Abstract

In this paper I prove the following theorems which are the converses of some results of Judah and Laver (1983) and of Judah and Marshall (1993).-IfKM+ATW is not an extension by definition ofKM (and the model involved is well founded), then the existence of two inaccessible cardinals is consistent with ZF.-IfKM+ATW is not a conservative extension ofKM (and the model involved is well founded), then the existence of an inaccessible number of inaccessible cardinals is consistent with ZF.whereKM is Kelley Morse theory andKM+ATW isKM with types of well-orders