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In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ $fragment$ from an infinitary lenguage, and a complete theory $T$ ).

He starts proving that $D(F,T)$ (the set of all possible $F$-diagrams of models of $T$) is Borel and then he constructed a continous map $\psi$ such that $S_n(F,T)$ is the image of $D(F,T)$ under this map.

The thing is that I don't understand quite well how this map works. Can anyone help me please.

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up vote 2 down vote accepted

I really don't know exactly what you are asking, but this is too long for a comment.

Looking at the mapping $\Psi$ on p.162 instead as a function $\mathcal{P} (F) \to \mathcal{P} (F)$, it is not too difficult to see that $$\Psi (A) = \{ \phi \in A : \phi \text{ has free variables } v_0, \ldots , v_{n-1} \}.$$ That is, $\Psi(A)$ is just the subset of $A$ consisting of those formulae with the "correct" free variables.

If $D$ is an $F$-diagram, then there is a countable model $\mathcal{M}$ (with universe $\omega$) such that $$D = \{ \phi (v_0, \ldots, v_m) \in F : \mathcal{M} \models \phi (0,1,\ldots,m) \}.^{\text{(*)}}$$ Then $\Psi (D)$ would consist of those $\phi \in D$ whose free varaibles are among $v_0, \ldots , v_{n-1}$. Adjusting some indices in p.159, it is clear that $\Psi (D)$ is an $F$-type: it is the set of all formulae satisfied in $\mathcal{M}$, above, by the $n$-tuple $(0,\ldots,n-1)$.

$^{\text{(*)}}$ I take it that formulae in an $F$-diagram can have free variables any (finite) initial segment of the variable symbols, which seems to better match common usage. Marker's presentation seems to imply that the formulae in an $F$-diagram have free variables only among $v_0, \ldots , v_n$, where $n$ is the same natural number as in the definition of the set $S_n(F,T)$ we are concerned with.

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thanks a lot, everything seems much more clear. – sanluc Sep 3 '12 at 15:16

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