FOM: Completion of previous posting

[Todd Wilson]

On Fri, 25 Feb 2000, JoeShipman at aol.com wrote:
>  1) Vopenka's Principle is one of my favorite large cardinal axioms,
>  since it is better motivated and easier to state than almost all of
>  them (inaccessibles and measurables are easier still, but I can't
>  think of any others that are easier).
>  
>  2) This is a use of large cardinals to settle an open problem in "ordinary 
>  mathematics" (excluding logic, foundations, set theory).  Have
>  there been any other good examples of this since Solovay's work on
>  real-valued measurables?

I don't know if this is exactly what Shipman was looking for, but,
since Vop^enka's principle is one of his favorites, he might be
interested in the book

    Ji^r'i Ad'amek, Ji^i'i Rosick'y, Locally Presentable and
    Accessible Categories, LMS Lecture Notes Series 189, Cambridge
    University Press, 1994,

especially

    Chapter 6 (40 pages): Vop^enka's Principle
    Appendix  (14 pages): Large Cardinals

and the list of 16 open problems with which the book ends, 12 of
which are settled in the affirmative using Vop^enka's Principle (or
the weak Vop^enka Principle).

The subject matter of the book is a very general but yet very useful
class of categories that includes all of the usual algebraic and
semi-algebraic categories (the varieties and quasi-varieties of
universal algebra) and all Horn classes of relational structures.

(I apologize for my ASCII attempts at the ^Cech diacritics;  I'm not
Unicode-enabled yet.)

-- 
Todd Wilson
Computer Science Department
California State University, Fresno