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Topic: Grothendieck universe
Replies: 5   Last Post: May 22, 2013 5:41 PM

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William Elliot

Posts: 1,556
Registered: 1/8/12
Grothendieck universe
Posted: May 21, 2013 4:51 AM
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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.

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