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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.)

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For further arguments on the true value of $2^{\aleph_0}$ (which is not what I am searching for here) see this related MO post: Solutions to the Continuum Hypothesis –  user45939 May 10 at 1:32
    
Is there an intuitionistic model coming from topos theory in which $\text{CH}\vee\neg\text{CH}$ is not true? If so, would this establish Feferman's conjecture? –  Joel David Hamkins May 10 at 1:42
    
@JoelDavidHamkins I don't know. I am new in this point of view. Personally I strongly believe in ontological maximalism also I like plural Platonism because it is a fundamentally new approach. As a consequence I am searching for a consistency between ontological maximalism and plural Platonism. –  user45939 May 10 at 1:53
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I think your question is a good one, but I disagree with your introduction. With the caveat that I'm not a set theorist, it has always seemed to me that $2^{\aleph_0}$ is a perfectly definite number — the question that ZF doesn't answer is rather "how big is $\aleph_1$?". –  Theo Johnson-Freyd May 10 at 5:14
    
@JoelDavidHamkins I don't know this for CH, at least for the AC, there must be one, since AC+extensionality+replacement imply TND. So I wouldn't be surprised if something similar holds for the AC. Anyway, I think the Question how many reals "really" exist is strange. I don't know much about philosophy, but in standard classical mathematics, reals and sets are basically defined by the axioms. And the question boils down to whether there exists a subset of the reals which has a surjection into N but no bijection into R and N. And this is just a question of definition. –  Christoph-Simon Senjak May 10 at 14:35

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