\NoBlackBoxes
\def\ac#1{#1}
\undefine\eth
%\input amstex
\documentstyle{surv}
\settabs\+\hskip.1in&\hskip.55in& \cr
\def\psc#1#2{{\tabalign & #1 & \vtop{\hsize=4.325in\parindent 0pt
#2\strut }&\cr}\vskip2pt}
\def\ps{\hangindent .3in\hangafter 1}
\def\iput#1{}
\topmatter
\endtopmatter
The changes listed here are mainly from a paper by
De la Cruz and Di Prisco:
\psc{}{ {\it Weak forms of the axiom of choice and partition
principles}, Set Theory, Di Prisco et al\. ed., Kluwer Academic Publishers,
Netherlands, 47-70.}
{\bf In part VI, the bibliography}, the following changes should be made.
\smallskip
\noindent 1. Change De la Cruz/Di Prisco \cite{1996} to
\psc{\cite{1998a}}{ {\it Weak choice principles}, Proc\.
Amer\. Math\. Soc\. {\bf 126}, 867-876.}
and make the corresponding changes in the other documents.
De la Cruz/Di Prisco \cite{1996} occurs as follows
\item{} In part I, the numerical list of forms, it occurs 7 times.
\item{} In part II, the topical list of forms, it occurs 7 times.
\item{} In part III, the models, it occurs 3 times.
\item{} In part IV, the notes, it occurs 0 times.
\item{} In part V, references for relationships between forms,
it occurs 3 times at positions (336,64), (378,132), and (376,377).
\noindent 2. Add to De la Cruz/Di Prisco
\psc{\cite{1998b}}{ {\it Weak forms of the axiom of choice and partition
principles}, Set Theory, Di Prisco et al\. ed., Kluwer Academic Publishers,
Netherlands, 47-70.}
\medskip
{\bf In parts I and II, the two lists of forms,} make the following changes.
\smallskip
\noindent 1. Add a comma to form 132 after "$PC(\infty$". The corrected form is:
\noindent{\bf FORM 132.} $PC(\infty, <\aleph_0,\infty)$: Every infinite
family of finite sets has an infinite subfamily with a choice
function. \ac{Blass} \cite{1977a} and \ac{Kleinberg} \cite{1969}.
\noindent 2. Add the following new forms.
\smallskip
\noindent{\bf FORM 11.} A Form of Restricted Choice for Families of
Finite Sets:
For every infinite set $A$, $A$ has an infinite subset $B$ such
that for every $n\in\omega$, $n>0$, the set of all $n$ element subsets
of $B$ has a choice function. \iput{restricted choice}
\ac{De la Cruz/Di Prisco} \cite{1998a}, \cite{1998b}.
\smallskip
\noindent{\bf FORM 12.} A Form of
Restricted Choice for Families of Finite Element Sets:
For every infinite set $A$ and every $n\in\omega$, there is an
infinite subset $B$ of $A$ such the set of all $n$ element subsets of
$B$ has a choice function. \iput{restricted choice}
\ac{De la Cruz/Di Prisco} \cite{1998a}, \cite{1998b}.
\smallskip
\noindent{\bf FORM 73.} $\forall n\in\omega$, PC$(\infty,n,\infty)$: For
every $n\in\omega$, if $C$ is an infinite family of $n$ element sets, then
$C$ has an infinite subfamily with a choice function. \iput{restricted choice}
\ac{De la Cruz/Di Prisco} \cite{1998a}, \cite{1998b}.
\smallskip
In part II put all of the new forms in the section
CHOICE FORMS, IV. Partial Choice
\smallskip
{\bf In the table of contents for part III}:
\smallskip
\noindent 1. Replace the entry for $\Cal N1$ with:
\toc
\head 1. $\Cal N1$: The Basic Fraenkel Model.\page{176}\endhead
\head $$ (6, 16, 23, 24, 37, 89, 112, 114, 115, 116, 127, 130,
133, 134, 135, 147, 191, 217, 232, 233, 263, 304, 305, 309, 313, 322,
325, 361, 363, 379, and 380 are true, but 15, 53, 64, 69, 76, 126,
128, 131, 146, 177, 200, 239, 267, 278, 292, and 344 are false.)\endhead
\endtoc
\smallskip
\noindent 2. Replace the entry for $\Cal N3$ with:
\toc
\head 3. $\Cal N3$: Mostowski's Linearly Ordered Model.\page{182}\endhead
\head $$ (6, 16, 23, 24, 37, 60, 91, 128, 130, 164, 165, 191,
305, 309, 313, 317, 325, 361, 363, 368, 369, 379, and 381 are true, but 15,
76, 84, 97, 118, 125, 126, 131, 147, 155, 156, 157, 200, 253, 290, 295,
296, 304, 355, and 376 are false.)\endhead
\endtoc
\smallskip
\noindent 3. Replace the entry for $\Cal N6$ with:
\toc
\head 6. $\Cal N6$: Levy's Model I.\page{185}\endhead
\head $$ (6, 37, 130, 191, 218, 305, 313, 361, and 363
are true, but 154, 164, 171, 308($p$), 314, 334, 344, 358,
and 379 are false.)
