Peano Axioms, it's possible construct a real number theory Is it possible to construct a coherent axiomatic theory to ground the integers, rational and irrational without induction?
 A: The Peano axioms are for constructing the natural numbers, not the real numbers.
As for answering your question, Robinson arithmetic is a theory of the natural numbers without induction, and because it lacks induction general statements over variables (e.g. x + y = y + x) are not provable. Induction is needed to prove statements involving arbitrary natural numbers represented by variables. Without induction, you are stuck with proving sentences that use only concrete values like 5 + 3 = 3 + 5. So no, there is no axiomatization of the natural numbers without induction that gives you the full expressive power of Peano arithemtic. This is justified by noticing that in order to prove that a proposition P holds for a variable, one must have a rule that has a consequence $\forall x P$... which is going to be induction if the rule is to be consistent with the property of the successor function.
However, as Andrés E. Caicedo pointed out, there may be some other rule (e.g. reflection principle) which serves as a replacement for induction and yet produces an axiomatic system equivalent to Peano arithmetic. I am not aware of any papers.
A: The usual second-order axiomatization of the real numbers is that they are a "complete ordered field".


*

*Fields axiomatize basic arithmetic and algebra

*Ordered fields axiomatize the order relation and how it relates to arithmetic

*Completeness describes the basic analytic features of the continuum. It implies (or is equivalent to) a number of basic results, such as the intermediate value theorem.


The analogous first-order theory is that of real closed fields.
