# FOM: McLarty's questions about my book

Randall Holmes holmes at catseye.idbsu.edu
Mon Mar 23 12:19:45 EST 1998

(Colin asked)

>(3) sets have Boolean unions
>(4) sets have unions in the usual sense
>
>       [4 is an infinitary version of 3]

Does this mean 4 does not imply 3? Is that due to a lack
of unordered pair sets, or something more subtle about sets of
sets?

(end query)

The reason that (4) does not follow from (3) is that I don't have a
pair set axiom; this is simply a matter of style: I have boolean union
and singleton set instead of pair set.  (4) could equally well have
been given at the end, along with the inclusion relation, because it
plays no role in the development before the proof of stratified
comprehension.

>(5) the equality relation and the projection relations of the pair exist

Can we define a "function from a set x to a set y" in
the usual set theoretic way (a single-valued relation between
elements of x and those of y, defined for each element of x), and
conclude (from existence of the equality relation) that every
set x has a one-to-one function to V?

(end query)

Certainly for any set X one can take (X \times X) \cap [=] (the
intersection of the Cartesian product of X and X with the equality
relation) and obtain the identity function on X (which is an injection
from X into V).  Is that all that is being asked?  The definitions of
relation and function are the obvious ones.  [=] itself, the equality
relation is also the universal identity function.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes