Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper:

S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. 401-446

John Steel makes the following remark in footnote 29:

There is the very remote possibility that one could show ZFC settles the questions without actually exhibiting the relevant ZFC-proofs. Goldbach's conjecture and the Riemann hypothesis are $\Pi_1^0$ statements, so one cannot prove them independent of ZFC without also proving them.

Could someone explain to me what kind of result/principle he is referring to here?

I'm guessing it's something of the form: ''For any $\Pi_1^0$ statement $\phi$ in PA (or, possibly, some weaker arithmetic) Ind(ZFC, $\phi$) iff some condition.'' The condition obviously can't be PA $\vdash \phi$ because, by soundness, that would imply that $\phi$ is not independent. But I can't think what other condition would justify Steel's remark.

Solution 1:

I think what John Steel (not Steele) was referring to is a lot simpler than forcing and $\Pi^1_2$ absoluteness. Consider Goldbach's conjecture. If it were false, so there's some even number $n>2$ that is not the sum of two primes, then this fact could be proved in ZFC, and indeed in far weaker systems. The proof would consist of just the calculations checking, for each $k<n$, that at least one of $k$ and $n-k$ is composite (and checking that $n$ is even and $>2$). This shows that, if the Goldbach conjecture is false then it's refutable in ZFC. Therefore, if I knew that the Goldbach conjecture was independent of ZFC --- meaning it's neither provable nor refutable --- then, since it's not refutable, I'd know that it can't be false.

Summary: If we prove that the Goldbach conjecture is independent of ZFC (or even prove half of that, namely that it can't be refuted in ZFC), then we'd have established that it's true.

As indicated above, the same applies with ZFC replaced by far weaker theories; Peano arithmetic is more than enough.

The same also applies to the Riemann Hypothesis, but the argument is more complicated. The point is that, just as in the case of the Goldbach conjecture, if the Riemann Hypothesis were false, then this could be proved in ZFC by just writing out an appropriate computation. Unlike the Goldbach case, though, it's not so obvious what the computation should be. I believe it amounts to computing a sufficiently accurate approximation to a contour integral, over a contour that goes around an off-the-crtical-line zero of the zeta function.

The property of "if false then refutable by a mere calculation" is basically what $\Pi^0_1$ means in the passage cited from Steel.

Solution 2:

To prove a $\Pi^0_1$ statement $\phi$ is independent of PA means we have to prove PA does not prove $\phi$ and that PA does not prove $\lnot \phi$. Now $\lnot \phi$ is $\Sigma^0_1$, and PA has the property that it can prove any true $\Sigma^0_1$ statement. So if PA cannot prove $\lnot \phi$, then $\lnot \phi$ is false, so $\phi$ is true. The same holds for ZFC, or any other theory whose arithmetical consequences include PA.

The key here is that the proof that PA doesn't prove $\phi$ would be in a system stronger than PA. So there is no contradiction; it's just that a sufficiently strong system, in proving $\phi$ is independent of PA, would also be able to prove $\phi$ is true.

I think the key to understanding the footnote is to look at the sentence to which it is attached:

However, we should recognize that until the problems are actually settled this will almost certainly never be more than an educated guess.²⁹

The two sentences in the footnote are separate comments on that phrase; the second sentence of the footnote is not an elaboration of the first sentence of the footnote.

Solution 3:

The reason is that $\Pi^1_2$ statement about the natural numbers is absolute between transitive models of ZFC (with the same ordinals, of course). That means that if $M\subseteq N$ are two "nice" models of ZFC which have the same ordinals, and $\varphi$ is a $\Pi^1_2$ statement about the natural numbers, then $\varphi$ is true in both of them, or false in both of them.

Our main method of proving independence of statements is by a technique called forcing. We start with a "nice" model of ZFC and we extend it by adding a new set, and the result is another "nice" model and they both have the same ordinals. For some statements we may assume that the first model satisfies them and show that the extended model does not, or vice versa. This way we prove a statement is independent of the axioms of ZFC, that is to say that ZFC cannot prove nor disprove it (if ZFC is consistent, of course).

So what we know is that relatively simple statements about the natural numbers cannot be forced to be true, or false. General statements, like the continuum hypothesis can be forced from one truth value to the other.

That been said, it might be another way of proving independence of statements, but currently we mostly know about forcing and compactness, and compactness is not always that useful.