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Around the classic Fraisse amalgamation theorem in model theory we have the following notions:

Definition (1): If $M$ be an $\mathcal{L}$-structure then define:

$age(M):=\lbrace N~|~N~\text{is finite and embedable in}~M\rbrace$

Definition (2): We say that a class $\mathcal{K}$ of finite $\mathcal{L}$-structures have $HP$ if for all $M\in \mathcal{K}$ we have $age(M)\subseteq \mathcal{K}$.

Definition (3): We say that a class $\mathcal{K}$ of finite $\mathcal{L}$-structures have $JEP$ if for all $M,N\in \mathcal{K}$ there is a $P\in \mathcal{K}$ such that $M$ and $N$ be embedable in $P$.

Definition (4): A class $\mathcal{K}$ of finite $\mathcal{L}$-structures called an $age$-class if there is an $\mathcal{L}$-structure $M$ such that $\mathcal{K}=age(M)$.

Theorem (1): If $K$ be a class of finite $\mathcal{L}$-structures with $HP$ and $JEP$ which has countable many members up to isomorphism then it is an $age$-class.

Now note to the following correspondence:

Definition (5): Let $\mathcal{L}$ be a countable first order language then define:

$U_{<\omega}(\mathcal{L}):=\lbrace M~|~M~\text{is a finite}~\mathcal{L}~\text{-structure} \rbrace$

Then $\frac{U_{<\omega}(\mathcal{L})}{\cong}$ is a countable set which forms a partial order with the following well defined order:

$\forall M,N\in U_{<\omega}(\mathcal{L})~~~~~[M]_{\cong}\leq [N]_{\cong}\Longleftrightarrow N~\text{is embedable in}~M$

Remark (1): Now consider the "countable" partial order $\mathbb{P}=\langle\frac{U_{<\omega}(\mathcal{L})}{\cong},\leq\rangle$ as a "forcing notion" (with the largest element $[\emptyset]_{\cong}$). Then for any class $\mathcal{K}$ of finite $\mathcal{L}$-structures with $HP$ and $JEP$ which has countable many members up to isomorphism (we call such collection a Fraisse class) the countable set $\frac{\mathcal{K}}{\cong}$ forms a "filter" over $\mathbb{P}$ and theorem (1) simply says that the "generic limit" of such filters is well defined. We call $\frac{\mathcal{K}}{\cong}$ a "Fraisse filter". Now one more step remains, "using Fraisse filters to produce generic models of $ZFC$ with special properties". In order to do this we first need to have a ground model of $ZFC$ which contains the partial order $\mathbb{P}$ as a member. So the first question is:

Question (1): Let $\mathcal{L}$ be a countable first order language. Is there a c.t.m of $ZFC$ like $M$ such that $\langle\frac{U_{<\omega}(\mathcal{L})}{\cong},\leq\rangle\in M$?

Question (2): Let $\mathcal{L}$ be a countable first order language and $M$ is a c.t.m of $ZFC$ such that $\langle\frac{U_{<\omega}(\mathcal{L})}{\cong},\leq\rangle\in M$ and $\mathcal{K}$ is a Fraisse class which $\frac{\mathcal{K}}{\cong}$ forms a $\mathbb{P}$-generic Fraisse filter over $M$. Then what kind of "model theoretic" statements could be true in generic extensions $M[\mathcal{K}]$ by varying $\mathbb{P}$ and $\mathcal{K}$? For example can we prove the independence of Vaught's conjecture using this method?

Remark (2): Note that there are some similarities (and even differences) between behavior of the function $\kappa \mapsto I(T,\kappa)$ for some complete theory $T$ in a "countable" language and the function $\kappa \mapsto 2^{\kappa}$. For example both are increasing (By Shelah's proof of Morley's conjecture). Even we have $2^{\aleph_{0}}=|P(\omega)|=|\frac{Mod_{\omega}(T)}{\cong}|$. So it seems that "Vaught's Hypothesis" ($VH$)(i.e. For any complete theory $T$ in a countable language if $I(T,\aleph_{0})>\aleph_{0}$ then $I(T,\aleph_{0})=2^{\aleph_{0}}$) is just a "Continuum Hypothesis" ($CH$) (i.e. For any subset $X$ of real numbers if $|X|>\aleph_{0}$ then $|X|=2^{\aleph_{0}}$) in a different "space" which its points are "equivalence classes of finite structures up to isomorphism" not real numbers.

Question (3): Is there any known independence result about "model theoretic statements" using forcing? For example the results like the statement "...there is a generic extension of a c.t.m of $ZFC$ like $M[G]$ such that: $M[G]\models ~\text{Any theory}~T~\text{with the property}~P_{1}~\text{has a model with the property}~P_{2}$ "

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What does the forcing you describe have to do with Vaught's Conjecture? I don't see the connection. –  Noah S Oct 6 '13 at 22:14
    
The Vaught's conjecture is just an example of "model theoretic statements" which I think these kind of forcings can give us an approach to them. Even I think that Vaught's hypothesis is very similar to continuum hypothesis and "if the theory of such forcings be developed" then one can solve it in a "simple" way as same as proving independence of $CH$ which is an exercise in the scope of new forcing methods. –  user36136 Oct 6 '13 at 22:23

2 Answers 2

up vote 3 down vote accepted

If I understand your definitions correctly, something more needs to be assumed about the language $\mathcal L$. If $\mathcal L$ is purely relational, then your forcing notion $\mathbb P$ is directed, in the sense that every two conditions have a common extension, because, given any two finite $\mathcal L$-structures, there is another in which they can both be embedded. That makes $\mathbb P$ trivial as a forcing notion; its separative quotient is the one-element poset, and its only $M$-generic filter is all of $\mathbb P$ (for any model $M$ that contains $\mathbb P$).

Even if $\mathcal L$ is not purely relational, it seems that $\mathbb P$ will be just the disjoint union of several directed pieces, one piece for each maximal satisfiable (in finite models) set of atomic sentences. That would imply again that forcing with $\mathbb P$ is trivial.

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Dear Andreas. Ok. You are right. But the soul of the question remains non-trivial too that is: "using partial orders which are a set of models with a certain binary relation as a forcing notion to produce generic models of $ZFC$ which satisfy special statements about models and reaching to independence results in model theory by set theoretic methods". Did you see any use of forcing in this way? –  user36136 Oct 7 '13 at 5:50

Under the only meanings of question 1 I can think of, the answer is "yes," using Scott's Trick (we only need to look boundedly far in the cumulative hierarchy for elements of equivalence classes). The answer to question 3 is certainly "yes:" for example, Chang's two-cardinal theorem holds under GCH, but can be forced to fail.

Question 2, on the other hand, I find too vague to admit an answer.

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