Set-Theoretic Blockchains

TL;DR

This paper introduces the blockchain construction method to analyze the structure of the generic multiverse in set theory, enabling embeddings of various posets and exploring the relationships among forcing extensions and ground models.

Contribution

It develops the blockchain construction technique to embed complex posets into the generic multiverse, improving upon previous results and extending to class forcing in second-order set theory.

Findings

Finite posets embed into the multiverse preserving nonexistence of upper bounds.

The blockchain method accommodates infinite posets and various forcing notions.

Existence of exact pairs in the generic multiverse.

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.