A Multiverse Axiom Induction Framework

A MULTIVERSE AXIOM INDUCTION FRAMEWORK
The multiverse paradigm in set theory does not only reflect philosophical preferences, or set up a new playground for mathematical investigation, but it also offers a powerful methodological tool for investigating the conceptual foundations of set theory by guiding the search for and the evaluation of new set-theoretic axioms or facts. The prototypical example is Friedman's Hyperuniverse Program (HUP). Our goal is to develop an abstract inferential framework generalizing the HUP whose (defeasible, inductive) inference methods are meant to identify or validate new axioms. We consider nonmonotonic consequence relations |~, parametrized by specifications of the multiverse and set-theoretic desiderata, which associate with any suitable ZFC + X new - not necessarily classically derivable - candidate truths. The desiderata could, for instance, consist of consistency conditions or maximization demands w.r.t. preorders over universes. There are a number of possible inductive strategies, but it doesn't seem that conceptual considerations at the level of set theory are sufficient to decide among them. The idea is therefore to also assess the inferential level and to use rationality postulates for nonmonotonic inference, heavily investigated within AI for modeling commonsense reasoning, to classify and evaluate such procedures. This is however a non-trivial task because of the special characteristics of axiom induction.