# Why did Hopcroft and Karp write $M_0, M_1, M_2, \cdots, M_i, \cdots$? (Hopcroft – Karp Algorithm)

I am reading “An $$n^{\frac{5}{2}}$$ Algorithm for Maximum Matchings in Bipartite Graphs” by Hopcroft and Karp.

Please see the image below.

Let $$s$$ be the cardinality of a maximum matching.
I think any of $$M_0, M_1, M_2, \cdots, M_s$$ is a matching and $$M_s$$ is a maximum matching.
So, I think $$P_s$$ doesn’t exist.
But the authors wrote $$M_0, M_1, M_2, \cdots, M_i, \cdots$$ and $$|P_0|, |P_1|, \cdots, |P_i|, \cdots$$.

Why?

Maybe I am confused.