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I believe there are at least several proofs of Gödel's incompleteness theorem. Nagel and Newman wrote a book (1958) that presents one in particular.

But considering the theorem itself exposes inherent limitations of every formal axiomatic system, how can we paradoxically use such systems to prove the theorem of incompleteness?

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    $\begingroup$ Seeing how your pot is not suitable for making fried eggs, how can any food be eaten? $\endgroup$ – Asaf Karagila Oct 11 '18 at 14:35
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    $\begingroup$ The incompleteness theorem states that there are claims that are impossible to prove, not that there are no claims that can be proven. $\endgroup$ – Vasya Oct 11 '18 at 14:56
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    $\begingroup$ To the OP: Why should knowing that there are questions a system can't answer suggest that we should worry that it's wrong about the questions it does answer? They are simply different issues. $\endgroup$ – Noah Schweber Oct 11 '18 at 21:39
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    $\begingroup$ There is no paradox here. You are misunderstanding the technical meaning of the word "complete" in this context. A formal system can be sound (i.e., incapable of proving false statements) without being complete (i..e., capable of proving all true statements). $\endgroup$ – Rob Arthan Oct 11 '18 at 22:37
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    $\begingroup$ @JoséCarlosSantos: "at least several" made me smile too $\ddot{\smile}$. $\endgroup$ – Rob Arthan Oct 11 '18 at 22:42
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The comments have already addressed this, but let me try to move this question off the "unanswered" list.

You write:

But considering the theorem itself exposes inherent limitations of every formal axiomatic system, how can we paradoxically use such systems to prove the theorem of incompleteness?

I think you're treating all "inherent limitations" as being the same sort of thing, but they're not. There's nothing paradoxical going on here: the limits imposed by the incompleteness theorem do not say that we cannot trust the things a theory does prove, merely that we can't expect the theory to prove or disprove everything. That is, the existence of gaps has nothing to do with the existence of errors.

So there's nothing paradoxical going on here. Or rather, if you think there is, the onus is on you to draw the connection.


Of course, I'm not disagreeing that Godel's theorem - not its precise content, but rather then general realization that logic is weird - might reasonably cause us to be more skeptical of the axiomatic theories we use in general. So perhaps now, whenever we have a proof of a statement $p$ in a theory $T$ which is at all complicated, we might assert "$p$ is true assuming $T$ is correct" rather than "$p$ is true." But this resulting skepticism is no more directed towards the incompleteness theorem than it is towards all other theorems. There's no paradox here.

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