Are P-names the codes used to define all possible membership relations that could be Forced from a Countable Model of Set Theory?

Question

Is the following anywhere close to a possible meaning of P-Names in Forcing, based upon a countable model M of Set Theory and the addition of a new set G and its associated model M[G] ?:

1) If the membership relation for each set in a model of set theory has been fully determined then the model is fully determined.

2) The Forcing Relation can be used to determine the complete membership relation for all sets in M[G] by specifying each element a in each set b in the model, where p is a forcing condition (i.e. an element p in a p.o. P $\supset$ G, with a,b,p,P $\in$ M) via a complete set of forcing conditions of the form:

p $\Vdash$ a $\in_G$ b ..................(i)

3) This Forced membership relation (i) can be coded in M as a triple in a,b,p:

< a,p > $\in$ b (i.e. b = { < a,p > , ...}) .................(ii)

within M. All possible sets of triples "< a,p > $\in$ b" that can be known to M are present within M, but as G isn't known to M, M will not know which of these triples will be part of the actual selected Forcing relation $\Vdash$. These sets of triples are equivalent to P-Names and represent all possible 'part' membership relations of any full $\in_G$ that could be forced and could be known to M.

4) The P-Names can be translated between M[G] and M by using the Shoenfield's Unramified Forcing paper definition, which 'looks' to be equivalent to relating (i) and (ii) above:

a $\in_G$ b $\leftrightarrow$ p $\in$ G AND < a,p > $\in$ b .........(iii)

where a,b,p are sets in M, the difference being in M[G] under $\in_G$ the elements of the sets can be different from $\in$ in M (which I thought was a particularly neat part of Shoenfield's Unramified Forcing paper).

Answer 1

Yes, it all looks good. I assume you're reading Shoenfield's paper, still IMO one of the best treatments of forcing in set theory (for the classical results). Let me add a few remarks.

For your point (1), "determined" here specifically means: the Mostowski collapse of the structure $(M,\in_G)$ is the model $M[G]$. (Shoenfield points this out.) The collapse function is called $K_G$ by Shoenfield (also $a\mapsto\bar{a}$).

Usually people don't apply the collapse to all of $M$, but to an inductively defined subset $V^P$: $$\begin{align*} V_0^P &= \varnothing\\ V_{\alpha+1}^P &= \mathcal{P}(V_\alpha^P\times P)\\ V_\lambda^P &= \bigcup_{\alpha<\lambda}V_\alpha^P\\ V^P &= \bigcup_{\alpha\in\Omega^M}V_\alpha^P \end{align*}$$ However $K_G(b)$ will ignore any elements not of the form $\langle a,p\rangle$; they are "chaff", "non-coding DNA".

If you're familiar with boolean-valued models, there one inductively defines a set of functions with values in the boolean algebra $B$: $F_0=\varnothing$, $F_{\alpha+1}$ is all functions (in $M$) with domain $F_\alpha$ and range in $B$, etc. Observe how close this is to the $V^P_\alpha$'s, thus: Take an element $f$ of $V^P_{\alpha+1}$; for any $a\in V^P_\alpha$, collect all the pairs $\langle a,p\rangle$ occuring in $f$; gather all these $p$'s to form a set of conditions. So $f$ is "essentially" a function with domain $V^P_\alpha$ taking values in the boolean algebra of sets of conditions.

Your point (3) gets at the heart of technique: each element of $V^P$ is a "name" of an element of $M[G]$. Inhabitants of $M$ can discuss elements of $M[G]$ using these names---they can say things like "$a$ is a potential element of $b$", "if condition $p$ is true, according to $G$, then $a$ really is an element of $b$". But since they don't know what's in $G$, they won't know the actual values of these names (i.e., they can't compute $K_G$).

Shoenfield's forcing language is just the standard language of ZF, augmented with all these names, and interpreting $\in$ as $\in_G$.

I wrote up a set of notes as a kind of gloss on Shoenfield's paper (and other things; $\S21$ is about forcing). It's at diagonalargument.com.