Weak prediction principles

TL;DR

This paper investigates the consistency of certain weak prediction principles at large cardinals, establishing the failure of one and the equivalence of another to a cardinal arithmetic statement.

Contribution

It proves the consistency of the failure of the weak diamond at strongly inaccessible cardinals and shows the equivalence of the very weak diamond to a cardinal exponentiation condition.

Findings

Failure of the weak diamond at strongly inaccessible cardinals is consistent.

The very weak diamond is equivalent to $2^{<\lambda}<2^\lambda$.

The very weak diamond holds at all strongly inaccessible cardinals.

Abstract

We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and hence holds at every strongly inaccessible cardinal.