[FOM] Tarski universes vs Grothendieck universes (correction)

joeshipman@aol.com joeshipman at aol.com
Thu Jan 10 08:18:07 EST 2008

In my previous post "Historical Queries on AC", I referred to Tarski's 
proof that a "Universes Axiom" implies the Axiom of Choice. However, I 
(and Alama in a post from January 4 on the same subject) should have 
noted that there are two kinds of Universe, a Tarski Universe and a 
Grothendieck Universe, and the proof that Grothendieck Universes 
satisfy Tarski's Axiom itself requires AC. (A Tarski Universe does not 
have to be transitive, and satisfies not only Replacement but also that 
any subset which is not equipollent to the whole Universe is an element 
of the Universe.)

Therefore Tarski's proof that any set contained in a Tarski Universe 
can be well-ordered does not have the metamathematical significance I 
attributed to it, since the more natural Grothendieck Universes axiom 
does not imply AC, as Solovay has pointed out.

-- JS
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