Known Errata in "Elementary Set Theory with a Universal Set"
I'm happy with the volume as a whole, and very grateful to the
editors and referees! None of the errors in the text, except the one
on p. 116, is anyone's fault but mine own...
The "blurb" on the back cover has some peculiar typos.
The worst error is:
p. 128 The object G used in the definition of sums and products of
indexed families of cardinals is not described correctly. Currently,
the text introduces G, incorrectly, as an element of the Cartesian
product of the indexed family F of cardinals. It is necessary to
stipulate further that the "index set" (the domain) of the indexed
family F of cardinals is a set of singletons; G is then correctly
specified as an element of SI^{-1}[\prod[F]]; i.e., SI{G}, not G
itself, belongs to the Cartesian product of F.
It would be even better to start with G: "Let G be an indexed family
of sets. Let F be the associated indexed family of cardinals, defined
by F({i}) = |G(i)|..." We could then define \prod[F] and \sum[F] in
the same forms given in the text.
In the proof of K\"onig's theorem on p. 132, the calligraphic A and B
functions are examples of the correct construction of G.
p. 132 It should be P-sub-1-super-2{B} in the proof of K\"onig's
Theorem, not P^2{B}. This is a mere "typo", but still distracting.
Other errors:
p. 71, repeated p. 74 There is an extra parenthesis in the definition
of Cartesian products of indexed families of sets, which might be
initially confusing.
p. 116 An obvious printer glitch; it should be possible to decipher.
p. 125 In the last proof, the occurrence of |A-Y|+|A| should be
|A-Y|+|Y|.
p. 148: The earlier comment given here is incorrect; the definition of
reals given on p. 148 succeeds. But there is an error: it is not correct
that every filter on bounded closed intervals in the rationals which
contains intervals of each positive length is an ultrafilter.
p. 173: The statement and proof of a theorem is missing here. I
assume without proving or even noting the assumption that for any rank
$X$ at or before $Z_0$, $T[X]$ is also a rank. This is true, and not hard
to prove, but it does need a proof (supplied on my web page).
p. 183 Both of the occurrences of T^2{Omega} in the proof of the
(correct) Theorem that No is an iterated cut system need to be
replaced with something else; in the first case we need to say that
the ranks are those indexed by elements of T^2[Ord] (the image of the
set of ordinals under the T^2 operation), and the second instance of
T^2{Omega} should be replaced by the limit of T^2[Ord], which is Omega
itself, not T^2{Omega}. The fact that lim T^2[Ord] = Omega is
discussed in the next chapter.
p. 190 In the definition of beth numbers, I neglected to stipulate
that each of the collections intersected to form the set of
beth-numbers must contain aleph-null.
Other notes for a purely hypothetical second edition [these are not
errors]:
p. 44: It might make sense to introduce the axioms of Set Union and
Singleton Images in this section along with Inclusion, since (as I
note on the next page) no essential use of them is made before this
point. This would require that some other definitions be brought
forward into this section as well. One could make a remark about the
fact that Set Union and Singleton Images together have the effect of
enforcing a precise parallelism between structure on objects and
structure on their singletons.
p. 82: It is silly, though not incorrect, to prove that A \times {0}
belongs to n when A does by mathematical induction; one should simply
exhibit the bijection and invoke stratified comprehension!
p. 84: The remark as to why m+n and mn are unique is really not
adequate, though the result is easy enough to be left as an exercise.
p. 110: When I read the proof at the top of this page for the first
time after receiving the printed copy, I found the step from the
existence of the map from {x} to seg{x} (x an ordinal) to the
existence of a similarity between <= on seg{\Omega} and SI^2{<=} to be
not as immediate as the author seems to expect it to be. The
intervening step is the observation that the map taking each {{x}} to
the order type of <= on seg{x} exists and is the similarity called
for.
p. 131: I would rather write in both of the Corollaries |P_1{V}| <
|P{V}| <= |V|, which makes it more obvious why it is a corollary.
This is just an esthetic point. It would also show a relationship
with Specker's theorem (which establishes that the second inequality
is also strict (indeed, more accurately written |P{V}| << |V|).
also p. 131: It isn't really an error, per se, but I should be talking
about binary expansions rather than decimal expansions here!
same chapter, no particular page: it might be nice to have more
cardinal arithmetic.
p. 173: The statement and proof of a theorem is missing here. I
assume without proving or even noting the assumption that for any rank
X at or before Z_0, T[X] is also a rank. This is true, and not hard
to prove, but it does need a proof!
Indication of Proof: First define T[S], for any set S of subsets of Z,
as the set of all T[X] for X in S.
Now consider H and T[H], where H is the set of ranks (written \cal H
in the book). If there is no X in H such that T[X] is not in H then
we have the desired result. Suppose on the contrary that X is the
first rank such that T[X] is not a rank, but each T[Y] for Y a rank
below X is a rank. X is either a successor rank, so X = P(Y), but
T[X] is not T[P(Y)] (this can only be true if Y is not a complete
rank, which puts X above Z_0 as desired) or a limit rank, in which
case X is the union of the Y's (this case is impossible, because T[X]
would then be the union of the T[Y]'s and itself a rank).
Since there _is_ an incomplete rank, there is in fact an X in H (the
successor P(Z_0) of the first incomplete rank Z_0) such that T[X] is
not in H (T[P(Z_0)] is a proper subset of P(T[Z_0])).