关于仅基于皮亚诺公理（无加减乘）的自然数集代数结构的问询

Great question! Let's unpack this idea clearly:

• Even when we define the natural number set $\mathbb{N}$ using only the Peano axioms (with no addition or multiplication defined yet), it’s far from being a plain, unstructured set. The key difference lies in the successor relationship between elements—something you never find in a generic set. For example, 4 is the successor of 3, 5 is the successor of 4, and this chain of dependencies gives $\mathbb{N}$ its inherent, non-trivial structure.
• We can formalize this structure using a successor function S: ℕ → ℕ. This function is defined after the natural numbers are constructed via the Peano axioms, and its sole role is to map each natural number to its immediate successor: think S(3) = 4, S(4) = 5, and so on for every element in $\mathbb{N}$.

内容的提问来源于stack exchange，提问作者kot