FOM: A slip comes to light

Adrian-Richard-David Mathias Adrian-Richard-David.Mathias at
Thu Aug 8 16:48:22 EDT 2002

Hazen's CONJECTURE is a variant of the Axiom H of Cole and Mitchell 
studied in my paper "The Strength of Mac Lane Set Theory" 
(APAL 110 (2001) pp 110-234). 

Axiom H says that for every set u there is a transitive set T 
such that every transitive set injectible into u is a subset of T. 

Proposition 3.2 of my paper says that over a certain weak base system 
which I call M_0, Axiom H is equivalent to the statement 
("Mostowski's principle") that every 
extensional well-founded relation is isomorphic to a transitive 

The forward direction is fine; but reflecting on the context with 
a proper class of Quine atoms, I now realise that the short derivation 
that I give in the paper, of Axiom H from Mostowski's principle, 
implicitly relies on the Axiom of Foundation , which is not among 
the axioms of M_0.  So my Proposition 3.2 is incorrect as stated. 
I *HOPE* that doesn't affect anything later on, but I haven't yet 
checked. Fortunately I am assuming Foundation for most of the time. 

A. R. D. Mathias

More information about the FOM mailing list