I've been going through some basic forcing and although I'm able to mechanically follow the steps I'm lacking any real intuition for it. In particular I'm confused about the evaluations used to describe elements in $M[G]$.

Suppose $M$ a class/set, $P \in M$ a poset, and $G$ a $M$-generic filter on $P$. Then we recursively define for $\sigma \in M$,

$i_G(\sigma) = \{ i_G(\delta) : \exists p \in G (\delta, p) \in \sigma \}$

Then we say $M[G] = \{i_G(\sigma) : \sigma \in M\}$

Finally we say $\tilde{y} = \{(\tilde{x}, p) : p \in P, x \in y\}$

and $\dot G = \{(\tilde{p}, p) : p \in P\}$

I get that $\dot G$ is defined so that $i_G(\dot G) = G$ and $i_G(\tilde{y}) = y$. But what is the reasoning for $i_G$ in the first place?

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    $\begingroup$ The idea is that names are a description in the ground model of the elements of the forcing extension, but the names only depend on the poset $P$, while the forcing extension can also depend on which $P$-generic filter $G$ you pick. $i_G$ is the function that tells you how exactly a name is interpreted as an element of $M[G]$, after $G$ has been chosen $\endgroup$ Feb 23, 2021 at 15:19
  • $\begingroup$ At a high level I get that, but why use this definition for a name? I mean, what's wrong with a name for $y$ being $\tilde{y} = \{(x, p) : x \in y$ and $p \in P \}$. Then the evaluation could just be $i_G(\sigma) = \{x : (x, p) \in \sigma $ for some $p \in P \}$ $\endgroup$ Feb 23, 2021 at 18:26
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    $\begingroup$ I suppose you could make such a definition work, but this way you'd only get sets in $M[G]$ all of whose elements are in $M$, think about how you'd write a name for the set $\{G\}$ for example $\endgroup$ Feb 23, 2021 at 18:35
  • $\begingroup$ Thanks, it turned out to be more intuitive than I thought; I just wasn't thinking hard enough. I have a notational question if you don't mind, about the difference between $\check y$ and $\dot y$ (i.e, checks vs dots). The explicit definition for $\check y$ is $\check y = \{(\check x, p) : p \in P, x \in y\}$ and is designed to be evaluated as $y$ of course. But we often talk about $\dot y$ which also seems to always refer to a name for $y$. Is there any difference between the two? I'm thinking that $\dot y$ is used when $y \not \in M$ and we're relying on $G$ to produce actually new sets. $\endgroup$ Feb 23, 2021 at 19:26
  • $\begingroup$ $\dot y$ denotes any name for $y\in M[G]$, while $\check y$ denotes a specific name for $y$ and only makes sense for $y\in M$. In general every element of $M[G]$ has lots of names (but there are results along the lines of "if $x\in M[G]$ is small in some sense that interacts well with $P$", then we can also find a small name for it). $\endgroup$ Feb 23, 2021 at 20:00


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