The Continuum Hypothesis say that there is no set with cardinality between that of the reals and the natural numbers. Apparently, the Continuum Hypothesis can’t be proved or disproved using the standard axioms of set theory.
In order to disprove it, one would only have to construct one counterexample of a set with cardinality between the naturals and the reals. It was proven that the CH can’t be disproven. Equivalently, it was proven that one cannot construct a counterexample for the CH. Doesn’t this prove it?
Of course, the issue is that it was also proven that it can’t be proved. I don’t know the details of this unprovability proof, but how can it avoid a contradiction? I understand the idea of something being independent of the axioms, I just don’t see how if there is provably no counterexample the hypothesis isn’t immediately true, since it basically just says a counterexample doesn’t exist.
I’m sure I’m making some horrible logical error here, but I’m not sure what it is.
So my question is this: what is the flaw in my argument? Is it a logical error, or a gross misunderstanding of the unprovability proof in question? Or something else entirely?
Here’s an example axiomatic system:
- There exist exactly three objects $A, B, C$.
- Each of these objects is either a banana, a strawberry or an orange.
- There exists at least one strawberry.
Let’s name the system $X$.
Vincent’s Continuum Hypothesis (VCH): Every object is either a banana or a strawberry (i.e., there are no oranges).
Now, to disprove this in $X$, you would have to show that one of $A, B, C$ is an orange (“construct a counterexample”). But this does not follow from $X$, because the following model is consistent with $X$: A and B are bananas, C is a strawberry.
On the other hand, VCH does not follow from $X$ either, because the following model is consistent with $X$: A is a banana, B is a strawberry, C is an orange.
As you can see, there is no contradiction, because you have to take into account different models of the axiomatic system.