Some considerations on Amoeba forcing notions

TL;DR

This paper explores various amoeba-based notions in tree forcing, introducing new variants like amoeba-Silver and amoeba-Sacks, analyzing their decision properties, and discussing applications and generalizations.

Contribution

It introduces novel amoeba notions for tree forcing, proves their decision and property characteristics, and discusses their applications and generalizations.

Findings

Amoeba-Silver satisfies quasi pure decision but not pure decision.

Amoeba-Sacks satisfies the Laver property.

Generalized amoeba concepts lead to new questions in the field.

Abstract

In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.