ON RESURRECTION AXIOMS

The resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone. who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection (which we call unbounded resurrection) and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin's Maximum⁺⁺. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, for example, failures of the weak square principle.