# Natural numbers but without induction?

If I recall correctly the Peano axioms of natural numbers includes the axiom that proofs of induction should be valid. I am curious about what properties these "not so natural" numbers could have if it did not hold?

• See Presburger arithmetic and Robinson arithmetic or Q for some ideas. Presburger arithmetic allows induction with addition formulas but lacks multiplication (and is decidable), while Robinson arithmetic lacks induction altogether (being finitely axiomatizable) but remains undecidable. Dec 20, 2015 at 15:15
• Robinson arithmetic, or Q is "essentially PA without the axiom schema of induction" (according to Wikipedia), so it should be your answer. Dec 20, 2015 at 15:18
• @user236182: Perhaps I should post an answer, but I interpreted the question to be, what sort of "integers" could we have without induction? So the models of Robinson arithmetic will of course fall into this topic, but I think it fair to bring up Presburger arithmetic, with its omission of multiplication, as this arguably (esp. in the original Peano axioms) illustrates what might happen. Dec 20, 2015 at 15:21
• Thanks guys. I would accept both those as answers. Presburger seems really interesting even though wikipedia says it does actually include a schema of induction and being "not as powerful", does it dodge Gödel incompleteness theorem? Dec 20, 2015 at 15:30
• Yes, that's kind of the point of Presburger arithmetic. Lacking multiplication, even though it has a first-order scheme of induction, the theory of Presburger arithmetic turns out to be decidable and therefore it does "dodge Gödel incompleteness theorem". Robinson arithmetic demonstrates the other side of the balance, that despite lacking a scheme of induction, the properties of addition and multiplication are finitely axiomatized in a way that gives an undecidable (essentially incompleteable, in first-order logic) theory. Dec 20, 2015 at 15:37