What does "decidability" of a Model mean exactly? I'm looking at the theorem concerning the Model of Arithmetic:

M arith = (Integers, +, *, <) is undecidable.

What does the "decidability" of a model mean exactly? Does that mean that "the problem of determining if the given model satisfies any FOL statement" is undecidable?
Thanks for the help!

In this case, the undecidability means there is no algorithm for deciding in general whether a sentence in our language is true in $\mathbb{Z}$.
To prove the undecidability, it is easiest to use the fact that there is no algorithm for deciding whether a sentence over the the usual language for arithmetic is true in the natural numbers $\mathbb{N}$. (Here it is convenient and customary, but not necessary, to have $\mathbb{N}$ include $0$.)
Then to prove the required undecidability result, we show that given any sentence $\varphi$ of "arithmetic," we can mechanically produce a sentence $\varphi'$ such that $\varphi$ is true in $\mathbb{N}$ if and only if $\varphi'$ is true in $\mathbb{Z}$.
This shows that if we had an algorithm for deciding truth in $\mathbb{Z}$, we could produce an algorithm for deciding truth in $\mathbb{N}$. But there is no such algorithm.
