Can ‘concrete’ statements be undecidable?

[u/Lehatan]

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.

[u/SOTGO]

I would avoid relying too heavily on intuition here. Take the Busy Beaver function, from Wikipedia "the n-state busy beaver game consists of finding the longest-running or highest-scoring Turing machine which has n states and eventually halts." By definition the longest running Turing machine of a given size that halts must run for a finite amount of time (or else it would never halt), so the busy beaver function must be finite for any n. But it has been shown that for large enough values of n (I believe at least for n > 748) the value of the busy beaver function is undecidable, even though we know for a fact that it is some finite value.

[u/GoldenMuscleGod → u/SOTGO]

You have partial understanding but it is incomplete and I think your answer may mislead OP. The value of the busy beaver function is independent of an axiomatizable theory for sufficiently large inputs, where the size depends on the theory. It isn’t undecidable. Undecidable and Independent are not synonyms. It is impossible (in classical math) for any specific proposition to be undecidable - only decision problems can be undecidable. A single proposition can be independent of a theory but there is no such thing as being “independent” in general - it can only be independent of a theory, not of all theories. Independence is a binary relation.

There is a fact about the busy beaver function related to their intuition: there exists exactly one explictly written n for which BB(1000)=n is consistent with, say, ZFC (assuming ZFC is consistent). This is the “true” value of BB(1000). We can consistently add BB(1000)=/=n as an axiom to ZFC, this theory then proves BB(1000)=/=k for any numeral k. In a model of this theory the value of BB(1000) is a nonstandard element, not an alternative standard natural number.

Edit: actually I think to justify that claim I have to assume more than that ZFC is consistent - enough is to assume that an inaccessible cardinal exists.

[u/SOTGO → u/GoldenMuscleGod]

My apologies, I am not an expert in this area. I called it undecidable based on the title of a paper linked from the Wikipedia page, "The Undecidability of BB(748)" (link). Looking back the actual statement on the page is "it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy beaver," which you are correct is different than the claim I made

[u/GoldenMuscleGod → u/SOTGO]

To be fair the terms are sometimes used synonymously even though they shouldn’t be.

To clarify the difference: a decision problem can be thought of as a case where we have a class of questions and we want a “yes” or “no” answer on each of them. It is decidable if there exists an algorithm taking the question as input and outputting the answer. If not it is undecidable. Note that this definition does not mention any theory.

With this definition (in classical math, using the law of the excluded middle) a single question cannot be undecidable: its answer is either “yes” or “no” and so one of the two algorithms “ignore the input and say yes” and “ignore the input and say no” must give the correct answer. Of course this doesn’t help find the answer, which is why the classical notion of decidability can be counterintuitive (this is one area where constructive theories better capture intuition).

A sentence is “independent” of a theory if that theory neither proves it nor its negation. This depends on the theory. No sentence is independent of every theory: consider the theory where we take that sentence as an axiom. Of course this does not help us to determine whether the sentence is true with the meaning we intend for it. This is similar to the decidability issue for single propositions.

The ideas are distinct but related: suppose we have a binary relation where the problem of whether it holds is undecidable. For example, “does BB(n)=m?” Where n and m are variables that must be set for us to have a specific question. Then that means for any sound axiomatizable theory there must be some specific case of it that is independent of the theory (such as “does BB(1000)=[some explicitly written enormous number]?)

However which of these statements is independent of the theory will depend on the theory, and there is no statement that is independent from all sound and axiomatizable theories.