[FOM] Graphs vs. Ord

Andrew Brooke-Taylor andrewbt at gmail.com
Thu Mar 28 02:27:41 EDT 2013

Dear Jan,

[A] is Vopenka's Principle (or equivalent to it, depending on your
definitions) and [B] is Weak Vopenka's Principle.  Vopenka's Principle
directly implies Weak Vopenka's Principle: adding the "not equals" relation
to structures turns a counter-example for WVP into a class of structures
with no non-identy homomorphisms, and then taking unions gives a
counter-example to VP (one does have this sort of liberty with the language
thanks to results you can find in Pultr & Trnkova's book "Combinatorial,
algebraic, and topological representations of groups, semigroups and
categories".  These results also imply for example that it doesn't matter
whether you mean "symmetric" or "directed" when you say "graph").  It
remains open whether WVP implies VP.

VP is at the upper end of the large cardinal hierarchy: it implies the
existence of a proper class of extendibles, and hence a proper class of
supercompact cardinals, but its consistency is implied by the existence of
an almost huge cardinal.  Note however that an inaccessible kappa such that
V_kappa satisfies VP need not even be weakly compact itself.  WVP remains
more mysterious: it implies the existence of a proper class of measurable
cardinals, but as far as we currently know could lie anywhere between there
and VP in consistency strength.

The best reference for all of this that I know of is Adamek & Rosicky's
book "Locally Presentable and Accessible Categories" (London Maths. Soc.
Lecture Notes #189), Chapter 6 and the Appendix.

Best wishes,
Andrew Brooke-Taylor

On 27 March 2013 04:34, <pax0 at seznam.cz> wrote:

> Hi All,
> what is the logical relationship between these two hypotheses:
> [A] Ordinals (considered as a thin category) cannot be fully embedded into
> the category of Graphs.
> (unordered, without loops, with morphisms graph homomorphisms)
> [B] The dual of Ordinals is such.
> And yet, what is the approximate set theoretic strength of these two
> claims?
> Thank you, Jan Pax
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> FOM at cs.nyu.edu
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Dr Andrew Brooke-Taylor
JSPS Postdoctoral Research Fellow
Kobe University
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