[FOM] Second-order choice principles

[Øystein Linnebo]

In light of the recent discussion of second-order and global choice
principles, I'd like to take the opportunity to ask a question that
has puzzled me for some time.

Let ZFCU be Zermelo Fraenkel set theory with urelemente, which are not
assumed to form a set. Consider the class theory that results from
adding to ZFCU second-order (or class) variables and quantifiers, as
well as some second-order (or class) comprehension scheme. Let
_Cardinal Comparability_ be the principle that any two classes X and Y
are comparable in size: either there is a class of ordered pairs which
code an injection of X into Y, or there is a class of ordered pairs
which code an injection of Y into X.

QUESTION: In a theory of the sort described, how strong is Cardinal
Comparability compared to more familiar second-order choice
principles?

It is easy to prove CC from Global Well-Ordering ("there is a class of
ordered pairs which code a well-ordering of the universe"). But
neither I nor several logicians I've consulted have been able to tell
whether the converse implication holds, or more generally where CC
falls among more familiar second-order choice principles.

The question is of obvious relevance to neo-Fregean approaches to
mathematics, which involve a comparison of classes (or  concepts ) as
to their size, as well as to limitation of size approaches to set
theory.


-- 
Dr Øystein Linnebo
Birkbeck, University of London