Weaker Choice of the Real Numbers In set theory, a set $A$ is a projective set if for any other sets $B, C$ and for any function $f:A\rightarrow B$ and surjective function $g:C\rightarrow B$, there exists a function $h:A\rightarrow C$ such that $g \circ h = f$. The Axiom of Choice is the statement that all sets are projective sets, while the weaker Axiom of Countable Choice is the statement that $\mathbb{N}$ is a projective set.
Suppose that the Axiom of Choice in classical set theory is replaced with the weaker axiom that the set of real numbers $\mathbb{R}$ is a projective set. Let us for the sake of this question call this new axiom the Axiom of Real Choice, for lack of a better term.

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*Is the Axiom of Countable Choice provable from the Axiom of Real Choice? If so, then:

*Which results in mathematics (such as in real analysis, group theory, etc) that are proven using the full Axiom of Choice and cannot be proven using the Axiom of Countable Choice are now provable using the Axiom of Real Choice?

*Which results in mathematics that are proven using the full Axiom of Choice and cannot be proven using the Axiom of Countable Choice still cannot be proved using the Axiom of Real Choice?

*Are there any commonly used axioms that are inconsistent with the full Axiom of Choice that are consistent with the Axiom of Real Choice?

 A: *

*Yes. If you have a countable family of sets, $\{X_n\mid n\in\Bbb N\}$, extend it to a family of size $\Bbb R$.


*Yes. "Every set of size $\Bbb R$ admits a choice function".


*The Axiom of Choice, and therefore all of its equivalents: e.g. every vector space has a basis; every commutative unital ring has a maximal ideal; etc. Including every equivalent of the Boolean Prime Ideal theorem (e.g. every commutative unital ring has a prime ideal).


*This is not particularly clear, but likely to be false.
This is Form 181 in the Howard–Rubin dictionary of choice principles. But the biggest problem with this principle is that it will follow from the conjunction of:

*

*$|\Bbb R|=\aleph_1$ and

*$\sf AC_{\aleph_1}$.

And while it's a somewhat nontrivial conjunction, it is also not particularly powerful or interesting (for example, it is easy to arrange a model where this conjunction holds, but there are sets which cannot be linearly ordered). You can find more information in the book, or in the online graphing website: https://cgraph.inters.co/.
