MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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: $M[G]\models ~\text{Any theory}~T~\text{with the property}~P_{1}~\text{has a model with the property}~P_{2}$ "

share|cite|improve this question
What does the forcing you describe have to do with Vaught's Conjecture? I don't see the connection. – Noah Schweber 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
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.

share|cite|improve this answer
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.