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

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fom

Posts: 1,969
Registered: 12/4/12
Re: Grothendieck universe
Posted: May 22, 2013 5:41 PM
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On 5/22/2013 12:02 AM, William Elliot wrote:
> On Tue, 21 May 2013, fom wrote:
>>>> 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.

>
> I'm a bearded prochoice mathematican who shaves with Occams
> razor. Thus ZFO, ZF + Occams razor proves GCH, hence AxC
> and no inaccessibles. I've yet to determine if ZFO proves
> V = L. Perhaps it does.
>
> How does ZFO jib with your views?
>


While I understand the motivation for Ockham's razor,
I do not take it as a guiding principle. It is, however,
obvious that it works for you.

My focus in foundational mathematics always revolves
around individuation and the role of the identity
relation (in contrast to general equivalence). By a
roundabout means, I have concluded that the appropriate
notion is V=OD (ordinal definability). This follows
from the relation between identity and definability.

You can find an "answer" to questions that I have asked
myself in the link:

http://mathoverflow.net/questions/55392/intended-interpretations-of-set-theories/55394#55394

However, there are some twists associated with my
views. I "get it" when it comes to the cumulative
hierarchy that follows from the axiom of foundation.
So, the next restriction would yield V=HOD.
(hereditarily ordinal definable)

Next, my theory would be considered "second-order" set
theory because I choose to define my language primitives
circularly with the sentences:

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

According to Kunen, L=HOD in second-order. Thus, I
have no problem accepting V=L. But, since I approach
these questions from a non-standard viewpoint, I am
trying to carefully put the pieces together. I wish
to understand my choice rather than simply agree with
some viewpoint.

Although I am too verbose for your temperament,
you would appreciate my views in the sense that
they are motivated by topology. In particular,
the various ideas I choose to emphasize converge
on uniformities and uniform spaces.

There is a final restriction on set theory that
may come into play because of the role of
definability. There are certain aspects of
definability that may involve a relation to
provability (Tarski has written a paper, and,
Kleene discusses the eliminability of descriptions
in such terms.) So, the last step in my thought
process will be to look at the "strongly constructible
sets":

http://books.google.com/books?id=RTsx5M0HbdMC&pg=PA185&lpg=PA185&dq=%22strongly+constructible%22+Cohen+%22minimal+model%22&source=bl&ots=Z8_-cX9tn4&sig=2RhGNeHjB7-QYTyuBRSVkhEG0KQ&hl=en&sa=X&ei=JzmdUcikLui7ygG4qoBw&ved=0CCoQ6AEwADgK#v=onepage&q=%22strongly%20constructible%22%20Cohen%20%22minimal%20model%22&f=false

http://www.ams.org/journals/bull/1963-69-04/S0002-9904-1963-10989-1/S0002-9904-1963-10989-1.pdf


So, your Occam's razor and my deliberations
seem to lead to similar places.







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