Can ‘concrete’ statements be undecidable?
Let me preface by saying that, while I’m a mathematician, I am not a logician. I’m simply curious about the field. This post is about things that have always bugged me about undecidability.
I’ll take the Collatz conjecture (CC) as an example. Suppose the CC is undecidable in ZFC. Then, if we believe ZFC is consistent, we cannot find a counterexample. If we did, this would contradict it being undecidable. This is strange to me: we would have a statement for which we can’t show its truth while knowing we cannot find a counterexample.
Compare this with the continuum hypothesis (CH), the statement people usually refer to as example of undecidable statement. If we choose it to be true, we know there is a subset S of the real numbers which has cardinality between countable and the continuum. CH being undecidable implies that if we believe ZFC is consistent, we will not stumble into such an S by construction (say with schema specification, etc.). If we did, this would contradict ZFC. But that’s ok, because not every subset of the reals can necessarily be obtained ‘constructively’ through schema specification etc. (I am being informal here, but I believe there is a formal way to say this). So us not being able to produce a counterexample of CH does not mean that one cannot exist.
To me, this seems like a different flavour to CC. There, it seems that any counterexample would be constructive (it’s just a natural number!), and hence, if CC is undecidable, such a counterexample cannot exist.
I informally define ‘concrete’ statements are ones which are about ‘constructive’ objects. Thus any counterexample to a ‘concrete’ statement would be constructive, and hence undecidability of a ‘concrete’ statement would ‘say’ that no such counterexample exists (otherwise we would be able to construct it!).
I am left with the question: should we expect ‘concrete’ statements, such as CC, to be decidable? Maybe there is a formal answer to this question, by formalising what a ‘concrete’ statement is, and showing that such statements are decidable. Though, to me, this seems unlikely to be possible, by Gödel.
Moreover Gödel’s incompleteness theorem holds just under Peano Arithmetic, and so it seems to be saying there is a priori no difference between a statement about the natural numbers such as CC and a statement about cardinalities such as CH. But my intuition cannot help but telling me these are of different nature. Am I wrong to assume this?
A more general version of my question is, why as mathematicians do we expect the questions we are working on to be decidable? We get a sense that a proof or counterexample exists, and it seems that outside of certain ‘non-constructive’ parts of mathematics, we never run into such issues.
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u/viking_ Logic 5d ago
One of the key notions in mathematical logic is that statements like "truth" and "proof" are relative to a given model. The former is not at all unusual: a statement like "all polynomials have a root" is true in C but not in R. The latter is a little more unusual, but the basic idea is familiar to most mathematicians. For example, since the intermediate value theorem is not true in Q, we know that proving it for R requires some fact about R that is not true about Q. If the only facts we had about R were also true about Q, then we know that we can't prove IVT (or that our system is inconsistent).
https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem is another example, a statement that is true in integers, but not provable in Peano arithmetic.
Similarly, proof by "X is independent of ZFC, therefore it is true in ZFC because no counterexamples exist" (Riemann is the standard value of X here) cannot, itself, be written down in ZFC. X is provable in some other system, but not ZFC, just like the strengthened finite Ramsey theorem is not provable in PA.