Relation of Gödel's incompleteness theorems and Karl Popper falsification

Question

Falsifiability is considered a positive (and often essential) quality of a hypothesis because it means that the hypothesis is testable by empirical experiment and thus conforms to the standards of scientific method.

The citation about Karl Popper's falsifiability.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

And the Gödel's incompleteness theorems.

It's well known that Karl Popper's falsifiability principle can't be falsificated with itself.

So, can the falsifiability principle can be somehow extended with the Gödel's theorems to prove the theory is a true theory, maybe by introducing some isomorphism between natural numbers and the results of the physical, for example, theory?

Answer 1

If you defined an isomorphism between the natural numbers and some element of a physical theory, this would imply that there exist statements about the physical theory could not be proven or disproven within the theory. It certainly doesn't say that every statement in the theory is beyond falsification; and it proves nothing whatever about what might happen experimentally. The physical theory could still be wrong, if it conflicted with experiment.

Note that if the physical theory were disproven, this would not represent a disproof of number theory; it would only prove that the isomorphism between number theory and the physical theory doesn't necessarily expose any number-theoretic structure of reality. That is to say, the pre-image of the physical theory via the isomorphism would be a number-theoretic expression of an invalid physical theory. Not that a priority of number theory is to directly express physical physical structures, of course.

Answer 2

To prove that a "theory is a true theory" we cannot use the "falsifiability principle".

Popper's does not seem to answer Hume's question. The theory that the sun rises every day may have survived falsification yesterday, but is this a reason for believing that it will survive falsification tomorrow? That was what Hume wanted to know.

Popper retorts that induction is not justifiable. That a theory has been corroborated in the past "says nothing whatever about future performance." Popper wants to say that it is possible to avoid assuming that the future will, or probably will, be like the past, and this is why he has claimed to have solved the problem of induction. We do not have to make the assumption, he tells us, if we proceed by formulating conjectures and attempting to falsify them.

He says that, as a basis for action, we should prefer "the best-tested theory." This can only mean the theory that has survived refutation in the past; but why, since Popper says that past corroboration has nothing to do with future performance, is it rational to prefer this? Fundamental is the question how, even in theory, we can possibly prefer one hypothesis to another, or take one as a nearer approximation to truth than the other, if past corroboration has no implications for the future. Without the inductive assumption, the fact that a theory was refuted yesterday is quite irrelevant to its truth-status today. So demising the inductive assumption makes nonsense of Popper's own theory of the growth of scientific knowledge.

Theories or hypotheses can only be subjected to empirical testing in groups or collections, never in isolation. The idea here is that a single scientific hypothesis does not by itself carry any implications about what we should expect to observe in nature; rather, we can derive empirical consequences from an hypothesis only when it is conjoined with many other beliefs and hypotheses, including background assumptions about the world, beliefs about how measuring instruments operate, further hypotheses about the interactions between objects in the original hypothesis' field of study and the surrounding environment, etc. For this reason when an empirical prediction turns out to be falsified, we do not know whether the fault lies with the hypothesis we originally sought to test or with one of the many other beliefs and hypotheses that were also needed and used to generate the failed prediction. It forms a criticism of methodological falsificationism.

Holist underdetermination ensures there cannot be any such thing as a “crucial experiment”: a single experiment whose outcome is predicted differently by two competing theories and which therefore serves to definitively confirm one and refute the other. Our response to the experimental or observational falsification of a theory is always underdetermined in this way. When the world does not live up to our theory-grounded expectations, we must give up something, but because no hypothesis is ever tested in isolation, no experiment ever tells us precisely which belief it is that we must revise or give up as mistaken. All of the beliefs we hold at any given time are linked in an interconnected web, which encounters our sensory experience only at its periphery.

references: Peter Singer, Quine, Pierre Duhem

Answer 3

Karl Popper is wrong. Falsifiability is not synonymous with testability or verifiability. Consider the following diagram:

In this very simple table, we see that the true state of a claim is verifiable if the true state matches a condition which is determinable. It does not matter if a true claim has a proof that is unfalsifiable; a "false" outcome is impossible regardless. If the true state is "true", it only needs to be "truifiable", which means it is possible to verify in finite time. Only untruifiable true states and unfalsifiable false states are unverifiable. All other states are verifiable. [Seth Stewart, original work, 2019]

Popper either cleverly or ignorantly downplays discussion of falsifiability's complementary attribute of "truifiability", presuming that the claim is false (he sees or wants us to see only the upper row of this table). A full solution to the question of verifiability takes both falsifiability and truifiability into account. This more general problem of verifiability is reducible to the problem of decidability in Computer Science: The challenge is whether we can write a program that determines whether a given (unknown) program will ever stop. If it stops on all inputs, it is called a decidable program. The problem of deciding whether any arbitrary program is decidable is an undecidable problem, because some programs can run forever. A program is nominally undecidable if there exists at least one input that will cause it never to halt. However, some problems are semi-decidable: "A problem is called partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set. This means that there exists an algorithm that halts eventually when the answer is yes but may run for ever if the answer is no. Partially decidable problems and any other problems that are not decidable are called undecidable." [Emphasis added]. Popper's glaring error is in conflating semi-decidability with non-semi-decidable undecidability.

One could argue that problems that are semi-decidable (either unfalsifiable or untruifiable) are not worth pursuing, because we have no guarantee of ever finding a result. To the experimenter, there is no way a priori to distinguish an unfalsifiable false claim from an unfalsifiable true claim until if or when the evidence has been encountered that proves the claim is true (which will never happen if it is false). This is the halting problem. However, the fact that we have not yet encountered a positive proof of a claim does not mean that it is false, even when the claim is unfalsifiable. There may exist unfalsifiable claims which are true, and those which are true and truifiable are verifiable. Such truths and their proofs are in the realm of patience. An example is the Collatz conjecture: If a single counterexample exists, all one has to do is have the patience to test enough numbers to find that counterexample. So in essence Popper is arguing that we should restrict our search solely to claims that are decidable, while misrepresenting all claims as either fully decidable or fully undecidable.

Understanding Popper's idea for what it is, the notion that falsifiability equals verifiability or provability is in general false, and we are left simply with the Decidability problem and Gödel's incompleteness theorems for certain axiomatic systems as our set of criteria for provability within a logical system, so there is no further reconciliation to be made. Popper's error is an example of the fallacy of presuming the negative result as a precondition to attempts at verification, and a fallacy of impatience.