Set-theoretic blockchains

  • M. E. Habič, J. D. Hamkins, L. D. Klausner, J. Verner, and K. J. Williams, “Set-theoretic blockchains,” ArXiv e-prints, pp. 1-23, 2018. (under review)  
    @ARTICLE{HabicHamkinsKlausnerVernerWilliams2018:Set-theoretic-blockchains,
    author = {Miha E. Habič and Joel David Hamkins and Lukas Daniel Klausner and Jonathan Verner and Kameryn J. Williams},
    title = {Set-theoretic blockchains},
    journal = {ArXiv e-prints},
    year = {2018},
    volume = {},
    number = {},
    pages = {1--23},
    month = {},
    note = {under review},
    abstract = {},
    eprint = {1808.01509},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    keywords = {under-review},
    source = {},
    doi = {},
    url = {http://wp.me/p5M0LV-1M8},
    }

Abstract. Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.

Nonamalgamation in the Cohen generic multiverse, CUNY Logic Workshop, March 2018

This will be a talk for the CUNY Logic Workshop on March 23, 2018, GC 6417 2-3:30pm.

Abstract. Consider a countable model of set theory $M$ in the context of all its successive forcing extensions and grounds. This generic multiverse has long been known to exhibit instances of nonamalgamation: one can have two extensions $M[c]$ and $M[d]$, both adding a merely a generic Cohen real, which have no further extension in common. In this talk, I shall describe new joint work that illuminates the extent of non-amalgamation: every finite partial order (and more) embeds into the generic multiverse over any given model in a way that preserves amalgamability and non-amalgamability. The proof uses the set-theoretic blockchain argument (pictured above), which has affinities with constructions in computability theory in the Turing degrees. Other arguments, which also resemble counterparts in computability theory, show that the generic multiverse exhibits the exact pair phenonemon for increasing chains. This is joint work with Miha Habič, myself, Lukas Daniel Klausner and Jonathan Verner. The paper will be available this Spring.