🚀 go-pugleaf

It should be noted that it _is_ possible to define "cardinal number of
a well-founded set" in ZF- (the usual definition without choice,
relativized to the well-founded sets).  This demolishes completely
Zeleny's assertion that the notion of set in the usual set theory
depends on the prior notion of the ordinals.  ZF- certainly does not
depend on the ordinals for the definition of its concept of set, but
successfully defines the class of well-founded sets, which is the
universe of ZF, and the class of ordinals itself.  It cannot define a
general concept of cardinal number, though it _can_ define the notion
of two sets having the same cardinality, but it can define the notion
of cardinal needed in ZF; one then restricts one's attention to the
well-founded sets and one is doing ZF.  One could restrict oneself
further to ZFC by making the further restriction to the constructible
sets.  This makes it clear that the notion of cardinal number (but not
the notion of having the same cardinal) in ZF depends on the notion of
"well-founded set", not on the narrower notion of "ordinal".

The definition of absoluteness goes as follows:  let M be a transitive
class (a class which contains all elements of its elements).  A
formula P is absolute for M if the formula is equivalent to its
version with all quantifiers restricted to M, for any choice of
parameters from M.

Any formula in which each quantifier is restricted to a set is clearly
absolute if it makes sense in M at all (because all elements of a set
in M are in fact in M; if the set to which the quantifier is
restricted does not exist in M, the formula doesn't make sense in M).

It is possible to define "ordinal" using a restricted formula: "x is
transitive and x is linearly ordered by membership" iff x is an
ordinal, and every quantifier in the definition can be restricted to x
itself.  It is not possible to define cardinality using a restricted
formula: |A| = |B| if there is a bijection f from A to B; the problem
is that the transitive class M might not contain any witness f to this
fact, so, relative to M, A and B might have different cardinalities
due to the fact that no witness f to their equinumerousness is in M
(it would not be the case that there was a bijection f _in M_ from A
to B).

The non-definability of the cardinal numbers in ZF- is a different
kind of problem.  Equinumerousness (having the same cardinal) is
perfectly definable in ZF-; the problem is that it is impossible to
pick a canonical set to represent each cardinality in ZF-.
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu