Second Principle of Finite Induction/One-Based

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Theorem

Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers.


Suppose that:

$(1): \quad 1 \in S$
$(2): \quad \forall n \in \N_{>0}: \paren {\forall k: 1 \le k \le n \implies k \in S} \implies n + 1 \in S$


Then:

$S = \N_{>0}$


Proof

Define $T$ as:

$T = \set {n \in \N_{>0}: \forall k: 1 \le k \le n: k \in S}$

Since $n \le n$, it follows that $T \subseteq S$.

Therefore, it will suffice to show that:

$\forall n \ge 1: n \in T$


Firstly, we have that $1 \in T$ if and only if the following condition holds:

$\forall k: 1 \le k \le 1 \implies k \in S$

Since $1 \in S$, it thus follows that $1 \in T$.


Now suppose that $n \in T$; that is:

$\forall k: 1 \le k \le n \implies k \in S$

By $(2)$, this implies:

$n + 1 \in S$

Thus, we have:

$\forall k: 1 \le k \le n + 1 \implies k \in S$


Therefore, $n + 1 \in T$.


Hence, by the Principle of Finite Induction:

$\forall n \ge 1: n \in T$

That is:

$T = \N_{>0}$

and as $S \subseteq \N_{>0}$ it follows that:

$S = N_{>0}$

$\blacksquare$


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