タイプの排除と順序型
坪井明人 (Akito Tsuboi)筑波大学数理物質科学研究科
(Graduate
School
of Pure and Applied
Sciences, University of
Tsukuba)
Abstract 可算言語 $L$ が順序のための2変数述語 $<$ を含むとする. $T$ を $L$ で 表現された理論とする. Lopez-Escobar の定理により, タイプ$p(x)$ を 排除する $T$ のモデルで, 順序型が $\omega_{1}$ のものが存在すれば, 同じタイプ $p(x)$ を排除して, 整列でないモデルが存在することがわかる. この結 果は証明方法を変えることなく, 可算個のタイプの集合の場合に拡張さ れる. 本稿においては, 完全なタイプを扱う限りにおいては, タイプの数 が連続濃度未満の場合でも同様な結果が成り立っことをを示す. これか ら Morley のタイプ排除定理のある種の変形が得られる.
Proposition 1 Let $L$ be a countable language $with<$
.
Let $T$ bean
L-theory and $p(x)$ a type. Let $M\models T$ be a model omitting $p(x)$ such
that $otp(<M)=\omega_{1}$
.
Then there is a model $N\models T$ with$\bullet$ $N$ omits $p(x)$, and
$\bullet$ $N$ has an
infinite
descending sequence
with respect $to<^{N}$.
Proof:
Weassume
$L$ contains all the Skolem functions. We preparea
countable set $X=\{x_{i} : i\in\omega\}$ of variables. Let $\{t_{i} : i\in\omega\}$ be anenumeration of all the L-terms whose variables belong to $X$
.
We mayassume
that the variables of $t_{n}$ is contained in $\overline{x}_{n}=x_{0},$ $\ldots,x_{n-1}$.
Sowe may
assume
$t_{n}=t_{n}(\overline{x}_{n})$.
By $otp(<M)=\omega_{1}$, we assume $(M, <)=(w_{1}, <)$
.
By induction on$n\in\omega$, we choose sets $S_{n}\subset\omega_{1}$ and formulas $\varphi_{n}(x)\in p(x)$ with the
following properties:
1. $S_{n}\subset\omega_{1}$ is
a
set ofdescendingsequences
oflength$n$
.
数理解析研究所講究録
2. For
each $i\in\omega_{1},$ $S_{n}|i=\{(a0, \ldots, a_{n-1})\in S_{n} : a_{n-1}>i\}$ has thecardinality $\omega_{1}$
.
3.
If $(a0, \ldots, a_{n-1}, a_{n})\in S_{n}$ then $(a0, \ldots, a_{n-1})\in S_{n-1}$,
4. For all $\overline{a}\in S_{n}$,
we
have $M\models\neg\varphi_{n}(t_{n}(\overline{a}))$.
Now we
assume
thatwe
have successfully chosen $S_{n}$ and $\varphi_{n}(x)$.
Foreach $i\in w$, using condition 2, choose $\overline{a}_{i}$ from $S_{n}|i$
.
Let $\overline{a}(i)$ denote thesequence $\overline{a}_{i},i$
.
Notice that $\overline{a}(i)$ isa
decreasing sequence of length$n+1$.
Now cQnsider elements $t_{n+1}(\overline{a}(i))(i\in w)$.
Since $M$ omits $p(x)$, wecan
choosea
formula $\varphi_{n+1}^{i}(x)\in p(x)$ with $M\models\neg\varphi_{n+1}^{i}(t_{n+1}(\overline{a}(i)))$.
Since there
are
only countably many such formulas,we can
choosean
uncountable set $U\subset\omega_{1}$ such that forany
$i\in U\varphi_{n+1}^{i}$ is thesame
formula. Let $\varphi_{n+1}$ be thefixed formula. Weput $S_{n+1}=\{\overline{a}(i) : i\in U\}$
.
Now it is easy to
see
that $S_{n+1}$ and $\varphi_{n+1}$ satisfyour
requirements.The following claim is easily proven, using $S_{n}’ s$
.
Claim A $\Gamma(x_{0},x_{1}, \ldots)=\{x_{0}>x_{1}>x_{2}> \}\cup\{\neg\varphi_{n}(t_{n}(\tilde{x}_{n}))$ : $n\in$
$w\}$ is a con8i8tent $8et$
.
Let $I=(b_{\dot{j}})_{i\in w}$ realize F. Let $N$ be the Skolemhullof I. $N\supset I$ has
an
inifinite descending sequence. Since $\{t_{n}\}$ is
an
enumeration of all theSkolem terms, we have $N=\{t_{n}(\overline{b}_{n}) : n\in\omega\}$, where $\overline{b}_{n}=b_{0},$
$\ldots,$$b_{n-1}$
.
Since each $t_{n}(\overline{b}_{n})$ satisies $\neg\varphi_{n}(x)$,
we
conclude that $N$ omits $p(x)$.
Corollary 2 Let $T$ be a countable complete theory and $R$
a
setof
complete types with $|R|<2^{w}$
.
Suppose thatfor
each $i<\omega_{1}$,
there $i_{8}a$model $M_{i}\models T$ with the following propertie8:
1. $|M_{i}|\geq 3(\omega)$,
2. $M_{i}$ omits each member
of
$R$.
Then
for
each $\kappa$ there $i_{8}$ a model $M$ omitting $R$ and with $|M|\geq\kappa$.
References
[1] Chen Chung ChangH. Jerome Keisler, Model Theory (Studies in
Logic and the Foundations ofMathematics),
1990.
[2] Saharon Shelah, Classification Theory and the Number of
Non-Isomorphic Models (Studies in Logic and the Foundations of
Mathematics), 1990.
[3] Newelski, Omitting types and the real line, Journal of Symbolic
Logic, vol. 62 (1987),
