[Math] Why doesn’t the independence of the continuum hypothesis immediately imply that ZFC is unsatisfactory

Question

I am risking a "possible duplicate of… " here, in particular with respect to this question, this question, and this question. Nevertheless here goes.

I am going to keep this question as simple as possible:

First note two facts:

1. It has been proven that CH is independent of ZFC.

2. ZFC is intended as a foundational system of mathematics.

Secondly: the independence of CH from ZFC has (if I understand it correctly) led many people to state that "CH is neither true nor false".

My response is: It seems to me that the independence of CH from ZFC cannot be "blamed" on CH, but should be ZFC's responsibility, for the simple reason that CH never had any pretention of being in any way related to the ZFC axioms, while the ZFC axioms claim to be a foundation of mathematics.

Therefore if ZFC and CH are independent, this says absolutely nothing about the truth-status of CH, and merely implies that ZFC is inadequate. ZFC is merely one possible axiomatizations that some smart people came up with a long time ago, so why don't we take ZFC with a grain of salt. Why do people disagree with my argument, and instead claim that the independence of ZFC and CH implies that CH is "neither true nor false"?

Answer 1

It is true that, if $\mathsf{ZFC}$ is consistent, $\mathsf{CH}$ is undecidable within it. This is just one example of a more general fact: any consistent recursively enumerable first-order theory at least as strong as Peano arithmetic contains a statement undecidable in that theory. This is the first of Gödel's incompleteness theorems; the second gives an example, namely a statement calling the theory consistent.

The real question isn't whether $\mathsf{ZFC}$ has known limitations of this kind; of course it does. The question is which other statements we should add as axioms. The $\mathsf{C}$ in $\mathsf{ZFC}$ is the axiom of choice, which is itself undecidable in $\mathsf{ZF}$. The history of set theory has seen much broader support in favour of adding $\mathsf{AC}$ than in favour of subsequently adding $\mathsf{CH}$.

Why? Well, let's look at some of the differences:

• Although $\mathsf{AC}$ was initially much more controversial than it is today, it has come to enjoy broad support for the simple reason that, although it has some counter-intuitive consequences such as Zermelo's well-ordering theorem, its negation has "even worse" consequences such as trichotomy violation.
• Not only is $\beth_1=\aleph_1$ (i.e. the $\mathsf{CH}$) undecidable in $\mathsf{ZFC}$; so is $\beth_1=\aleph_n$ for any positive integer $n$. Why would we adopt the first as an axiom? By contrast, $\mathsf{AC}$ doesn't have an infinite family of obvious counterparts that feel equally feasible.
• One good thing about the $\mathsf{CH}$ is that it's a special case of a more general idea that looks nice, the $\mathsf{GCH}$ ($\beth_\alpha=\aleph_\alpha$ for all ordinals $\alpha$). There is some interest in adding $\mathsf{GCH}$ to $\mathsf{ZFC}$, but just $\mathsf{CH}$ on its own? That's an unpopular compromise between $\mathsf{ZFC}$ (which is weak enough for some people's tastes) and $\mathsf{ZFC+GCH}$ (which is strong enough for some other people's tastes).
• Lastly, outside of set theory $\aleph_1$ usually doesn't even come up as a concept, and so $\mathsf{CH}$ has no obvious benefit when founding any other areas of mathematics. (There are exceptions, e.g. nonstandard analysis uses $\mathsf{CH}$ to consider real infinitesimals.) By contrast, $\mathsf{AC}$ has uses all over the place, e.g. proving every vector space has a basis.