# Relation of Gödel's incompleteness theorems and Karl Popper falsification

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?

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I don't understand the question: What is the link between falsifiability an incompleteness which you are trying to establish? –  DBK Jan 14 '13 at 17:44
Is there a connection between theory that can explain everything is not a scientific theory due to Popper principle and it is not a formal system in Gödel sense? –  m0nhawk Jan 14 '13 at 18:01
Falsafiability and provability belong to two different orders. However, you may find this other link between Gödel and what it means for physical theory interesting: u.arizona.edu/~aversa/misc/… –  danielm Jan 14 '13 at 20:48

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.

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And what if we make isomorphism between (all possible true statements in number theory) <=> (physical reality) and (number theory itself) <=> (some general physical theory)? –  m0nhawk Jan 14 '13 at 15:41
How do you create an isomorphism between a formal theory, and reality, given that the latter is a sequence of phenomena and not a formal theory? –  Niel de Beaudrap Jan 14 '13 at 15:52

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

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