I've just started studying forcing. Currently, I am struggling to understand what is a $\mathbb{P}$-name in the first chapter in the Shelah's book page 6 (https://projecteuclid.org/euclid.pl/1235419817).
The definition goes like this:
Assume the axiom of foundation. Define the rank of any $a \in V$: $rk(a) = \bigcup \{rk(b)+1:b\in a\}$.
Define a $\mathbb{P}$-name by induction on $\alpha$. $\tau$ is a name of rank $\leq \alpha$ if it has the form $\tau = \{(p_i, \tau_i): i < i_0\}, p_i \in \mathbb{P}$ and each $\tau_i$ is a name of some rank $< \alpha$. The interpretation $\tau[G]$ of $\tau$ is $\{\tau_i[G]:p_i \in G, i<i_0 \}$.
The definition looks almost the same as the one in Kunen (1980) where he defined a $\mathbb{P}$-name $\tau$ iff $\tau$ is a relation and $\forall (\sigma, p) \in \tau$ [$\sigma$ is a $\mathbb{P}$-name and $p\in \mathbb{P}$]. However, I have trouble understanding what is the $i_0$ (which is not mentioned in the book). What is $i_0$? Is the $i_0$ in the definition of $\tau[G]$ the same as the $i_0$ in the definition of $\tau$? Also, why do we need it?
Thank you in advance.
It seems to me that $i_0$ is just some indicator that this is a set. Namely, it is some ordinal, and $p_i$ and $\tau_i$ are just conditions and names indexed by that ordinal.
I would strongly suggest, however, learning forcing from Kunen. While Shelah's book does have a very good chapter about rudimentary forcing facts, the rest of the book is far from rudimentary or introductory to forcing. On the other hand, Kunen's book is much more appropriate for this. You can also check out Halbeisen's "Combinatorial Set Theory" (which you can also find freely on his website) for a very thorough introduction of forcing.