If the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S4.2 and whose modal logic of grounds is S5.

**Theorem**. If the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S5 and whose modal logic of grounds is S4.2.

**Theorem**. There is no model of set theory such that both its modal logic of forcing and its modal logic of grounds are S5.

The current article is a brief extended abstract (10 pages). A fuller account with more detailed proofs and further information will be provided in a subsequent articl

eprints: ar$\chi$iv | NI12059-SAS | Hamburg #450

Dear Joel, could you sketch a bit why L-generic Cohen subsets of different regular cardinals are mutually generic?

Sure. The reason is that if $\gamma<\kappa$ are regular, then forcing to add a Cohen subset to $\kappa$ over $L$ adds no new dense subsets to the forcing to add a Cohen subset to $\gamma$. Thus, if $G$ and $H$ are each $L$-generic for $\text{Add}(\gamma,1)$ and $\text{Add}(\kappa,1)$, respectively, then $G$ remains $L[H]$-generic, and so they are mutually generic over $L$.

Thanks for the hint!