Boolean Prime Ideal Theorem
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Theorem
Let $\struct {S, \le}$ be a Boolean algebra.
Let $I$ be an ideal in $S$.
Let $F$ be a filter on $S$.
Let $I \cap F = \O$.
Then there exists a prime ideal $P$ in $S$ such that:
- $I \subseteq P$
and:
- $P \cap F = \O$
Proof from the Axiom of Choice
Let $T$ be the set of ideals in $S$ that contain $I$ and are disjoint from $F$, ordered by inclusion.
Let $N$ be a chain in $T$.
Then $\ds U = \bigcup N$ is clearly disjoint from $F$ and contains $I$.
Let $x \in U$ and $y \le x$.
Then:
- $\exists A \in N: x \in A$
By the definition of union:
- $x \in U$
Let $x \in U$ and $y \in U$.
Then:
- $\exists A, B \in N: x \in A, y \in B$
By the definition of a chain:
- $A \subseteq B$
or:
- $B \subseteq A$
Hence $U$ is also an ideal.
of what, and why
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Without loss of generality, suppose $A \subseteq B$.
Then:
- $x \in B$
Since $y$ is also in $B$, and $B$ is an ideal:
- $x \vee y \in U$.
By Zorn's Lemma, $T$ has a maximal element, $M$.
It remains to show that $M$ is a prime ideal:
Every Boolean algebra is a distributive lattice.
So by Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice, $M$ is a prime ideal.
Axiom of Choice
This proof depends on the Axiom of Choice, by way of Zorn's Lemma.
Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.
Proof from the Ultrafilter Lemma
We need this proof to show that the Boolean Prime Ideal Theorem is equivalent to the Ultrafilter Lemma in ZF
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Proof from Stone's Representation Theorem for Boolean Algebras
We need this proof to show that the Boolean Prime Ideal Theorem is equivalent to Stone's Representation Theorem for Boolean Algebras in ZF
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Significance
The Boolean Prime Ideal Theorem is weaker than the Axiom of Choice, but is similarly independent of ZF theory.
It is sufficient to prove a number of important theorems, although such proofs are often more involved than ones relying on the Axiom of Choice.