For uni I'm studying intuitionism, and I came across the following disproof for the IVT:
The thing I'm trying to understand is why this disproof is not valid in classical mathematics. In my research on intuitionism I read about some properties regarding infinity cannot be used in intuitionistic proofs, and think it has to do with this problem, but I'm not sure. Could anyone maybe clarify if this is the case, and if not, what oyther reason is there for this disproof to not count classically? Any help is appreciated.
I do not like the term "disproof", because it suggests that intuitionistic mathematics would say that the intermediate value theorem (IVT) is false, which it does not. What happens here is that we show that IVT would allow us to recover part of the law of excluded middle. In particular, it would decide the parity of $k_0$ (if it exists). What this means is that we cannot prove IVT in intuitionistc mathematics.
Remember, everything you prove in intuitionistic mathematics also holds in classical mathematics. This is because every logical step you take in intuitionistic mathematics is also classically valid. It is the other way around that does not work.
So from a classical point of view all that this "disproof" gives us is that there is some continuous function $f: [0, 3] \to \mathbb{R}$ that has a zero $x$ for which the exact value of $x$ is tied to the parity of $k_0$ (if it exists). However, classically if we assert the existence of such an $x$ we make no claims about being able to locate it.