Here is the most "tangible" distinctive feature of models of $PA$ that satisfy the negation of Gödel's true but unprovable sentence.

Let $\phi$ be a true $\Pi^0_1$ arithmetical sentence that is not provable from $PA$, e.g., $\phi$ can be chosen as the sentence expressing "I am unprovable from $PA$" [as in the *first* incompleteness theorem], or the sentence "$PA$ is consistent [as in the *second* incompleteness theorem].

Thanks to a remarkable theorem of Matiyasevich-Robinson-Davis-Putnam, known as the MRDP theorem, see [link text][1], there is a *diophantine* equation  $D_{\phi}$ that has no solutions in $\BBb{N}$, but has the property that for any model $M$ of $PA, $D_{\phi}$ has a solution in $M$ iff $M$ satisfies the negation of $\phi$.

[note: $D_{\phi}$ is allowed to have negative coeffients, but $D_{\phi}$ can be re-expressed as an equation of the form $P=Q$, where $P$ and $Q$ are polynomials [of several variables] with coeffiecients in $\Bbb{N}$; hence it makes perfectly good sense to talk about $D_{\phi}$ having a solution in $M$].

Let me close by recommending two good sources for the study of nonstandard models of $PA$.

Richard Kaye, *Models of Peano arithmetic*. Oxford Logic Guides, 15. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.

Roman Kossak and James H. Schmerl, *The structure of models of Peano arithmetic*, Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006.


  


  [1]: http://en.wikipedia.org/wiki/Diophantine_set