3 deleted 6 characters in body

We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion axiom "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion axiom "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply are provable from the existence axiom of an infinite Dedekind-finite setcountable choice, all of which and are weaker than consistent with the assertion assumption "Every There exists an infinite Dedekind-finite" , one of these assertion (but do not imply it), for example ) is "Every infinite set can be split into two disjoint infinite sets".

Generally speaking, there are several "orthogonal" choice principles (and their obvious extensions, etc.) for example axiom of choice for families of size $\kappa$ is generally independent from choice principles like the Boolean Prime Ideal Theorem (and its consequences).

The generalized Kinna-Wagner principle, while related to the Boolean Prime Ideal theorem can be shown to hold even when BPIT fails badly (e.g. there exists an amorphous set which cannot be linearly ordered). This principle is also a good example for something independent from the "choice-related" choice principles (again, if there is an amorphous set countable choice fails badly).

2 added 671 characters in body

We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply the existence of an infinite Dedekind-finite set, all of which are weaker than the assertion "Every Dedekind-finite", one of these assertion (for example) is "Every infinite set can be split into two disjoint infinite sets".

Generally speaking, there are several "orthogonal" choice principles (and their obvious extensions, etc.) for example axiom of choice for families of size $\kappa$ is generally independent from choice principles like the Boolean Prime Ideal Theorem (and its consequences).

The generalized Kinna-Wagner principle, while related to the Boolean Prime Ideal theorem can be shown to hold even when BPIT fails badly (e.g. there exists an amorphous set which cannot be linearly ordered). This principle is also a good example for something independent from the "choice-related" choice principles (again, if there is an amorphous set countable choice fails badly).

1

We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply the existence of an infinite Dedekind-finite set, all of which are weaker than the assertion "Every Dedekind-finite", one of these assertion (for example) is "Every infinite set can be split into two disjoint infinite sets".