Boolean Prime Ideal Theorem

From ProofWiki
Jump to: navigation, search

Theorem

If $(S, \le)$ is a Boolean algebra, $I$ is an ideal in $S$, $F$ is a filter on $S$, and $I \cap F = \varnothing$, then

there exists a prime ideal $P$ in $S$ such that $I \subseteq P$ and $I \cap F = \varnothing$.


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 $U=\bigcup N$ is clearly disjoint from $F$ and contains $I$. It is also an ideal:

Suppose $x\in U$ and $y \le x$.
Then for some $A \in N$, $x \in A$.
By the definition of union, $x \in U$.
Suppose $x \in U$ and $y \in U$.
Then for some $A,\,B \in N$, $x \in A$ and $y\in B$.
By the definition of a chain, $A \subseteq B$ or $B \subseteq A$.

Suppose without loss of generality that $A \subseteq B$.

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


Proof from Stone's Representation Theorem for Boolean Algebras


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.