Is it always possible to determine the "truth" of a proposition from an independence result?

Question

Let me start off with a bit of background information. Despite (unfortunately) not having any formal education in metamathematics / logic / set theory, etc... due to never having attended schools where professors have had that sort of research interest, I am very interested in these topics.

I am also coming to this question from a bit of a unconventional place. I am not a mathematical platonist, and lean more towards the intuitionistic camp. As such, when I say "truth", I do not mean it in the sense most common amongst platonist set theorists.

Given those caveats, I will try to explain to the best of my ability what I am looking to understand.

Truth

To me, truth is equated with provability. However, if there does not exist a proof of something (i.e. it is not in principle possible to derive a proof of something), I consider it to be false.

Essentially, the notion of "mathematical proof" I find most compelling is provability + some metamathematical reasoning principles. For example, if we have a base formal system $T$, and we consider a statement $\phi$ that is independent from $T$, then I'd consider $\phi$ to be "false" from the perspective of $T$ if $T + \phi$ was $\omega$-inconsistent, or non-constructive (in the sense of this math stack exchange post).

I am not sure if there is anyone else that expresses a similar view, or if there may be philosophical problems with such a view, but that is probably out of scope for the math stack exchange.

Examples

This is something I've been trying to formalize for a long time, but it's been difficult to see if there's any existing literature on the topic or not, given my idiosyncratic philosophy and background.

It seems to me in an intuitive sense, that given a fixed formal theory T we've already committed to working with (for instance, CZF, HA, or some form of dependent type theory with universes, since the previously mentioned principle of being constructive already rules out classical theories like PA and ZF), there should be some sense in which a formally independent statement $\phi$ of $T$ can still be determined to be "true" or "false".

For example, let's take a $\Sigma^0_1$ statement in $T$ such as $\phi = \exists (n : \mathbb{N}) p(n)$. If $p$ is independent from $T$, then it follows that there does not exist a natural number $n_0$ such that we can prove $p(n_0)$ holds in $T$. Intuitively then, we have thus shown that $\neg \exists (n : \mathbb{N}) p(n)$ from a "metamathematical perspective". Or a bit more formally, that out of $T + p$ and $T + \neg p$, only $T + \neg p$ is "true" from my "metamathematical proof" perspective, since $T + p` would not be a "constructive" theory.

I believe we can show something similar for a $\Pi^0_1$ statement $\phi = \forall (n : \mathbb{N}) p(n)$ following this principle. If this statement is independent from $T$ (and we assume consistency of all of the theories we are working with), then it must be the case that there does not exist a natural $n_0$ such that $\neg p(n_0)$ is provable in $T$, otherwise out theory would be inconsistent. It follows then, that out of the alternatives $\phi$ and $\neg\phi$, $\phi$ must be the "true" one.

Question

Are there any statements that we cannot apply metamathematical reasoning to in the sense derived above to derive from any statement of a constructive theory which one of the alternatives $T + \phi$ or $T + \neg\phi$ is "constructively true"?

In other words, is it possible for a statement to be "absolutely independent" in the sense of "constructive truth"?

Perhaps a bit more concretely: Does there exist a constructive theory $T$, with statement $\phi$ independent of $T$, such that both $T + \phi$ and $T + \neg\phi$ are constructively acceptable theories? (i.e. constructive in the sense stated previously, as well as perhaps ensuring that the new theories satisfy the disjuction and existence properties)

Answer 1

I believe the answer to your question is that there are "absolutely independent" statements by your criteria. Note that your examples have only considered $Π_1$ and $Σ_1$ statements, which are particularly simple. And indeed, there are arguments that PA/HA will prove every 'true' $Σ_1$ statement about the 'standard' natural numbers, so that a $Σ_1$ statement can only be independent if it is 'false' in that sense. However , this doesn't extend to higher complexity statements or more complex theories.

Perhaps it's instructive to look at the way the argument breaks down. A key piece is that HA is (syntactically) complete for $Δ_0$ statements. Such statements contain only quantifiers that are ultimately bounded by numerals, so there are finitely many cases to check, and HA will contain a corresponding proof for whatever the check decides. So, at least for $Δ_0$ statements, HA both proves every true statement and refutes every false statement.

Then a $Σ_1$ statement contains a prefix of existentials, so if it holds for some instantiation to numerals, which is a $Δ_0$ statement, HA will prove that, and thus the $Σ_1$ statement. Similarly I suppose, if some instantiation of a $Π_1$ statement is false it will be refuted, and so will the $Π_1$ statement.

However, HA is not complete for $Σ_1$ or $Π_1$ statements. For instance, the standard inconsistency statement is $Σ_1$, and it is (hopefully) independent. So, we aren't in a comparable situation at $Δ_1$. We only know that true $Σ_1$ statement are proved, but not that false ones are refuted, and a complementary fact for $Π_1$ statements (and if you union them to get $Δ_1$, neither holds).