\endhead
\endtoc
\smallskip
\noindent 4. Replace the entry for $\Cal N49$ entry with
\toc
\head 49. $\Cal N49$: De la Cruz/Di Prisco Model.\page{217}\endhead
\head $$ (6, 9, 37, 63, 91, 130, 191, 305, 309, 313, 361,
and 363 are true, but 47($n$), 106, 163, 167, 344, 379, and 380
are false.)\endhead
\endtoc
\smallskip
{\bf In part III, models,} make the following changes.
\noindent 1. In $\Cal N1$ replace the part after
Therefore, form
328 ($MC(WO,\infty)$) is false because $122 + 328\to 40$.
with:
De la Cruz and Di Prisco have shown that every infinite collection
of non-empty well orderable sets has an infinite subfamily with a
choice function (380 is true) and that every infinite collection of
non-empty sets has an infinite subfamily with a Kinna-Wagner selection
function (379 is true). It is shown in
\ac{Howard/Keremedis/Rubin/Stanley} \cite{1997} that form 232
(Every metric space $(X,d)$ has a
$\sigma$-discrete basis.) is true. Since form 165 ($C(WO,WO)$) is
true, (133 implies 165) it follows from note 2(8 and 9) that 16 and
24 are also true.
\smallskip\noindent
$\Cal N1\models$ 6, 16, 17, 23, 24, 31, 37, 63, 89, 91,
112, 114, 115, 116, 127, 130, 133, 134, 135, 147, 191,
217, 232, 233, 263, 273, 304, 305, 309, 313, 322,
325, 361, 363, 368, 369, 379, and 380, but 15, 53, 64, 68, 69,
76, 106, 126, 128, 131, 146, 177, 200, 239, 267, 278, 292,
323, 328, and 344 are false.
References include \ac{Blass} \cite{1977a}, \ac{Brunner} \cite{1981a},
\cite{1982a}, \cite{1983d}, \cite{1984b}, \cite{1984f}, \cite{1985a},
\ac{Dawson/Howard} \cite{1976}, De la Cruz/Di \ac{Prisco} \cite{1998a},
\ac{Fraenkel} \cite{1922}, \ac{Felgner/Jech} \cite{1973}, \ac{Felgner}
\cite{1971a}, \ac{Jech} \cite{1973b}, \ac{Jech/Sochor} \cite{1966a},
\ac{Halpern} \cite{1964}, \ac{Harper/Rubin} \cite{1976}, \ac{Hickman}
\cite{1976}, \ac{Hodges} \cite{1974}, \ac{Howard/Keremedis/Rubin/Stanley}
\cite{1997}, \ac{Howard/Rubin} \cite{1977}, \ac{L\"auchli} \cite{1962},
\ac{Levy} \cite{1958}, \ac{Pincus} \cite{1969}, \ac{Specker}
\cite{1957}, \ac{Stavi} \cite{1975}, notes 2(8,9), 18, 41, 46, 52, 64,
66, 88, 89, 105, 109, 116, 120(56) and 123.
\smallskip
\noindent 2. In $\Cal N3$ replace the part after
Since form 165 ($C(WO,WO)$) is true, (60
implies 165) it follows from note 2(8 and 9) that 16 and 24 are also true.
with:
De la Cruz and Di Prisco have shown that every infinite family of
non-empty sets has an infinite subset with a Kinna-Wagner
selection function (379 is true). They have also pointed out
that the atoms have no infinite subset $B$ such that collection of
all subsets of $B$ of cardinality greater than 2 has a Kinna-Wagner
selection function (376 is false). Ramsey's theorem (325) is true
by an argument very similar to the proof that 325 is true in $\Cal N1$
given by \ac{Blass} in \cite{1977a}.
\smallskip\noindent
$\Cal N3\models$ 6, 14, 16, 23, 24, 31, 37, 60, 83, 91, 128,
130, 164, 165, 191, 273, 305, 309, 313, 317, 325, 361, 363,
368, 369, 379, and 381, but 15, 84, 90, 97, 106, 118,
125, 126, 131, 147, 155, 156, 157, 200, 253, 290,
295, 296, 304, 355, and 376 are false. References \ac{Mostowski} \cite{1939},
\ac{Blass} \cite{1977a}, \ac{Brunner} \cite{1982a}, \cite{1983a},
\cite{1983c}, \cite{1983d}, \cite{1984c}, \cite{1984f}, \cite{1985c},
\ac{Dawson/Howard} \cite{1976}, De la Cruz/Di \ac{Prisco} \cite{1998b},
\ac{Gonzalez} \cite{1995a}, \ac{Halpern} \cite{1964}, \ac{Howard}
\cite{1973}, \ac{Howard/Keremedis/Rubin\slash Rubin} \cite{1997b},
\ac{Howard/Yorke} \cite{1989}, \ac{Jech} \cite{1973b}, \ac{Krom}
\cite{1986}, \ac{L\"auchli} \cite{1964}, \ac{Levy} \cite{1958},
\ac{Pincus} \cite{1969}, \cite{1997}, \ac{Sageev} \cite{1981},
notes 2(8,9), 18 and 120(34 and 45).
