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$Qustions$

Suppose $\mathcal{M_n} = (M_n ,I_n )$ is an $L$-model, $\mathcal{M_n}$$\preceq$$\mathcal{M_{n+1}}$ for each $n∈\mathbb N$. Let$M = \bigcup_{n \in \mathbb N} M_n $

1.Define an interpretation $I$ on $M$ properly so that each $\mathcal{M_n}$ is a substructure of $\mathcal{M} = (M,I)$.

2.Show $\mathcal{M_n}$$\preceq$$\mathcal{M}$ for each $n$ and the above $\mathcal{M}$.

3.Show if $\mathcal{N}$ is such that $\mathcal{M_n}$$\prec$$\mathcal{N}$ for all $n$, then $\mathcal{M}$$\preceq$$\mathcal{N}$.

$Answer$

First part

I have finished the first part.

1.For constant symbol $c$, $I(c)=I_0(c)$

2.If $F$ is an $n$-ary function symbol, $a_1,...,a_{n+1}∈M_i$, then $I(F)(a_1, \ldots, a_n) =a_{n+1}$ iff$ I(F)(a_1, \ldots, a_n)=$$a_{n+1}$.

3.If $R$ is a $n$-ary relation symbol, $a_1,...,a_n∈M_i$, then$(a_1, \ldots, a_n)∈I(R)$ iff$ (a_1, \ldots, a_n)∈$$I(R_i)$.

The second part

I think we need $Tarski's$ $Criterion$ in this part. Suppose $A$ is definable in $\mathcal{M}$ with parameters from $M_n$, then we need to show $A∩M_n≠∅$. But the difficult is how to find a element which belongs to both $A$ and $M_n$?

the third part

Intuitively, it is obvious. But how to prove? Still use $Tarski's Criterion$?

Summary

I hope you can give some help in $Question 2,3$

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Remember that $M$ is the union.

Hence if you have a witness in $M$, then there must be a witness in some $M_{k}$ for some $k$. Now using the fact that $M_{n}\leq M_{k}$

I'm not sure about your question 3, unless you mean "then $M\leq N$". In which case once again you just need to check the definition: certainly $M$ is a substructure of $N$, and if $N$ have a solution then $M_{n}$ have a solution, hence $M$.

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  • $\begingroup$ There is a typo in question 3. I have amended it. Even though the witness belongs to $M_k$, why must it belong to $M_n$? $\endgroup$ – Marvin Dec 29 '16 at 8:30
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    $\begingroup$ This answer is quite incomplete. For example: Given a formula $\exists x\, \varphi(x,\overline{a})$ satisfied in $M$, there is (as you say) some $M_k$ such that the tuple $\overline{a}$ and a witness $b$ is in $M_k$. So $M\models\varphi(b,\overline{a})$, but how do you know that $M_k\models\varphi(b,\overline{a})$? [Note that in the Tarski-Vaught test, $\varphi$ may have quantifiers...] $\endgroup$ – Alex Kruckman Dec 31 '16 at 22:47
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    $\begingroup$ Instead, try doing induction on the number of quantifiers. $\endgroup$ – Alex Kruckman Dec 31 '16 at 23:11
  • $\begingroup$ @AlexKruckman Thanks. I already understand it. $\endgroup$ – Marvin Jan 1 '17 at 13:19

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