# [FOM] Consistency strength order

pax0@seznam.cz pax0 at seznam.cz
Wed Apr 30 13:22:48 EDT 2008

Can someone of Dear FOMers make the notion of  "Consistency strength order" sufficiently clear to me?
For first order theories S, T (in the same language) we define that T has higher consistency strength then S:
S \le_{Cons} T iff P_S \subseteq P_T, where P_S and P_T are Pi^0_1 consequences of S or T, respectively.
Second definition could be
S  \le_{Cons} T iff U |-- Con(T) --> Con(S), where U is some weak base theory.
What is the relation between these two definitions, and how should the right definition read?
Thank you, J.P.