Cohen-like first order structures

TL;DR

This paper explores uncountable structures similar to Fra"issé limits, using forcing to construct generic structures that are sensitive to set-theoretic universe changes, and demonstrates their rigidity and automorphism properties.

Contribution

It introduces a forcing-based method to construct uncountable structures akin to Fra"issé limits, revealing their set-theoretic sensitivity and automorphism characteristics.

Findings

Most uncountable generic structures are rigid.

Existence of a dense set of reals with automorphism group of integers.

Structures depend on the set-theoretic universe.

Abstract

We study uncountable structures similar to the Fra\"iss\'e limits. The standard inductive arguments from the Fra\"iss\'e theory are replaced by forcing, so the structures we obtain are highly sensitive to the universe of set theory. In particular, the generic structures we investigate exist only in generic extensions of the universe. We prove that in most of the interesting cases the uncountable generic structures are rigid. Moreover, we provide a (consistent) example of an uncountable, dense set of reals with the group of integers as its automorphism group.