
Grothendieck universe
Posted:
May 21, 2013 4:51 AM


A Grothendieck universe is a set G of ZFG with the axioms:
for all A in G, B in A, B in G, for all A,B in G, { A,B } in G, for all A in G, P(A) in G, I in G, for all j in I, Aj in G implies \/{ Aj  j in I } in G.
The following are theorem of G:
for all A in G, {A} in G, for all B in G, if A subset B then A in G, for all A in G, A /\ B, A / B in G, for all A,B in G, (A,B) = { {A,B}, {B} } in G.
Are the following theorems or need they be axioms? If theorems, what would be a proof?
For all A,B in G, AxB = { (a,b)  a in A, b in B } in G.
I in G, for all j in I, Aj in G implies prod_j Aj in G.
For all A in G, A < G.
Is there any not empty Grothendieck universes.


