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lo.logic – Intuition behind Boolean-valued fashions of clique principle Answer

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lo.logic – Intuition behind Boolean-valued fashions of clique principle

$DeclareMathOperatorCard{Card}$The bespeak Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann offers an instinct behind boolean valued fashions of clique principle which I’ll elucidate beneath. But when I attempt to make employ of the instinct, as I grasp it, I speed into issues.

On pages 272-275 the next instinct is offered:
Consider the boolean algebra $B:=mathcal{P}(lbrace 1,2,3 rbrace)$.
The sole parts $1$, $2$ and $3$ can breathe considered three workable “worlds”.
When contemplating a boolean clique, i.e. an component of $V^{(B)}$ love $x mathrel{:=} lbrace(emptyset,lbrace2,3rbrace),(lbrace(emptyset,lbrace1,3rbrace)rbrace,lbrace1,3rbrace)rbrace$ it might probably breathe written as $x = commence{circumstances}
1: lbracelbraceemptysetrbracerbrace
2: lbraceemptysetrbrace
3: lbraceemptyset,lbraceemptysetrbracerbrace
aim{circumstances}$

exhibiting which “normal” clique hides behind x in accordance with every world.
Also the boolean veracity worth $lVertvarphirVert in B$ of a sentence $varphi$ in set-language (together with constants for every boolean clique) can now breathe seen because the clique of worlds by which $varphi$ holds. The following instance is offered:
Along with $x$, deem the boolean clique $y mathrel{:=} lbrace(emptyset,lbrace1,3rbrace),(lbrace(emptyset,lbrace1,2,3rbrace)rbrace,lbrace1,2,3rbrace)rbrace = commence{circumstances}
1: lbraceemptyset,lbraceemptysetrbracerbrace
2: lbracelbraceemptysetrbracerbrace
3: lbraceemptyset,lbraceemptysetrbracerbrace.
aim{circumstances}$

Then from the instinct behind $Vert cdot Vert$, it instantly follows:
$lVert x in yrVert = lbrace2rbrace$,
$lVert x subseteq yrVert = lbrace1,3rbrace$,
$lVert x = yrVert = lbrace3rbrace$.

From this, I understood the given instinct as follows:
For a given boolean algebra $B$ we deem the clique $W_B$ of worlds.
This ought to breathe workable by the illustration theorem of Stone.
(And every of those worlds can breathe seen as a “copy of the universe $V$“.)
A boolean clique $x in V^{(B)}$ then induces a map $x’: W_B rightarrow V$ mapping every world to the clique that “hides behind $x$“.
(As completely different boolean units can induce the identical map, I doubt that the employ of the equality image between a boolean clique and the respective illustration distinction, as carried out within the bespeak, is exquisite.) For an capricious sentence $varphi$ we’d get (in accordance with the instinct) $lVertvarphirVert = lbrace w in W_B mid textual content{“$V models varphi_w$”} rbrace$, the place $varphi_w$ is obtained by changing the constants in $varphi$ with the arrogate clique in accordance with world $w$ (e.g. $x in V^{(B)}$ as constant-symbol in $varphi$ is changed by $x'(w)$).

Analogously, the identical can breathe carried out when contemplating the boolean clique universe $mathcal{M}^{(B)}$
of a countable, transitive benchmark mannequin $mathcal{M} fashions ZFC$, the place $B in mathcal{M}$ is a boolean algebra that’s full in $mathcal{M}$. Now worlds can breathe seen as “copies of $mathcal{M}$” and $x in mathcal{M}^{(B)}$ induces a map $x’: W_B rightarrow mathcal{M}$. As above, we get $lVertvarphirVert^{mathcal{M}^{(B)}} = lbrace w in W_B mid mathcal{M} fashions varphi_w rbrace$.

The drawback:
I’m properly cognizant that my judgement of the instinct given within the bespeak is missing of one thing. This can breathe seen when contemplating outcomes love (web page 367):
If B satisfies CCC in $mathcal{M}$, then $forall x in mathcal{M}: mathcal{M} fashions Card(x) Leftrightarrow lVert Card(bridle{x}) rVert^{mathcal{M}^{(B)}} = 1_B$. We would get
the conclusion immediately from our instinct (while not having $B$ to answer CCC in $mathcal{M}$) because the benchmark consultant $bridle{x}$ induces a fixed map and $1_B$ is the clique of all worlds.

As the bespeak doesn’t refer into mighty additional element regarding this instinct, I might breathe very happy if any person may ameliorate me out. What has to breathe modified in organize for every part to breathe sound once more? Is the instinct solely genuine for very specific sentences, maybe solely atomic sentences? Is my judgement of ‘world’ too unostentatious? Are specific boolean algebras needful? Is the instinct offered within the bespeak restricted to very particular circumstances? (This is the primary time I’ve encountered this instinct.)

I’m a set-theory newbie and value any ameliorate — plane whether it is patent.

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