A wrong definition of infinity and a doubt about its modification In this answer, @Hyperplane defined an infinite set as follows:

Well a set $M$ is infinite if for every finite subset $U \subsetneq M$ there exists a $x \in M, x\notin U$. Consequently, you will be able to construct a sequence $(x_n)_{n \in \mathbb{N}} $ of distinct elements in $M$. Therefore $|M| \ge \aleph_0 $ 

Then @Andrés E. Caicedo commented here:

This is not true as written. Replace $\subsetneq$ with $\subseteq$, and it will be fine now.


I think that @Hyperplane's definition of an infinite set is wrong. My reasoning is:

For every subset $U \subsetneq M$, we always have $\exists x \in M, x \notin U$ by the definition of $\subsetneq$. So @Hyperplane just restated the property of $\subsetneq$ and said nothing about the definition of infinite set.

When I replace $\subsetneq$ with $\subseteq$ in @Hyperplane's answer as suggested by @Andrés E. Caicedo, I even find the new definition more awkward. My reasoning is:

For $U=M$, there does NOT exist any $x \in M, x\notin U$. Thus the statement Well a set $M$ is infinite if for every finite subset $U \subseteq M$ there exists a $x \in M, x\notin U$ is wrong. Let alone define an infinite set.

Could you please check whether my above reasonings are correct? 

You are correct to think that the first given definition is wrong and you have gavin a good reason for it. However, for the corrected definition, you are not. Indeed, in order to chose $U=M$, $M$ must be finite.
