Why does it suffice to show that X is an inductive set?

[evinda]

Hi! (Smile)

We want to show that the elements of the natural numbers are natural numbers, i.e. $(n \in \omega \wedge x \in n) \rightarrow x \in \omega$

Could you explain me why, in order to show this, it suffices to show that $X=\{ n \in \omega: (\forall y \in n)(y \in \omega)\}$ is an inductive set? (Worried)

 

[evinda]

I am looking again at the proof..

It suffices to show that $X=\{ n \in \omega: (\forall y \in n) (y \in \omega)\}$ is inductive because we know that $X \subset \omega$ and because of the following sentence:

For each subset $X$ of $\omega$ it holds the following:

$$0 \in X \wedge (\forall n (n \in X \rightarrow n' \in X)) \rightarrow X=\omega$$

we conclude that $\{ n \in \omega: (\forall y \in n) (y \in \omega)\}=\omega$ and thus $(\forall y \in n) (y \in \omega)$ holds for all $n \in \omega$, i.e. the elements of natural numbers are natural numbers, right? (Smile)