I hope to clarify the difference between the following two statements:
$S\subset \mathbb{N}$ (set of natural numbers),
- $\forall n \in \mathbb{N}, \text{ if } n\in S \text{, then } n+1\in S$
- $\forall n\in S, n+1\in S$
I am currently studying the definition of principles of mathematical induction (PMI). PMI assumes (1.) (https://www.siue.edu/~jloreau/courses/math-223/notes/sec-induction.html) but I am wondering if (2.) means the same thing. They seem to be the same for me. Thank you!