Boolean prime ideal theorem

The Boolean prime ideal states that every Boolean algebra contains a prime ideal. The proof of this theorem cannot be done without transfinite methods, which means that it cannot be proved from the axioms of set theory without the axiom of choice . Conversely, the axiom of choice cannot be proven from the Boolean prime ideal theorem, so this theorem is weaker than the axiom of choice. In addition, the theorem (relative to the axioms of the Zermelo-Fraenkel set theory ) is equivalent to some other theorem, such as Gödel's completeness theorem . (This means that one can prove the same theorems from the axioms of set theory plus the Boolean prime ideal theorem as from the axioms of set theory plus Godel's completeness theorem.)

If the Boolean algebra is replaced by its dual Boolean algebra, the Boolean prime ideal theorem becomes the ultrafilter lemma.

Definitions

In a Boolean algebra, an order can be introduced naturally :

${\displaystyle u\leq v:\Leftrightarrow u\lor v=v}$

An ideal of a Boolean algebra is a true subset of with the following properties: ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle B}$

• ${\displaystyle (u\in I}$ and ${\displaystyle v\leq u)\Rightarrow v\in I}$
• ${\displaystyle (u\in I}$ and ${\displaystyle v\in I)\Rightarrow u\lor v\in I}$

An ideal is a prime ideal if it has the additional property that for every element from that either or contains. ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle u}$ ${\displaystyle B}$ ${\displaystyle I}$ ${\displaystyle u}$ ${\displaystyle \neg u}$

${\displaystyle I}$can not both and include, as otherwise ${\displaystyle u}$ ${\displaystyle \neg u}$

${\displaystyle 1=u\lor \neg u\in I}$

would, and always there for any element ${\displaystyle v\in B}$

${\displaystyle v\leq 1}$

would then also apply to everyone , i.e. contrary to the definition of an ideal. ${\displaystyle v\in I}$ ${\displaystyle v\in B}$ ${\displaystyle I=B}$

sentence

The statement of the Boolean prime ideal theorem is:

• Every Boolean algebra has a prime ideal.

This statement is only apparently weaker than the following:

• Every ideal of a Boolean algebra lies in a prime ideal. ${\displaystyle B}$

Because if an ideal is an equivalence relation can be defined: ${\displaystyle I}$ ${\displaystyle B}$

${\displaystyle u\sim v:\Leftrightarrow (u\wedge \neg v)\lor (\neg u\wedge v)\in I}$, so

${\displaystyle u\sim 0\Leftrightarrow u\in I}$

The quotient according to this equivalence relation (or the ideal) carries through the definitions ${\displaystyle B/I}$

${\displaystyle [u]\lor [v]:=[u\lor v],\quad [u]\land [v]:=[u\land v],\quad \neg [u]:=[\neg u]}$

a natural structure as Boolean algebra and the canonical homomorphism maps exactly to . Hence the archetype of a prime ideal of is a prime ideal of that contains. ${\displaystyle \phi \colon B\to B/I}$ ${\displaystyle I}$ ${\displaystyle 0}$ ${\displaystyle \phi ^{-1}(P)}$ ${\displaystyle P}$ ${\displaystyle B/I}$ ${\displaystyle B}$ ${\displaystyle I}$

proof

The proof is a standard application of Zorn's lemma and thus the axiom of choice. The set of all ideals is ordered via the subset relation and the union of a chain is again an ideal. So there is a maximum element.

Now proof by contradiction: Assume that this maximal ideal is not a prime ideal. Then there is one with . ${\displaystyle M}$ ${\displaystyle u\in B}$ ${\displaystyle u,\neg u\notin M}$

If it is for one , then applies to all : Because if it were , it would also be ${\displaystyle \neg u\lor w=1}$ ${\displaystyle w\in B}$ ${\displaystyle u\lor s<1}$ ${\displaystyle s\in B}$ ${\displaystyle u\lor s=1}$

${\displaystyle 1=(\neg u\lor w)\wedge (u\lor s)=(\neg u\lor u)\wedge (\neg u\lor s)\wedge (w\lor u)\wedge (w\lor s)=(\neg u\lor s)\wedge (w\lor u)\wedge (w\lor s)}$

Since there is an ideal, lie and in , therefore also what cannot be. ${\displaystyle M}$ ${\displaystyle (\neg u\lor s),(w\lor u)}$ ${\displaystyle (w\lor s)}$ ${\displaystyle M}$ ${\displaystyle 1\in M}$

Therefore it is true for all or for all ${\displaystyle w\in B:\neg u\lor w<1}$ ${\displaystyle w\in B:u\lor w<1}$

It applies to everyone without restricting the general public

${\displaystyle w\in B:u\lor w<1}$

The smallest ideal that includes and contains ( ) is really greater than . is therefore not maximally in opposition to the assumption, i.e. contradiction. ${\displaystyle M^{*}}$ ${\displaystyle M}$ ${\displaystyle u}$ ${\displaystyle M^{*}:=M\cup \{s\lor u|s\in M\}}$ ${\displaystyle M}$ ${\displaystyle M}$

${\displaystyle M}$ is therefore a prime ideal.

Equivalent statements

The following statements are equivalent to the Boolean prime ideal theorem if only ZF is assumed:

• The Stonesian representation theorem :

Every Boolean algebra is isomorphic to a set algebra.

• The Gödel completeness theorem :

Every consistent theory has a model.

• The compactness theorem :

A set of statements of first order predicate logic has a model if and only if every finite subset has a model.

• The ultrafilter lemma :

Each filter can be expanded to an ultrafilter.

• The set of Lindenbaum :

Any consistent theory of first-order predicate logic can be expanded to a maximally consistent theory.

Inferences

From the axioms of set theory without a choice axiom, but with the Boolean prime ideal theorem, one can deduce, among other things:

• Marczewski-Szpilrajn's theorem : Every partial order can be expanded to a linear order .
• Artin-Schreier theorem : An order can be introduced on every body in which −1 is not a sum of squares.
• Tychonoff's sentence for Hausdorff rooms : Every product of compact Hausdorff rooms is compact. Tychonoff's general theorem, however, is equivalent to the axiom of choice.

literature

• Thomas Jech : The Axiom of Choice . North Holland, 1973, ISBN 0-7204-2275-2 .
• Horst Herrlich : Axiom of Choice . Springer, 2006, ISBN 978-3-540-30989-5 .

Web links

• Frithjof Dau: Diploma thesis on the Boolean prime ideal theorem. Retrieved on December 26, 2016 (contains, among other things, a detailed compilation of various statements equivalent to the Boolean prime ideal theorem).