```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 revolvesaround individuation and the role of the identityrelation (in contrast to general equivalence).  By aroundabout means, I have concluded that the appropriatenotion is V=OD (ordinal definability).  This followsfrom the relation between identity and definability.You can find an "answer" to questions that I have askedmyself in the link:http://mathoverflow.net/questions/55392/intended-interpretations-of-set-theories/55394#55394However, there are some twists associated with myviews.  I "get it" when it comes to the cumulativehierarchy 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" settheory because I choose to define my language primitivescircularly 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, Ihave no problem accepting V=L.  But, since I approachthese questions from a non-standard viewpoint, I amtrying to carefully put the pieces together.  I wishto understand my choice rather than simply agree withsome viewpoint.Although I am too verbose for your temperament,you would appreciate my views in the sense thatthey are motivated by topology.  In particular,the various ideas I choose to emphasize convergeon uniformities and uniform spaces.There is a final restriction on set theory thatmay come into play because of the role ofdefinability.  There are certain aspects ofdefinability that may involve a relation toprovability (Tarski has written a paper, and,Kleene discusses the eliminability of descriptionsin such terms.)  So, the last step in my thoughtprocess will be to look at the "strongly constructiblesets":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=falsehttp://www.ams.org/journals/bull/1963-69-04/S0002-9904-1963-10989-1/S0002-9904-1963-10989-1.pdfSo, your Occam's razor and my deliberationsseem to lead to similar places.
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