Incompleteness theorem
Chapter 1 - Worldview
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Welcome to the Incompleteness theorem page
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As the foremost logician of the 20th century, Kurt Gödel is well known for his incompleteness theorems and contributions to set theory, the publications of which changed the course of mathematics, logic and computer science. When he was awarded the Albert Einstein Prize to recognise these achievements in 1951, the mathematician John von Neumann gave a speech describing Gödel's achievements in logic and mathematics as so momentous that they will remain visible far in space and time.
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Core ideas
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A theorem postulated by Kurt Gödel
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| The solution of a final problem in a closed system can never be found in the system itself, but only in a higher order system. |
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Suppose an axiomatic system is at least as strong as the standard axiomatisation of arithmetic. In that case, it cannot prove its consistency (unless it is inconsistent, in which case it can prove anything, including its consistency). We must show that our basic principles are accurate and our basic inference rules are valid.
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Practical examples
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Self-steering teams
The theorem explains the root cause of all the problems we experience with self-steering teams. They are a CAS (Complex Adaptive System), aka a system. If they can't or, worse, are not allowed to create a META system, they cannot solve the problems occurring in their team. We can't expect a smooth-running team if we only care about what they DO within the group.
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Strategy
Strategy is on every level of your organisation, especially on the highest level, no more or no less than a META system (beliefs) about interests, values and goals, able to create solutions for the operational problems occurring or the realisation of your dreams.
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Deep dive
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On being certain
One reason the theorem is philosophically essential is that it can be used to argue that we can never be absolutely certain about the truth of our theories. It tells us that there is a precise sense in which no finite computer could possibly encapsulate the whole of arithmetical (or communication) truth and has far-reaching philosophical consequences.
Sometimes, our best theories of the natural world are mistaken. The geocentric theory of the universe turned out to be mistaken. So did Newtonian physics. We now have different theories of the natural world. But how can we be sure they will be correct, too? Certainty would appear elusive.
I used to think that even if certainty remained forever elusive in our theories of the natural world, mathematical theories were different. When it comes to mathematics, we really can aspire to absolute certainty. Alas, my youthful self was mistaken. It is a consequence of Gödel's Theorem that absolute certainty is no more possible in our mathematical theories than in our theories of the natural world.
When Copernicus proposed the heliocentric theory of the universe, he defended his hypothesis by arguing that it was simpler than rival theories. But, of course, the fact that a theory is simple does not guarantee that the theory is true. Copernicus's theory is false as he proposed it since we now know that the planetary orbits are not circular, as he claimed, but rather elliptical. In contrast, when Euclid proposed that there are infinitely many prime numbers, he justified his hypothesis with proof from basic principles. Since Euclid's proof is valid, it guarantees that if his basic principles are correct, his conclusion is also accurate.
It's natural to assume that Euclid, unlike Copernicus, established his result conclusively. However, there's a crucial point to consider. A mathematical proof is always conditional. It demonstrates the validity of its conclusion, provided that the basic principles on which the proof is based are true. This means that we must not only prove the validity of our basic principles but also ensure the soundness of our basic inference rules.
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What science can tell you
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| Stanford Encyclopedia of Philosophy |
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| Raatikainen, Panu, "Gödel’s Incompleteness Theorems", The Stanford Encyclopedia of Philosophy (Spring 2022 Edition), Edward N. Zalta (ed.), |
| https://plato.stanford.edu/entries/goedel-incompleteness/ |
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