Am I thinking of this correctly? In PA, you start with zero and a successor function, such that > is transitive and succ(a) > a. In nonstandard models, you start with a successor function and zero, but also at least one other weird entity not reachable by any number of applications of the successor function to zero. You can apply the successor function to that entity to get new numbers that are all greater than anything reachable from zero.

OR is it that there have to be infinite starting objects? OR should we think of it as having zero and both a successor function and a weirdSuccessor function (whose applications commute) such that > considers the number of weirdSuccessor applications before it looks at the number of successor applications?