Modifying the Peano Axioms to allow multiple successors

Question

If you took the familiar Peano Axioms and replaced the axiom

$x \in \mathbb{N} \implies \exists y\in \mathbb{N}(y =S(x))$

with

$x \in \mathbb{M} \implies (\exists y_1\in \mathbb{M})(\exists y_2\in \mathbb{M})(y_1 =S_1(x)\wedge y_2 =S_2(x) \wedge y_1\ne y_2)$

and the other axioms (including the ones defining addition and multiplication) modified accordingly.

The structure here ($\mathbb{M}$) would seem to resemble a tree of numbers, which each "level" $n$ containing $2^n$ elements. I was wondering if there is a name for this set of numbers.

Answer 1

Assuming I have the correct interpretation of other axioms modified accordingly, you're describing an infinite binary tree, or more precisely an infinite, rooted, complete binary tree.