The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more than "*it could be everything under current weak foundation of mathematics*". In fact Hilbert's first question is still open and even more open than the first time which he announced it because "*How many real numbers do we have "really"?*" is a completely different question from "*How many real numbers "can" we have?*" The set theoretic results tells many things about the last question but a few about the first one which is Hilbert's first question.

A well-known conjecture about the value of $2^{\aleph_0}$ is Continuum Hypothesis ($\text{CH}$) which says $2^{\aleph_0}=\aleph_1$.

Some set theorists (e.g. Goedel) who mainly believe on ontological maximalism, think $\text{CH}$ is false. Some others who believe in smaller universe of mathematical objects think $\text{CH}$ is true. Recently a third point of view is growing up by some set theorists like Hamkins and Feferman. It says the value of continuum is "indefinite" and the "true value" of $2^{\aleph_0}$ doesn't exist at all.

The following is quoted from Wikipedia.

Solomon Feferman (2011) has made a complex philosophical argument that $\text{CH}$ is not a definite mathematical problem. He proposes a theory of "definiteness" using a semi-intuitionistic subsystem of $\text{ZF}$ that accepts classical logic for bounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition $\phi$ is mathematically "definite" if the semi-intuitionistic theory can prove $(\phi \vee \neg\phi)$. He conjectures that $\text{CH}$ is not definite according to this notion, and proposes that $\text{CH}$ should therefore be considered not to have a truth value.

**Question:** What is known about Feferman's conjecture on indefinite value of continuum? References on the philosophical arguments about this conjecture are also welcome. (The only paper which I am aware of is Koellner's article.)

Solutions to the Continuum Hypothesis– user45939 May 10 '14 at 1:32