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Gödel’s Incompletness Theorem By Njegos Nincic

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Overview Set theory: Background, History Naïve Set Theory Axiomatic Set Theory Icompleteness Theorem Godel’s Proof Simplified Explanation Consequences Summary

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Set Theory Underlaying foudations of all mathematics All proofs must be compliant with the axioms of the theory Analogy: Assembly : Visual Basic :: Set Theory : Calculus Language of logic symbols x[Ø x y(y x → {y,{y}} x)]

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Naïve Set Theory Always implied Formalized by Cantor in the 19 th century Set is any collection of objects Allows construction of “Set of All Sets” Georg Cantor

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Russell’s Paradox S = { x : x x } S S S S If the barber shaves all those who do not themselves shave, does he shave himself? Responded to by introduction of rigorous axioms in set theory (Zermelo-Fraenkel axioms) Bertrand Russell

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ZFC Axioms: Extensionality: x y[ z(z x ≡ z y) → x=y] Null Set: x¬ y(y x) Pairs: x y z w(w z ≡ w=x w=y) Unions: x y z[z y ≡ w(w x & z w)] Power Set: x y z[z y ≡ w(w z → w x)] Infinity: x[Ø x y(y x → {y,{y}} x)] Regularity: x[x≠Ø → y(y x & z(z x → ¬(z y)))] Replacement Schema: z 1v … z k [ x !yφ(x,y,z ) → u v r(r v ≡ s(s u & φ x,y [s,r,z]))] Separation Schema: z 1 … z k [ u v r(r v ≡ r u & ψ x, [r, z])] Axiom of Choice: y z w[(z w & w x ) v u[ t((u w & w t) & (u t & t y) u = v)] (optional) “Set of all sets” impossible to construct from axioms

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Incompletness Theorem After axiomatization, formal mathematics revitalized by Hilbert’s Program to prove consistency of a finite comlete set of axioms In 1930, Incompletness Theorem introduced and proved by Kurt Goedel States that: “In any consistent formalization of mathematics one can construct a statement that can neither be proved nor disproved within that system”

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Incompletness Theorem Formalization states: “No consistent system can be used to prove its own consistency” Hilbert’s program stopped No mathematical system can be complete and consistent at the same time Kurt Godel

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Proof Compute (encode) the Godel number for a given statement by using the following table: SymbolNumberSymbolNumber 01X9 S 11 *4 12 =5 13 (6 14 )7 15,8 16 [ example: x¬ y(y x) ]

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Proof … by raising the corresponding prime from the list of primes (2, 3, 5, 7, 11, …) – where the first symbol gets the first prime, the second symbol the second prime and so on – to the power of the number we get from the table for that symbol and multiplying everything together: x¬ s(s x) = 2^13 3^9 5^11 7^13 11^2 13^6 17^2 19^16 23^9 29^7

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Proof We can assign Godel numbers to groups of statements (GN “Godel number”): Statement 1 (GN1) Statement 2 (GN2) Statement 3 (GN3) Let R “proves” relationship: R(s, x) “x proves s” R(GN3, GN2) = “Statement with Godel number GN2 proves statement with Godel number GN3” } = 2 GN1 3 GN2 5 GN3

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Proof We can construct the following statement: x, ¬R(v,x) “no proposition of type v can be proved.” Calculate the Godel number for the statement and feed it back into the statement: x, ¬R(GN(above statement),x) “no proposition that says 'no proposition of type v can be proved' can be proved.” “this proposition cannot be proved.”

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Proof – informal account Suppose that a Universal Truth Computer (UTC) is built. It claims that it can prove (true or false) any statement that can possibly be made. Name the system on which the UTC is built S. Let P = “Machine based on system S will never prove statement R” Let R = P: “Machine based on system S will never prove this statement” If the UTC proves the statement true, that means that it will never prove it; if it does not prove it, it must prove it. Therefore we, outside of system S, can see that the above statement is indeed true (the UTC will never prove it) but this can not be shown within the system.

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Undecidable statements Continuum hypothesis: The Halting Problem: “Does a given algorithm, when executed on an initial input, ever halt?” Axiom of choice: y z w[(z w w x ) v u[ t((u w w t) (u t t y) u = v)]

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Summary All math theorems must be provable within set theory Problems in set theory: one fixed (Russell’s paradox), one can never be fixed Naïve set theory : inconsistent ZFC : incomplete Godel’s theorem holds for all consistent systems Many theorems proved undecidable based on Godel’s theorem

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References Godel’s Theorem: Axiom of Choice: Godel and paradoxes: 100 years of Russel’s paradox: Metamath: Rudy Rucker: “Infinity and Mind” Simon Singh: “Fermat’s Enigma”

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