Date: May 22, 2013 5:41 PM Author: fom Subject: Re: Grothendieck universe 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.