Boolean prime ideal theorem and the axiom of choice

Question

The Boolean prime ideal theorem is strictly stronger than ZF, and strictly weaker than ZFC. I'm looking for nice examples (like the existence of non-measurable set) that request at least that theorem (ZF+BPI), but weaker than AC (ZFC).

The context is that I'm trying to gain some intuition, but most literature I could get access to is very technical and being lost in the details I can't see the big picture. Any help (like examples or your intuition behind it) or references that provide examples would be greatly appreciated.

Answer 1

The Boolean Prime Ideal theorem has a lot of useful equivalents. Two important ones are:

1. The completeness theorem for first-order logic.
2. The compactness theorem for first-order logic.

What you wrote in your question, however, is not fully accurate. The existence of a non-measurable subset does not require "at least" the Boolean Prime Ideal theorem. It is in fact much much weaker than that; and is implied by weaker principles (e.g. Hahn-Banach theorem) as well very different principles (e.g. $\aleph_1\leq2^{\aleph_0}+\sf DC$ implies the existence of a non-measurable set).

If you are looking for consequences of $\sf BPI$ which are unprovable from $\sf ZF$ itself then there are plenty. Here are a few:

1. Every set can be linearly ordered.
2. Every infinite set has a non-trivial ultrafilter.
3. If $V$ is a vector space, and $V$ has a basis $B$ then every basis of $V$ has the same cardinality as $B$.
4. Marshall Hall's marriage theorem.
5. Every partial order can be extended to a linear order.
6. Hahn-Banach theorem.
7. Every field has an algebraic closure, which is unique up to isomorphism.
8. Every family of finite non-empty sets admits a choice function.

And many many more.

Some of these examples you can find in the surprisingly not-very-technical book by Herrlich, The Axiom of Choice.

If you are looking for principles which are unprovable from $\sf BPI$, but true in $\sf ZFC$, then there are plenty of these as well:

1. The axiom of countable choice.
2. Every infinite set is Dedekind-infinite.
3. More generally, $\sf DC_\kappa$, for any $\kappa$.
4. The statement "For every infinite cardinal $\frak a$, $\frak a+a=a$".

And many many others.

Answer 2

An example from topology: $\mathsf{BPI}$ is equivalent to the Tikhonov product theorem for compact Hausdorff spaces, while $\mathsf{AC}$ is equivalent to the full Tikhonov product theorem.

Answer 3

There is a book, Equivalents of the Axiom of Choice (two volumes) by Rubin and Rubin that has an exhaustive catalog of principles equivalent to AC. The book Consequences of the Axiom of Choice by Howard and Rubin has many more principles that are weaker than AC. There is also a book The Axiom of Choice by Jech that discuses the prime ideal theorem.

Answer 4

A great reference for Consequences of the Axiom of Choice and their relations to each other is the book by Paul Howard and Jean Rubin, Consequences of the Axiom of Choice.