Date: May 21, 2013 10:44 PM Author: fom Subject: Re: Grothendieck universe On 5/21/2013 9:21 PM, William Elliot wrote:

> 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_i|i in I}

>> Then prod_j A_j would be a subset of P(P(\/{A_i|i 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

> non-trivial Grothendeick universe. Doesn't |V_omega0| = aleph_omega0

I believe this is correct.

> which is almost always big enough for mathematics?

>

Well, that depends on what you mean by "mathematics".

I wrote a set theory that includes a universal class.

I believe it is minimally modeled by an inaccessible

cardinal (when the axiom of infinity is included). My

argument for such a structure is that the philosophy

of mathematics ought to be responsible for the ontology

of its objects. So, I reject predicativist views that

take "numbers" as given.

My views, however, are non-standard and I am still working

at how to understand them in relation to standard paradigms.