Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If so, why does it appear that nobody investigating it?


To contextualize this question, I refer to discovery of the equiconsistency of the non-Euclidean geometries. The geometry has flourished (or at least, developed) to its modern state with manifold theories and later developments was only possible with the recognition of non-Euclidean geometries. For a long time, Euclidean geometry and the Parallel Postulate was seen as the only geometry, very much as today's standard model for number theory ${\mathbb N}$ and the induction axiom is seen as the only number theory. (at least to the extent that i'm aware of, so that's why I'm asking this question.) I wonder why, therefore, that there is only one number theory, and no alternatives like non-Induction number theory.


Note: I am aware that recursion theorists study fragments of induction. However, in the sense of which I'm asking, this does not count at alternative axioms to induction since ${\mathbb N}$ is still a model of these these fragments. Using my geometry analogy, these fragments amount to saying that the Parallel Postulate does not hold everywhere throughout space, something like space is not homogeneous but still has some symmetry.



Alternative axiom to induction
A crucial difference between non-Euclidean geometries and "non-inductive" models of PA- is that any model of PA- contains a canonical copy of the true natural numbers, and in this copy of $N$, the induction schema is true. In other words, PA is part of the complete theory of a very canonical model of PA-, and as such it seems much more natural (so to speak) to study fragments of PA rather than extensions of PA- which contradict induction. 


To make this a little more precise (and sketch a proof), the axioms of PA- say that any model $M$ has a unique member $0\_M$ which is not the successor of anything, and that the successor function $S\_M: M \to M$ is injective; so by letting $k\_M$ (for any $k \in N$) be the $k$-th successor of $0\_M$, the set $\{k\_M : k \in N\}$ forms a submodel of $M$ which is isomorphic to the usual natural numbers, $N$. Any extra elements of $M$ not lying in this submodel lie in various "Z-chains," that is, infinite orbits of the model $M$'s successor function $S\_M$.


(In the language of categories: the "usual natural numbers" are an initial object in the category of all models of PA-, where morphisms are injective homomorphisms in the sense of model theory.)


So, while PA seems natural, I'm not sure why there would be any more motivation to study PA- plus "non-induction" than there is to study any of the other countless consistent theories you could cook up, unless you find that one of these non-inductive extensions of PA- has a particularly nice class of models.


