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The finite Boolean expansion contradiction in the supported-ideal proof

Statement

Let $B$ be a Boolean algebra, let $I$ be a proper ideal of $B$ , let $y$ be a finite set, and let $d : y \to B$ . For $z \subseteq y$ define its truth-assignment atom by

$e_{z} = (⋀_{a \in z} d (a)) \land (⋀_{a \in y ∖ z} \neg d (a)) .$

If $e_{z} \in I$ for every $z \subseteq y$ , then the finite Boolean expansion of $1$ puts $1$ in $I$ , contradicting propriety.

Here is the exact conditional clopen pattern used in the unary supported-ideal calculation. Let $K$ be a finite nonempty index set. For each $i \in K$ , let $U_{i}$ contain pairwise disjoint cells $U_{i, j}$ indexed by a finite nonempty set $J_{i}$ , with all the cells $U_{i, j}$ pairwise disjoint as $i, j$ vary. Suppose $y$ contains exactly one point of every $U_{i, j}$ and no other points. Assume

whenever $z \subseteq y$ meets every $U_{i}$ , one has $⋀_{a \in z} d (a) \in I$ ; and
for every $i \in K$ , whenever $w \subseteq y$ meets every $U_{i, j}$ with $j \in J_{i}$ , one has $⋀_{a \in w} \neg d (a) \in I$ .

Then every $e_{z}$ belongs to $I$ , so the contradiction follows. These hypotheses isolate the final finite calculation; they do not assert that the unresolved A-page primality lemma has constructed such a clopen pattern.

Facts & Assumptions

Given: The Boolean algebra, proper ideal, finite set, map, and, for the second assertion, the displayed finite clopen-incidence hypotheses.

[F1]

Boolean algebras and their order supplies distributivity, $b \lor \neg b = 1$ , and the conventions that the empty join is $0$ and the empty meet is $1$ .

[F2]

Boolean ideals, filters, prime ideals and ultrafilters says that an ideal is downward closed and closed under binary joins, and that propriety means $1 \notin I$ .

Proof

technique · induction on the finite set $y$

1.1

If $y = \emptyset$ , its only subset is empty, $e_{\emptyset} = 1$ by the empty-meet convention, and therefore $⋁_{z \subseteq y} e_{z} = 1$ .

F1base

1.2

Suppose the expansion identity holds for a finite set $y_{0}$ , take $a \notin y_{0}$ , and put $y = y_{0} \cup {a}$ . [ih, construct] Every subset of $y$ is uniquely either $w$ or $w \cup {a}$ for a subset $w \subseteq y_{0}$ . With $e_{w}^{0}$ denoting the atom formed over $y_{0}$ , the two corresponding atoms over $y$ are

$e_{w} = (\neg d (a)) \land e_{w}^{0}, e_{w \cup {a}} = d (a) \land e_{w}^{0} .$

2.1

Finite distributivity and the induction hypothesis give the following identity.

F1step 1.2ih

$\begin{aligned} \underset{z \subseteq y}{⋁} e_{z} & = \underset{w \subseteq y_{0}}{⋁} ((\neg d (a) \land e_{w}^{0}) \lor (d (a) \land e_{w}^{0})) \\ = (\neg d (a) \lor d (a)) \land \underset{w \subseteq y_{0}}{⋁} e_{w}^{0} \\ = 1 \land 1 = 1. \end{aligned}$

3.1

Steps 1.1 and 2.1 prove for every finite $y$ the exact identity below.

step 1.1step 2.1

$1 = ⋁_{z \subseteq y} [(⋀_{a \in z} d (a)) \land (⋀_{a \in y ∖ z} \neg d (a))] .$

4.1

With the identity of step 3.1 established, now assume the displayed clopen pattern and fix $z \subseteq y$ . If $z$ meets every $U_{i}$ , the first hypothesis puts $p_{z} = ⋀_{a \in z} d (a)$ in $I$ , and $e_{z} \leq p_{z}$ puts $e_{z}$ in $I$ by downward closure. Otherwise choose an $i \in K$ with $z \cap U_{i} = \emptyset$ . The unique point of $y \cap U_{i, j}$ then lies in $y ∖ z$ for every $j \in J_{i}$ , so the second hypothesis puts $n_{z} = ⋀_{a \in y ∖ z} \neg d (a)$ in $I$ . Again $e_{z} \leq n_{z}$ gives $e_{z} \in I$ . These two cases are exhaustive, including $z = \emptyset$ and $z = y$ .

F2givenstep 3.1cases

5.1

Thus every term in the finite join of step 3.1 lies in $I$ . Repeated binary join closure, with the singleton case covering the empty $y$ boundary, puts their join $1$ in $I$ . This contradicts the propriety condition $1 \notin I$ and proves both assertions.

F2step 3.1step 4.1discharge-induction∎

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Miroslav Repický, A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem, final calculation, pp.545-546 (standard reference, not scraped)