Natural Numbers under Addition form Inductive but not Strictly Inductive Semigroup

Theorem
Let $\struct {\N, +}$ denote the algebraic structure consisting of the set of natural numbers $\N$ under addition $+$.

Then $\struct {\N, +}$ forms an inductive semigroup, but not a strictly inductive semigroup

Proof
Recall the definition of inductive semigroup:

Recall the definition of strictly inductive semigroup:

The natural numbers $\N$ can be considered as a naturally ordered semigroup.

From Naturally Ordered Semigroup is Unique, $\struct {\N, +}$ is unique up to isomorphism.

Hence from Naturally Ordered Semigroup forms Peano Structure, $\N$ is a Peano structure.

Let $S$ be a subset of $\N$.

Then by Peano's axioms:


 * $\paren {0 \in S \land \paren {\forall z \in S: z + 1 \in S} } \implies S = \N$

The latter condition is that which defines an inductive semigroup, where $0$ is identified with $\alpha$ and $1$ with $\beta$.

However, there is no element in $\struct {\N, +}$ which can be identified with $\beta$ in the definition of strictly inductive semigroup.

Indeed, the element $0$ can be expressed in the form $x \circ 1$ for any $x$.