TITLE: Last Status of Feferman's Conjecture on Indefinite Value of Continuum
QUESTION [11 upvotes]: 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.)

REPLY [6 votes]: Recently M. Rathjen have proved, that CH (Continuum Hypothesis) is a indefinite problem in system SCS. In my last work I prove the same for two-sorted intuitionistic Zermelo-Fraenkel set theory ZFI2C with the Collection. This theory is much stronger than SCS. In fact, the classical theory ZF2 = ZFI2C+LEM by some variant of Friedmans version of Goedels negative translation. Of course, the same holds for one-sorted theory ZFIC with the Collection schema. So, CH is an indefinite problem in a rather strong sense.
Sincerely yours,
Alexey G. Vladimirov