based on this ordering, does it have a smalleat element or not? I'm reading a research article http://www.sciencedirect.com/science/article/pii/S0097316512001677
Now, a question pops up. In section 2, the author defined an ordering $<_{M}$ on $\mathcal{P}(n)$, which is the family of all the subsets of $\{1,2,3,\cdots, n\}$ as the following. 
$A<_{M}B$ if and only if $\max(A) > \max(B)$ or ($\max(A) = \max(B)$ and $\max(A\Delta B)\in B$. 
Then, it continues to talk about that "on the set of all finite subsets of all the integer set, this ordering does not have a smallest element, so does not have finite initial segments." I was deeply confused about this. Based on my understanding, since it is a total ordering, any two sets can compare to each other. Thus, on the set of all finite subsets of all the integer set, this ordering should have a smallest element. While on the other hand, while I was locating the smallest element, it seems like I can not find one. This is very strange. Did I miss something? Any comments are greatly appreciated. 
 A: 
Based on my understanding, since it is a total ordering, any two sets can compare to each other. Thus, on the set of all finite subsets of all the integer set, this ordering should have a smallest element. [emphasis mine]

While the first statement is true, since the set in question (set of all finite subsets of $\mathbb{N}$) is not finite, there may or may not exist a smallest element. A simple example is the real numbers, which are totally ordered. Is there a smallest real number? No.
In this case, one can construct a simple example of an infinitely descending chain with $$\ldots <_M \{3\} <_M \{2\} <_M \{1\} <_M \{0\}$$
where all the inequalities are true because $\ldots > 3 > 2 > 1 > 0$. Of course, this by itself does not mean that there is no smallest element, but gives an idea of how you might keep getting smaller elements. 
More generally, if you have some general $S\subseteq N$, then ${\max(S) + 1} <_M S$ so $S$ cannot be the smallest element. Since $S$ is arbitrary, the ordering does not have a smallest element.
