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.