Moving up and down in the generic multiverse

  • 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

3 thoughts on “Moving up and down in the generic multiverse

  1. 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 $L$.

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