Gödel's incompleteness theorems
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| - | :''see [[axiom]], [[philosophy]], [[propositional knowledge]]'' | + | '''Gödel's incompleteness theorems''' are two [[theorem]]s of [[mathematical logic]] that establish inherent limitations of all but the most trivial [[axiomatic system]]s for mathematics. The theorems, proven by [[Kurt Gödel]] in 1931, are important both in mathematical logic and in the [[philosophy of mathematics]]. The two results are widely interpreted as showing that [[Hilbert's program]] to find a complete and consistent set of [[axiom]]s for all of [[mathematics]] is impossible, thus giving a negative answer to [[Hilbert's second problem]]. |
| - | The [[Age of Reason]] sought to establish [[axiomatic philosophy]] and [[Enlightened absolutism|absolutism]] as [[foundation]]s for [[knowledge]] and [[stability]]. [[Epistemology]], in the writings of [[Michel de Montaigne]] and [[René Descartes]], was based on extreme skepticism and inquiry into the nature of "knowledge." The goal of a philosophy based on self-evident axioms reached its height with [[Baruch Spinoza|Baruch (Benedictus de) Spinoza]]'s ''[[Ethics (book)|Ethics]]'', which expounded a [[pantheistic]] view of the universe where God and Nature were one. This idea then became central to the Enlightenment from [[Newton]] through to [[Jefferson]]. | + | The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the [[natural number]]s. 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 shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself. |
| - | ==Issues== | + | |
| - | :"We're tired of [[tree]]s. We should stop believing in trees, [[root]]s, and radicles. They've made us suffer too much. All of arborescent culture is founded on them, from [[biology]] to [[linguistics]]" --''[[A Thousand Plateaus: Capitalism and Schizophrenia]]'' --[[Deleuze]] & [[Guattari]] | + | |
| - | Not every consistent body of propositions can be captured by a describable collection of axioms. Call a collection of axioms [[recursive set|recursive]] if a computer program can recognize whether a given proposition in the language is an axiom. [[Gödel's incompleteness theorems|Gödel's First Incompleteness Theorem]] then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. If the computer cannot recognize the axioms, the computer also will not be able to recognize whether a proof is valid. The result is that one will not know which propositions are theorems and the axiomatic method breaks down. An example of such a body of propositions is the theory of the [[natural number]]s. The Peano Axioms thus only partially axiomatize this theory. | + | |
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| - | == See also == | + | |
| - | * [[Vico]]'s [[114 axioms]] | + | |
| - | *[[Axiomatic system]] | + | |
| - | *[[Spinoza]]'s ''[[Ethics (book)|Ethics]]'' | + | |
| - | *[[Wittgenstein]]'s ''[[Tractatus Logico-Philosophicus]]'' | + | |
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Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. 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 shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.
