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!
 A: If you mean the effectiveness of $PA$ (i,e., the first order theory of arithmetic with $+,*,<$ primitive recursive), then this link http://www.math.cornell.edu/~shore/papers/pdf/effmod32.pdf covers a great deal of direct properties of effectively decidable first order theories and their corresponding models.
But, regarding your question in particular, the decidability of a model $M$ of arithmetic (a model of $PA$) can mean the following:
1) The language $L_M$ is recursively enumerable, in the sense that every term, formula, and sentence can be effectively associate with a Goedel numbering.
2) By Goedel's incompleteness theorem, since $L_M$ cannot be both complete and consistent, then there are some arithmetical truths (in the sense of a partial truth predicate $Tr_1$) for which there is no proof sequence leading from the axioms of $PA$ to all valid sentences of the natural numbers. This is, in essence, the content of Goedel's first incompleteness theorem coupled with the compactness theorem, which asserts that $PA$ (a set of first-order sentences) has a model iff every finite subset of $PA$ has a model.
The problem you cite originates in the proof of (2) that assets that $PA$ is not both consistent and complete. This is a case of Goedel's first incompleteness theorem. Formulated in terms of computability, it simply is the proof of the fact that no finite program $e$ of a Turing machine $T^e$ can decide whether a $\Delta^0_1$ sentence $\forall x\phi(x)$ or $\exists x \phi(x)$ is true (halts if 0 if $Tr_1$ holds for $x$, and 1 if otherwise) in a denumerable domain (a model $M$) or whether it is refutable. 
This is clear if one has an enumeration of the partial recursive functions $f_1(z),..,f_e(z)$ where $e$ is the code of $T_e$ on arbitrary input $z$. Since if all such sentences $\phi$ where decidable by $T^e$ then one could effectively deduce a proof of $0=1$, which proves everything from $PA$, including its own inconsistency (which is the content of Goedel's second incompleteness theorem).
Hope this helps. As a hint for a proof, think of the diagonal argument in terms of proving that that set of all valid sentences $\forall x \psi(x)$ is not in the range of a total recursive function, since if it were, then out could deduce a contradiction, consequently proving such a function indeed does not exist for the set in question.
A: 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.
