Is this definition consistent and correct (infinite set)

Question

A set $S$ is infinite iff $\forall x\in S \exists x_0\in S$ s.t. $x \leq x_0$. Obviously, by that definition we can say, since $x_0\in S \implies\exists {x_0}_{0}\in S$ s.t. $x_0\leq {x_0}_0$. Ad infinitum.

This is obviously for well-ordered sets. I can't seem to find anything wrong with this definition. Any help appreciated!

Answer 1

Nope, $x\leq x$, so this is true for any non-empty set with an ordering. There are two canonical definitions of infinite sets

1. $S$ is infinite if there exists an injection $f:\mathbb N\to S$.
2. $S$ is infinite if for some $s\in S$, there exists an injection $f:S\to S\setminus\{s\}$.

Answer 2

Consider the set $\{1,2,3\}$ which is finite,but check that your definition still holds fo it.There are two equivalent definitions:

$1$.A set$S$ is said to be infinite iff it is not a finite set i.e. there does not exist any bijection $\phi:S\to\{1,2,...,n\}$ for any $n\in \mathbb N$.

$2$.A set $S$ is said to be infinite iff there exists a proper subset of $S$ equivalent to $S$.