\smallskip
\noindent 3. In $\Cal N6$ replace the part after
but for each $n\in\omega$,
$n>0$, $C(\infty,n)$ (61) is true.
with
(It is also clear that $\{P_n: n\in\omega\}$ has no infinite subset
with a Kinna-Wagner selection function so
$KW(\aleph_0,<\aleph_0)$ (358) and $PKW(\infty,\infty,\infty)$ are
also false.) Levy, also proves the
axiom of Multiple Choice (67) is true. However, Bleicher has shown that
$(\forall n\in\omega)MC(\infty, \infty,$ relatively prime to $n$) (218)
is equivalent to 61 + 67, so 218 is also true. Since 218 implies 333
($MC(\infty,\infty,\hbox{ odd})$) and Keremedis has shown that $333 +
334 (MC(\infty,\infty,\hbox{ even})) \leftrightarrow$ AC, it follows
that 334 is false. Since 218 is true, and therefore, [218 A] (Existence
of Complementary Subspaces) is also true, it follows that 95($F$)
(Existence of Complementary Subspaces over a Field $F$) is true.
\par
Shannon proves if $T =\{f: \exists n f\hbox{ is a choice function on }
\{P_0,P_1,\cdots,P_n\}\}$ with the partial order $f\le g$ iff $g\subseteq
f$, then $T$ is a denumerable union of finite sets, all antichains are
finite, and there is a denumerable family of dense sets for which there in
no generic filter (171 is false). See note 47 for definitions. In
\ac{Howard/Yorke} \cite{1987} it is shown that for any prime $p$, there
is a set of finite groups $\{ G_y : y\in Y\}$ such that the weak direct
product has no maximal $p$-subgroup
(308($p$) is false in $\Cal N6$ for any prime
$p$). Since form 106 (Baire category Theorem for compact Hausdorff spaces)
is true (218 implies 106) and 43 (Principle of Dependent Choices) is
false (43 implies 171), form 154 (the Tychonoff theorem for countably many
$T_2$ spaces.) must be false. (106 + 154 implies 43. See \ac{Brunner}
\cite{1983c}) Degen has shown
that if $\rho_i = \pi_i/P_i$, $\rho_i$ is not defined on $A-P_i$, and
$\phi=\bigcup_ {i\in\omega}\rho_i$, then $\phi$ cannot be expressed as
the product of two reflections. ($\phi$ is a reflection if $\phi^2=\phi$.)
Consequently, form 314 (Every permutation on a non-empty set can be
expressed as a product of two reflections.) is false.
\smallskip\noindent
$\Cal N6\models$ 6, 37, 61, 67, 95, 130, 191,
218, 273, 305, 313, 361, and 363, but 10, 154, 164, 171, 308($p$),
314, 334, 344, 358, and 379 are false. References \ac{Bleicher} \cite{1965},
\ac{Brunner} \cite{1984b}, \ac{Degen} \cite{1988}, \ac{Howard/Yorke}
\cite{1987}, \ac{Jech} \cite{1973b} (Theorem 7.11 and prob 7.15),
\ac{Keremedis} \cite{1996a}, \ac{Levy} \cite{1962},
\ac{H.~Rubin/J.~Rubin} \cite{1985}, \ac{Shannon} \cite{1990},
notes 18, 35, 95, and 120(2 and 56).
\smallskip
\noindent 4. In $\Cal N49$ replace the part after:
$S$ is the set of finite
supports. De la Cruz and Di Prisco have shown that every infinite set
in this model is Dedekind infinite (9 is true) and that
$\{ A_i : i\in\omega \}$ is a countable family of well orderable sets
such that no infinite subfamily has Kinna-Wagner selection function.
with
It follows that 167, 379, and 380 are false. In addition De la Cruz
and Di Prisco have shown that for every $n\in\omega$, $n\ge 2$,
there is a set of $n$ elements sets in the model with no choice function
(47(n) is false).
\smallskip\noindent
$\Cal N49 \models$ 6, 9, 37, 63, 91, 130, 191, 305, 309,
313, 361, and 363, but 47(n), 106, 163, 167, 344, 379 and 380 are false.
References De la Cruz/Di \ac{Prisco} \cite{1998a}, notes 18 and
120(20, 23, 55 and 56).
\smallskip
{\bf In part V, references for relationships between forms} a long list
of changes were made. The corrected TeX version of part V can be
downloaded from the ``changes'' web page from which you accessed this file.
\end