Why is (1) a copy of $\mathbb{N}$ "followed by" a copy of $\mathbb{Z}$ not a (non-standard) model of arithmetic, neither (2) a copy of $\mathbb{N}$ followed by an infinite sequence of copies of $\mathbb{Z}$, but (3) a copy of $\mathbb{N}$ followed by infinitely many densely ordered copies of $\mathbb{Z}$ is? (see the Wikipedia entry on non-standard models)
Can this intuitively be seen, or be explained in laymans terms?
The axioms concerning the successor function hold in all of these (pseudo-)models, don't they?
But the induction axiom really puzzles me! Naively, it can be interpreted as describing essentially an infinite row of dominoes: knocking over the first will let fall all of them. How can this be understood in the non-standard model (3), where there is no immediate "contact" between the building blocks? What's the "true" interpretation of induction then? And why does it work in (3) but not in (2) or (1)?