Set theory is much more complicated than "common" mathematics in this aspect, it deals with things which you can often prove that are unprovable.
Namely, when we start with mathematics (and sometimes for the rest of our lives) we see theorems, and we prove things about continuous functions or linear transformations, etc.
These things are often simple and have a very finite nature (in some sense), so we can prove and disprove almost all the statements we encounter. Furthermore it is a good idea, often, to start with statements that students can handle. Unprovable statements are philosophically hard to swallow, and as such they should usually be presented (in full) only after a good background has been given.
Now to the continuum hypothesis. The axioms of set theory merely tell us how sets should behave. They should have certain properties, and follow basic rules which are expected to hold for sets. E.g., two sets which have the same elements are equal.
Using the language of set theory we can phrase the following claim:
If $A$ is an uncountable subset of the real numbers, then $A$ is equipotent with $\mathbb R$.
The problem begins with the fact that there are many subsets of the real numbers. In fact we leave the so-called "very finite" nature of basic mathematics and we enter a realm of infinities, strangeness and many other weird things.
The intuition is partly true. For the sets of real numbers which we can define by a reasonably simple way we can also prove that the continuum hypothesis is true: every "simply" describable uncountable set is of the size of the continuum.
However most subsets of the real numbers are so complicated that we can't describe them in a simple way. Not even if we extend the meaning of simple by a bit, and if we extend it even more, then not only we will lose the above result about the continuum hypothesis being true for simple sets; we will still not be able to cover even anything close to "a large portion" of the subsets.
Lastly, it is not that many people "believe it is not a simple deduction". It was proved - mathematically - that we cannot prove the continuum hypothesis unless ZFC is inconsistent, in which case we will rather stop working with it.
Don't let this deter you from using ZFC, though. Unprovable questions are all over mathematics, even if you don't see them as such in a direct way:
There is exactly one number $x$ such that $x^3=1$.
This is an independent claim. In the real numbers, or the rationals even, it is true. However in the complex numbers this is not true anymore. Is this baffling? Not really, because the real and complex numbers have very canonical models. We know pretty much everything there is to know about these models (as fields, anyway), and it doesn't surprise us that the claim is true in one place, but false in another.
Set theory (read: ZFC), however, has no such property. It is a very strong theory which allows us to create a vast portion of mathematics inside of it, and as such it is bound to leave many questions open which may have true or false answers in different models of set theory. Some of these questions affect directly the "non set theory mathematics", while others do not.
Some reading material:
- A question regarding the Continuum Hypothesis (Revised)
- Neither provable nor disprovable theorem
- Impossible to prove vs neither true nor false
In mathematics, and in particular when talking about the incompleteness phenomenon, there is a grave danger of confusing the two terms "True" and "Provable".
Being "true" is a semantic property of a statement. Statements are true in a particular interpretation of the language, or a particular model of a theory. Whereas being provable is a syntactical property which means that there is a sequence statements which are axioms, or inferences from the axioms which is both finite, and its final statement is the statement we wanted to prove.
The completeness theorem tells us that $T$ proves a statement if and only if the statement is true in every model of $T$. So we sometimes abuse the meaning and say things like "Cantor's theorem is true in $\sf ZF$" when we mean that $\sf ZF$ can prove Cantor's theorem.
Sometimes, as in the case of arithmetic, there is an intended interpretation or a standard model for the theory. With the integers we have a standard model that we know to characterize. That is the model we care about. So in the context of Peano's axioms of arithmetic we say that $\varphi$ is true if it is true in the standard model, and provable if it is true in every model. However $\sf ZFC$ is far from $\sf PA$ in this context, and it does not have a standard or intended interpretation (in set theory the term "standard model" means something which is far more arbitrary than in $\sf PA$).
So finally we ask what does it mean for $\sf CH$ to be true? Since we don't have a standard model for set theory, we can't associate some "Platonic truth" to the statement in the language of set theory. Some will be true in some models, and some will be false in other models. The continuum hypothesis - and its negation - are both such statement. And when we say that the continuum hypothesis is not provable from the axioms of $\sf ZFC$, we mean that mathematically we have proved that this statement does not have a proof from the axioms. How did we do that? We showed that there models where it is true, and models where it is false (assuming there are models to begin with, of course).
So saying that $\sf CH$ is true, but not saying where is meaningless in the sense that it doesn't offer sufficient information to properly evaluate the claim. It is true in $L$, which is Godel's constructible universe (i.e. in every model of set theory satisfying the statement $V=L$ the continuum hypothesis is true). But it need not be true in other models of set theory (e.g. models of $\sf PFA$).
Edit:
From the question's comments it shows that the question rose after reading a statement of the form "the continuum hypothesis is true for all practical purposes" and "the continuum hypothesis is true for Borel sets". The statement simply say that if we are only concerned about Borel sets (or some other indicated family of sets) then they are either countable, or have the cardinality of the continuum.
Historically when Cantor set to prove his continuum hypothesis, it was simply to find a bijection between open intervals and the real numbers; and using a clever method he showed that uncountable closed sets must have the cardinality of the continuum. Cantor expected that these proofs can carry on on some "complexity" of sets, and eventually cover all the sets. However the construction of taking complement and countable unions only gives us the Borel sets, and indeed shortly after leaving the Borel sets one can already run into classes which do not have to satisfy the continuum hypothesis.
Further reading:
- Why is the Continuum Hypothesis (not) true?
- Unprovable things
- Neither provable nor disprovable theorem
Best Answer
Cohen's result is that from a certain set of axioms ($\mathsf{ZFC}$) we cannot prove the continuum hypothesis. Gödel's result is that from the same set of axioms, we cannot refute the continuum hypothesis. This only means that the set of axioms under consideration is not strong enough to settle this question. If you manage to exhibit a set of intermediate size, your argument necessarily uses axioms beyond those in $\mathsf{ZFC}$, or is not formalizable in first-order logic. Similarly if you manage to show that there is no such set. It is worth pointing out that there are standard axioms beyond those in $\mathsf{ZFC}$ that settle the continuum problem. These axioms are not universally accepted yet, but this illustrates that, as explained, their results are not absolute, but relative to a very specific background theory.