How to interpret the definition of inductive set?

Question

I can't understand the sentence below: "A subset Y ⊂ X will be called inductive if, for every x ∈ X such that y ∈ Y for all y ∈ X such that y < x, we have x ∈ Y." please tell me what's the meaning, thanks!

Answer 1

Yeah, that's not a very well-worded sentence, is it?!

Try this: Given $x\in X$, define $X_{{}<x} = \{ z\in X\colon z<x\}$. With this notation, a subset $Y\subset X$ is inductive if the following implication always holds for every $x\in X$: $$ X_{{}<x} \subset Y \implies x\in Y. $$ (If $Y$ contains everything in $X$ less than $x$, then it contains $x$ as well.)