Recognizable sets and Woodin cardinals: Computation beyond the constructible universe

TL;DR

This paper explores the concept of recognizable sets of ordinals via ordinal time Turing machines, establishing their connection to large cardinal axioms like Woodin cardinals and inner models such as M-infinity.

Contribution

It introduces the notion of recognizability for sets of ordinals and links it to large cardinal hypotheses, showing the extent of recognizable objects in the hierarchy.

Findings

Recognizable objects with infinite time computations reach up to Woodin cardinals.

Computable sets are contained in the constructible universe L.

Recognizable closure corresponds to the inner model M-infinity.

Abstract

We call a subset of an ordinal $\lambda$ recognizable if it is the unique subset $x$ of $\lambda$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $\lambda$ as input, halts with the final state $0$. Equivalently, such a set is the unique subset $x$ which satisfies a given $\Sigma_1$ formula in $L[x]$. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. (1) Computable sets are elements of $L$, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals. (2) A subset of a countable ordinal $\lambda$ is in the recognizable closure for subsets of $\lambda$ if and only if it is an element of $M^{\infty}$, where $M^{\infty}$ denotes the inner model obtained by iterating the least measure of $M_1$ through the ordinals, and where the recognizable closure for subsets of $\lambda$ is defined by closing under relative recognizability for subsets of $\lambda$.