Definability in Infinite structure I am wondering something about definability :
Suppose we have an infinite set of finite structures $\mathcal{A}^i$ such 
that $\forall i \geq 0, \mathcal{A}^i \subseteq \mathcal{A}^{i+1}$, i.e for each 
$i \geq 0, $ $\mathcal{A}^i$ is a substructure of $\mathcal{A}^{i+1}$.
Suppose that I can define a set $S_i$ in each $\mathcal{A}^i$ by a formula of first order logic $\varphi$
such that $S_i= \{ \vec{a} \in A^i, \mathcal{A}^i \vDash \varphi(\vec{a}) \}  \subseteq
  \{ \vec{a} \in A^{i+1}, \mathcal{A}^{i+1} \vDash \varphi(\vec{a}) \} =S_{i+1}$, then 
can I find a first order formula defining the set 
$\bigcup \{ \vec{a} \in A^i, \mathcal{A}^i \vDash \varphi(\vec{a}) \} =\bigcup S_i$ in the structure 
$\mathcal{A}=\bigcup \mathcal{A}^i $? 
P.S: Note that $|\mathcal{A}^i| < \omega$ and $|\mathcal{A}|=\omega$.
 A: No. Let $(A,\le,P)$ be the structure of a partial order consisting of countably many disjoint copies of $\omega$, and a unary predicate $P$ which is satisfied by copies of all even points. That is, $A=\omega\times\omega$, $(a,n)\le(b,m)$ iff $a=b$ and $n\le m$, and $P((a,n))$ iff $n$ is even. Let $S=\{(a,n):a\text{ even}\}$. Obviously, $S$ is not definable in $A$. However, let $A^i=\{(a,n)\in A:a+n\le2i\}$, and $S_i=S\cap A^i$. Then $S_i$ is definable in $A^i$ by the formula $\varphi(x)=\exists y\,(x\le y\land\forall z\,(y\le z\to y=z)\land P(y))$.
A: The fact is that I have a property that it is preserved by union of chains, 
but I can express it by a Higher logic than First order , now what I am wondering is 
if there exists a FO formula, will it be a $\forall \exists$ , I mean could I have a FO formula that defines correctly my property over finite structure but gives a different 
interpretation in the union of finite structure ...? It seems strange to me because then it 
means that the formula will not be the "good definition", since my property has to be preserved ... Well there is something I can understand , i don't If I make my problem clear ... in fact i cannot really understand if the semantic which says that the property has to be preserved implies that the Fo formula (if it exists) has to be preserved ....  
A: The latest question (presented as an answer) is not very clear.  In particular, it contains two entirely different questions connected by an "I mean" suggesting that thanyon didn't really intend to ask both (and might have intended something different from both).  Nevertheless, here's some information that answers some of what was asked (intended or not).
If a property of (arbitrary, not necessarily finite, first-order) structures is preserved by unions of chains and is expressible by a first-order sentence $A$, then there is an $\forall\exists$ sentence $A'$ logically equivalent to $A$ (i.e., defining the same property).
It is entirely possible for a property, even a first-order definable one, to be preserved by unions of chains, to be defined on finite structures by a sentence $A$, but not to be defined by the same $A$ on infinite structures.  For example, take the property of being a field and let $A$ be a sentence defining division rings.  On finite structures, $A$ also defines fields, since all finite division rings are fields, but on infinite structures $A$ defines a broader class.  
