I'm working on a mathematical logic question.
Suppose $<$ belongs to $S$ and $\Phi \subseteq L_{0}^{S}$. Assume that for any $m \in \mathbb{N}$ there is a model $\mathfrak{A}$ of $\Phi$ such that $(A, <^{A})$ is a strict lineaire order containing at least $m$ elements. Then there is a model $\mathfrak{B}$ of $\Phi$ such that $(B, <^{B})$ is a strict lineair order containing an infinite descending chain.
I thought maybe i can show that $(B, <^{B})$ is a model for Th($\mathfrak{N}^{<}$) where Th($\mathfrak{N}^{<}$) = {$\varphi \in L_{0}^{S}| \mathfrak{N}^{<}\models\varphi$}. Because i know those models contains infinite descending chains. But I have no idea how to do this.
Please can someone help me?
This is an easy application of the compactness Theorem.
Let $\psi$ be a sentence asserting that $<$ is a strict linear order and $\varphi_n(x) \in L^S_1$ a formula with the free variable $x$ asserting that $x$ has $n$ $<$-predecessors.
Now consider the theory $\Gamma = \Phi \cup \{\psi\} \cup \{\varphi_n(x) | n\in\mathbb{N}\}$
By assumption every finite subset of $\Gamma$ has a model, therefore $\Gamma$ has a model $\mathfrak{B}$ by the compactness Theorem. But then the interpretation of x in $\mathfrak{B}$ has infinitly many predecessors.