NON-FINITE AXIOMATIZABILITY (Generic structures and their applications)
Byunghan Kim · Institutional Repositories DataBase (IRDB) · 2004
We prove non-finite axiomatizability of some rank-l cj-categorical structures.Fix an $\omega$ -categorical infinite structure $M$ having a simple theory $T$ such that the universe itself is a solution set of (unique) rank-l Lascar strong 1-type $p^{1}(x)$ .Without loss of generality, we can assume the language $\mathcal{L}$ has only relational symbols.For $A\subseteq M$ , $acl(A)$ is an algebraic closure of $A$ in $M$ , and $ad^{eq}(A)$ is that in $M^{eq}$ .Now given $n>1,$ there are $i_{n}$ many $n$ -(independentObviously in $M$ , given $p_{j}^{n}(x_{1}$ , ..., $x_{n})$ , there is non-empty finite set $F(n,j)$ $\subseteq\{1$ , ...,Definition 0.1.Let $N$ be a subset of M. We say that $N$ is $k$ -generic substructure of $M$ for $k\geq 1$ if $N$ is an algebraically closed subset of $M$ such $that_{f}$ for any $m<k,$ and any tuple $(a_{1}, \ldots, a_{m})$ from $N$ with $M\models p_{j}^{m}(a_{1}, \ldots, a_{m})$ , and $l\in F(m,j)$ , there is $b\in N$ such that $M\models p_{l}^{m+1}(a_{1}, \ldots, a_{m}, b)$ .Lemma 0.2.There is a function $bd$ : $\omega$ $arrow$ $i$ ) (depending on $T$ ) satisfying the following: Let a be a sentence in $T$ having $k$ quantifiers (in its Prenex norrmal for $7m$).Suppose that $N$ is $bd(k)$ -generic substructure of M. Then $N\models\sigma$ .Proof.Let the function $pbd$ be defined in such a way that for any $j\leq m<k,$ and any tuple $\overline{e}=$ ($e _{1}$ ...e $m$ ) from $M$ , if $\overline{e}'=\{\mathrm{z}\mathrm{i}$ , ..., $e_{i_{j}}$ ) is the maximal independent subtuple, then there are at mostNow, to prove the lemma, it obviously suffices to prove the following.Claim) For $m