Consider the following axioms which I'll call the "Ultrafinite Peano Axioms":
$0 \in \mathbb N$.
$\forall x,y \in \mathbb N,$ $S(x)=S(y)$ implies $x=y$.
$\forall x \in \mathbb N, S(x) \neq 0$.
There exists a $Z \in \mathbb N$ (a zillion) such that $S(Z) \notin \mathbb N$. $\forall x \in \mathbb N - \{Z\}, S(x) \in \mathbb N$.
Suppose $K$ is a set such that $0 \in K$ and ($\forall x \in \mathbb N-\{Z\}$, $x \in K$ implies $S(x) \in K$). Then $\mathbb N \subseteq K$.
Axioms 1 to 3 are no different than the their corresponding three Peano Axioms. Axiom 4 says that every number except $Z$, a zillion, has a successor and $Z$ has no successor. Axiom 5 is the induction axiom with the restriction that $x \neq Z$. These axioms model the ultrafinite view of math, that there are only a finite number of numbers, so not every number has a successor.
Gödel's Theorem says that the Peano axioms cannot be proven consistent. My question is can we prove that these axioms are consistent?