This means the above argument doesn't extend to $Π_2$ or $Σ_2$ statements. For HA to 'prove every true $Σ_2$ statement' (by the same argument), it would have to be the case that it also 'proves every true $Π_1$ instantiation.' But we only know that false $Π_1$ statements are disproved. Also, in something like a set theory, the bounded quantifiers in a $Δ_0$ sentence can include all of arithmetic (because the natural numbers are a particular set). So there is no longer a nice, finite situation to bootstrap from.

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Now, from what you've written, it seems to me like you expect that we can fix this by adding to our theory based on the above reasoning. E.G. we add axioms for all the false $Σ_1$ and true $Π_1$ statements. Then we are complete for $Δ_1$ and can proceed to $Δ_2$ and so on. But, this won't work in a few ways.

For one, it isn't really possible to 'normalize' statements into $Π_n/Σ_n$ forms constructively. So at the very least you need to extend the notions to non-prenex forms, and probably have proper $Δ_n$ formulas which consist of e.g. a disjunction of a $Π_1$ and $Σ_1$ formula. Perhaps this is not so large a problem, because you are intending to decide the generators for these formulae, and the $Δ_n$ formulas would be decidable by induction (you hope).

Two, even if this all worked, I believe it would result in a very poorly behaved theory—something like "true arithmetic." You won't be able to figure out what all the axioms even are systematically, because they can only be answered by infinite searches, or provably impossible decision problems.

But finally, the expected results contradict some known constructive principles. Take for instance the Halting problem mentioned in the comments. This is the (extended¹) $Π_2$ statement:

$$∀M, I. (∃T. \mathsf{H}(M,I,T)) ∨ (∀ T. ¬ \mathsf{H}(M,I,T))$$

where $M$ ranges over 'machines', $I$ ranges over 'inputs', $T$ ranges over 'traces' (all encoded as numbers in arithmetic), and $\mathsf{H}$ expresses that running $M$ with input $I$ halts with trace $T$.

Now, if we follow the pattern from the first section, we would expect that either this statement is already refuted, or we can add it "because it's true." But this is not going to work out. To see this, we can consider the (Turing machine) realizability interpretation of constructive mathematics. This interprets mathematical formulas as statements about machines in classical computability to endow them with more constructive content. And in it:

1. The $Π_2$ formula above is false. For it to be true, there would need to be a Turing machine that decides the halting problem for Turing machines, which is well known to be impossible. So, as you have ascertained, this is not a constructive principle.
2. Every instantiation of the formula to numerals is true. This is because every particular machine classically does or does not halt, and so there is some machine that just reports the right information for each case. It is only when the statement is quantified that the realizability interpretation gains its force.

So, the standard realizability interpretation is actually '$ω$-inconsistent' in this sense. I'd understand if you don't like that, because I too think it is a fundamental flaw with realizability/classical computability (#2 above is nonsense). However, rejecting this flaw is of no help, because it is necessarily the case that no instantiation of the formula can be refuted. Each instantiation is of the form:

$$P ∨ ¬ P$$

where $P = ∃T. \mathsf{H}(M_0,I_0,T)$. And $¬¬(P ∨ ¬P)$ is a constructive theorem. Here it is equivalent to the statement that the concrete case cannot both not halt and not diverge. So, we cannot have a concrete case that is refuted, in order to refute the $Π_2$ statement, because all the instantiations are irrefutable.

There are other statements where both the affirmation and denial are considered constructive (by different schools). For instance, Markov's principle is true in the above realizability interpretation. Even from an intuitive but more constructive perspective, Markov's principle is the plausible idea that if we can decide $P(n)$, and we have shown that $P$ cannot fail for all $n$, then an unbounded search will eventually find an $n$ for which $P$ holds. However, there are other constructive perspectives from which this explanation does not make sense, because $P$ is not necessarily determined just by a self-contained algorithm. So, in that setting, the denial of Markov's principle is sensible. Both are considered constructive, but with different criteria for what sort of constructive witness justifies 'truth.'

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Incidentally, I would suggest you stop conflating constructive truth with proof. The tendency to do so is unfortunately endemic to discourse on constructive mathematics, but it doesn't make sense even there. It is better to use some other word like "witness" or "evidence" to talk about whatever objects justify a constructive truth, because they are generally not just proofs in a particular formal system. Conflating the former with the latter leads to the idea that formal systems are only for reasoning about their own syntactically definable objects, which is not the case, and rules out various useful perspectives.

[1]: I think if you wanted you could turn this into a more traditional $Π_2$ statement by having $T$ indicate the halting status instead, but I haven't thought about it in detail.