Iterated club shooting and the stationary-logic constructible model

TL;DR

This paper explores the construction and manipulation of the stationary-logic inner model $C(aa)$ through forcing, demonstrating the ability to produce models with complex iterated structures and preserving key set-theoretic properties.

Contribution

It introduces methods to force over $L$ to achieve models where $V=C(aa)$ and the sequence of iterated $C(aa)$ models can have arbitrarily large order types, using novel stationary-set preservation techniques.

Findings

Successfully forced models with $V=C(aa)$ from $L$

Constructed models with decreasing iterated $C(aa)$ sequences of large order types

Developed new distributivity results for club-shooting forcings using mutually stationary and fat sets

Abstract

We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated $C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.