I was asked the following question:
for every $x\in \mathbb R$ we define the set $Q(x)=\{q \in \mathbb Q|q\leq x\}$, the set of all rational numbers less or equal to $x$.
Let $M=\{Q(x)|x\in \mathbb R\}$ the set of such sets $Q$, $M$ is a subset of the power set of $\mathbb Q$.
1) is $M$ linearly ordered with the relation $\subseteq$?
2) Find the ordinal $[(M,\subseteq)]$
I managed to show that 1) is true,$\subseteq$ defines a linear order on $M$, but how do I find the ordinal? it's infinite, but it isn't anything related to $\omega$ because $\omega$ has a beginning, it has a smallest value. $M$ doesn't have minimal element with respect to $\subseteq$.