关于含整数常规加法公理与归纳法的公理系统的三项技术问询

a) Does this system have a specific name? #

Yep, this is essentially the integer version of Presburger arithmetic. The standard Presburger arithmetic focuses on natural numbers (0, 1, 2, ...) with addition, commutativity, associativity, and first-order induction. Your system extends this to all integers (positive, negative, zero) while keeping only addition and induction as core components—no multiplication, exponentiation, or other operations built in. It's often referred to as Presburger arithmetic over the integers in logic circles.

b) Can we define multiplication (ab) and prove (ab=ba)? #

Short answer: No, you can't define multiplication in this system. Here's why:

• To formalize multiplication as "repeated addition" (like (a \times b) being (b) added to itself (a) times), you need a way to count how many times the addition happens. But your system only has addition and induction tailored strictly to additive properties—there's no mechanism to encode that counting behavior.
• More formally, Presburger arithmetic (including its integer variant) is a decidable theory: there's an algorithm that can always determine if any statement in the system is true or false. If we could define multiplication here, the system would gain enough expressive power to encode primitive recursive functions, making it undecidable (like Peano Arithmetic). Since we know Presburger arithmetic is decidable, multiplication can't be defined within it.
• Without being able to define multiplication at all, proving its commutativity is impossible.

c) Does Gödel's Incompleteness Theorem apply to this system? #

Nope, Gödel's theorems don't apply here. Let's recap the key prerequisites for Gödel's First Incompleteness Theorem: it requires a consistent, recursively axiomatized system that's strong enough to encode basic arithmetic (meaning it can define both addition and multiplication).

• Your system can't define multiplication, so it fails the "sufficiently strong" criterion. In fact, Presburger arithmetic (the core of your system) is complete—every true statement in the system can be proven, and every false statement can be disproven. There are no undecidable sentences here, which directly means Gödel's incompleteness results don't apply.

内容的提问来源于stack exchange，提问作者Dave