If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive}\}$ is inductive. Hence every $n\in\mathbb{N}$ is transitive.
If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive and }x\notin x\}$ is inductive. Hence, $n\notin n$ and $n\neq n+1$ for each $n\in\mathbb{N}$.
Some set theory, inductive sets
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elementary-set-theory
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3Well, what about them? Are you just letting us know about interesting stuff you've discovered? If you have a question, how about *asking* it? – 2011-02-20