
Re: Grothendieck universe
Posted:
May 21, 2013 10:21 PM


On Tue, 21 May 2013, fom wrote: > On 5/21/2013 3:51 AM, William Elliot wrote:
> > 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. > > ... and closure under unions, right? See FOM post listed > at bottom. It's a theorem. A,B, { A,B } in G, \/{ A,B } in G. In fact, the last in equivalent to for all A in G, \/A 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. > > Are not Cartesian products sometimes explained > as subsets of P(P(A \/ B))?
(a,b) = { {a,b}, {b} }. Ok, that's simple.
Also if a,B in G, then {a}xB = \/{ (a,b)  b in B } in G and if also A in G, then AxB = \/{ {a}xB  a in A } in G
> > I in G, for all j in I, Aj in G implies prod_j Aj in G.
> Along the same lines, wouldn't this involve > an application of replacement to form the {A_ii in I} > Then prod_j A_j would be a subset of P(P(\/{A_ii in I})) > No, P(P(I \/ \/{ Aj  j in I }))
> > For all A in G, A < G. > Wouldn't the closure axiom on power sets make this true? > > Closure under power sets. Elements of an element > is an element. So, if any element were equipollent > with the universe, the universe would have the > cardinal of its own power set. Right? Ok
> > Is there any not empty Grothendieck universes. > > ...any favorite set theory > > http://en.wikipedia.org/wiki/Grothendieck_universe > > There are two simple universes discussed (empty set > and V_omega). The rest are associated with the > existence of strongly inaccessible cardinals.
The latter don't exist in ZFO. So V_omega0 is the only nontrivial Grothendeick universe. Doesn't V_omega0 = aleph_omega0 which is almost always big enough for mathematics?
> You might find this to be of interest, > > http://www.cs.nyu.edu/pipermail/fom/2008March/012783.html > > http://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory >

