Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

Question

I heard that the P vs NP question is equivalent to a $\Pi_2^0$ sentence, and that the Riemann hypothesis is equivalent to a $\Pi_1^0$ sentence. Many known mathematical theorems state that some specific $\Pi_2^0$ sentence is actually true, take Goodstein's theorem for example. The related Paris-Harrington theorem states that the strengthened finite Ramsey theorem (another $\Pi_2^0$ sentence) implies the consistency of Peano arithmetic, which is equivalent to a $\Pi_1^0$ sentence.

I wonder whether there could be a sort of progress towards a challenging conjecture by showing it equivalent to a $\Pi_1^0$ sentence. The examples above indicate that such equivalence might only hold under additional assumptions, like that $\epsilon_0$ is well ordered.

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The following comment made me wonder about the meaning of such equivalences:

"Equivalent" in what sense? Since the model existence theorem is true, it is equivalent to lots of $\Pi_1^0$ sentences - for example, the sentence asserting that the Turing machine which on any input halts automatically, halts on input zero. Presumably, you mean equivalent over some base theory - but then you need to specify that theory! – Noah Schweber Mar 12 at 5:39

The corresponding question used the following definition:

DEFINITION 4.4. A $\Pi_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the empty input tape. A $\Pi_2^0$ sentence is a sentence asserting that some given Turing machine halts at every input tape.

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Edit This is a serious question about provable $\Pi_2^0$ sentences. The rather random examples of $\Pi_2^0$ sentences (P vs. NP, Goodstein's theorem, and the strengthened finite Ramsey theorem) given in the question are enormous statements, and I feel the only way how such enormous statements can be true is by being provable. But being provable of course means being provable in some specific formal theory, and the consistency of such a theory is equivalent to a $\Pi_1^0$ sentence. But the examples also show a connection to ordinal numbers, and ordinal numbers are used to measure the proof strength of certain formal theories. I had hoped that somebody would be able to explain whether there is a connection between ordinal numbers and $\Pi_k^0$ sentences.

The answers and comments focused on the meaning of "equivalent" in this context. I would have no problem with interpreting this as equivalent over PA (or rather ACA₀), if a base theory must be specified. Also PRA would be fine for me, if it helps (or simplifies) explaining the connection to ordinal numbers. The intended informal meaning of "equivalent" in this context is probably closer related to the existence of Turing reductions between the two statements, but I admit that one also needs to prove that a given reduction indeed works as claimed, and hence a base theory is still needed, even for this informal definition of "equivalence". But the base theory should not be used to discuss away the existence of Turing machines. They are part of the meta-system, and their (idealized) existence is part of the explicit ontological commitments.

Answer 1

The problem is with your notion of logic and truth. Before you even can talk about a sentence, you need to specify your language, and before you can talk about whether it is provable or disprovable (or neither), you need to specify the formal system (which must use the same language or larger), and before you can talk about truth, you need to have a model. So simply put all your statements are not well-defined, and that is precisely what is causing you not to understand.

It so happens that all your examples can be expressed as sentences over PA, at least in some sense. What do I mean? Well, for example what does it mean to say that a formal system $T$ is consistent? To even state that, you need to work in a meta-system, because you need to be able to form the collection of all sentences generated by the inference rules of first-order logic from the axioms of $T$ (in some set theory), or at least to state the provability relation (in other meta-systems). A common choice for the meta-system is ZFC or some type theory. This cannot be done directly in PA, and the best you can do is to construct a sentence over PA that you can prove in the meta-system would be true in the standard model $\mathbb{N}$ if and only if PA is consistent (as viewed in the meta-system). We usually denote this sentence by $Con(T)$. Remember, this notion of $Con$ only exists in the meta-system, not $T$! But $Con(T)$ is a sentence over PA.

Let's now check whether you understand the above. We (in the meta-system) shall assume that PA is consistent. So is $Con(PA)$ true? Try answering before looking!

This is a quite meaningless question. You need to specify a model to talk about truth. $Con(PA)$ would be true in the standard model of PA but may not be true in some other models of PA. Our assumption that PA is consistent does not tell us anything directly about the truth value of $Con(PA)$ in all models of PA.

Godel showed essentially that any formal system $T$ that can prove all the sentences that PA can prove (with suitable translation if necessary) cannot prove $Con(T)$ unless $T$ is inconsistent! Note that if $T$ is inconsistent then it proves every sentence, including its negation, and so is useless. If we apply this theorem to PA, we conclude that PA does not prove $Con(PA)$, and hence by the completeness theorem (note that this notion of completeness is totally not the one in the incompleteness theorems) $Con(PA)$ is false in some model of PA.

So although we say that the consistency of a formal system $T$ is expressed by a $Π_1$-sentence over $PA$, always remember that although $T$ is either consistent or not consistent in the meta-system, $Con(T)$ may not have a fixed truth value in the models of PA! There is however a 'consolation' prize; if $T$ is inconsistent then $\neg Con(T)$ would be a $Σ_1$-sentence over $PA$ that is true in the standard model, and hence there is a natural number that witnesses it, which is expressed by a term over PA, and PA is strong enough to prove every true quantifier-free sentence!

Similarly if Goldbach's conjecture (a $Π_1$-sentence over PA) is false (in the standard model of PA), it can be disproven in PA (you can even take the counter-example and run a computer program to verify that it is not the sum of two primes). So if Goldbach's conjecture is independent (neither provable nor disprovable) over PA, then it must be true. Notice that people don't usually mention "the standard model" here simply because this conjecture is about the natural numbers in ZFC, which is the standard model of PA.

Now it should be clear what your examples mean. We say that a conjecture $C$ is equivalent to a sentence $C'$ over PA if and only if $C$ is equivalent to $\mathbb{N} \vDash C'$ (which says "$C'$ is true in $\mathbb{N}$). You may think that this is silly if $C$ is itself a statement about $\mathbb{N}$, since there appears to be no difference between the two notions. But as $Con(PA)$ shows, these two notions can be very different, and this is because $C'$ must be a sentence in the language of PA, so it cannot talk about the domain of quantification itself.

Just for illustration, Goodstein's theorem is proven in ZFC and equivalent (in the above sense) to a sentence that is independent of PA. Now you might object that this is useless, after all if say a conjecture $C$ is stated in ZFC then I can define $C'$ to be $\top$ if $C$ is true (in ZFC) and $\bot$ otherwise. $C'$ would be well-defined in ZFC and indeed $C$ would be equivalent to $C'$ in the above sense. But this is not constructive, in that we want an explicit $C'$ written down, not one that is defined by invoking the law of excluded middle! That is what makes these results interesting, because they actually wrote down (or showed how to write down) such equivalent sentences. Better still, there is sometimes an algorithm to generate these for a whole class of statements. For example there is an program that given any input $V$ that is a program that checks validity of a proof in some formal system $T$ will output a sentence over $PA$ that has the above properties of $Con(T)$.

It is unlikely that showing that something is equivalent to an explicit $Π_1$-sentence over PA will help much to prove or disprove it, since the non-existence of a solution to any given diophantine equation is also a $Π_1$-sentence, and diophantine equations are incredible monsters! (See Fermat's last theorem. and note that exponentiation can be encoded in PA.)

Also, you cannot even refer to the ordinal $ε_0$ without it already being well-ordered, by definition of ordinal. Most of the time people work in ZFC, in which case such ordinals already exist. The interesting question is what ordinals can be proven to exist if you restrict the comprehension axiom (in ZFC) but you add some axiom corresponding to some theorem, such as "PA is consistent" or "Goodstein's theorem is true".