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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 capable of doing arithmetic. 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 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, a corollary of the first, shows that such a system cannot demonstrate its own consistency

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- 1. Gödel’s Theorem:An Incomplete Guide to Its Use and Abuse by Torkel Franzén Summary by Xavier Noria August 2006 fxn@hashref.com
- 2. This unique exposition of Kurt Gödel’s stunning incompletenesstheorems for a general audience manages to do what no other hasaccomplished: explain clearly and thoroughly just what the theoremsreally say and imply and correct their diverse misapplications tophilosophy, psychology, physics, theology, post-modernist criticismand what have you. Solomon Feferman
- 3. Introduction
- 4. Gödel’s incompleteness theorem is probably theone that has arose more interest among non-mathematicians, it was published in 1931 byGödel, and strengthened in 1936 by RosserThat’s a result of logic about the consistency andcompleteness of formal systems, those terms have aprecise technical meaning and so care is neededwhen interpreting the theorem outside logic
- 5. The implications of this theorem are oftenoverstated, for example: “turned not onlymathematics, but also the whole world ofscience on its head”Gödel’s incompleteness theorem brought norevolution whatsoever to mathematics, ingeneral it plays no role in the work ofworking mathematiciansHowever, the theorem raises a number ofphilosophical questions concerning thenature of logic and mathematics
- 6. The First Incompleteness Theorem
- 7. TerminologyGiven a formal language L, a formal system S in L isa set of axioms and syntactical rules of reasoning (alsocalled inference rules) expressed in L.A theorem of S is any statement T of S which isobtainable by a series of applications of theinference rules of S starting from the axioms of S,such a sequence is called a proof of T in SThose concepts are purely symbolic, nothing is saidabout any meaning or soundness
- 8. A formal system S is said to be consistent if it hasno sentence A such that A and ¬A are boththeorems of SA sentence A of S is said to be undecidable in S ifneither A nor ¬A are theorems of SA formal system is said to be complete if it has noundecidable sentences, otherwise it is incompleteIf A is a sentence of S, we denote by S+A theformal system obtained by adding A as a newaxiom to the ones of S, and with the same rules
- 9. ObservationsBy ex falso quodlibet an inconsistent theory has noundecidable statements, everything is a theoremA is probable in S iff S+¬A is inconsistentA is undecidable in S iff both S+A and S+¬A areconsistent
- 10. First Incompleteness TheoremFirst incompleteness theorem (Gödel-Rosser). Any consistent formal system S withinwhich a certain amount of elementary arithmeticcan be carried out is incomplete with regard tostatements of elementary arithmetic
- 11. Goldbach-like StatementsGoldbach’s conjecture states that every evennumber greater than 2 is the sum of two primes,it is unknown whether it holdsFor any given even number greater than 2 we cancheck that property applying an algorithm, that’sa computable propertyWe call Goldbach-like those statements of the form“Every natural number has property P”, where Pis a computable property
- 12. Goldbach-like statements have an importantcharacteristic, if they are false they aredisprovable because there’s an algorithm thatﬁnds a counterexampleHowever, we do not know in advance whichmathematical methods will give a proof of atrue Goldbach-like statement, in case there isone
- 13. The Twin Prime ConjectureThe Twin Prime conjecture states there areinﬁnitely many primes p such that p+2 is alsoprime, it is unknown whether it holdsThat is not a Goldbach-like statement: given anatural n the procedure that systematically looksfor pairs p, p+2 with n ≤ p will not terminate ifthere’s none
- 14. An Important PropertyIf a Goldbach-like sentence in a consistent systemS incorporating some basic arithmetic is provableor undecidable in S then it is necessarily true, forif it was false a proof of its negation would exist,and so S wouldn’t be consistentIn the traditional proof of the incompletenesstheorem a Goldbach-like and undecidablesentence is shown
- 15. Hilbert’s non ignorabimusGödel’s Incompleteness theorem does not refuteHilbert’s view that mathematicians can ﬁndsolutions to any mathematical problem by meansof pure reasonInstead, it establishes that Hilbert’s optimismcannot be justiﬁed by exhibiting any single formalsystem within which all mathematical problemsare solvable
- 16. Unprovable TruthsIt is often said that Gödel demonstrated that therea truths that cannot be proved. This is incorrect,“provability” is always relative to a formal systemIf an arithmetic sentence was undecidable in ZFCwe still could select a stronger set of axioms validas foundations and try to prove it thereThat is not to say that such a switch would bewithout controversy
- 17. Complete Formal SystemsThe incompleteness theorem does not imply thatevery consistent formal system is incompleteThe theory of the real numbers is complete, andit comprises the arithmetic of real numbersAlthough the natural numbers are a subset of thereal numbers they are not deﬁnable within thetheory of real numbers, and so the premise of theincompleteness theorem do not hold
- 18. ApplicabilityThe incompleteness theorem guarantees theexistence of undecidable arithmetical statements incertain kinds of formal systemsIt says nothing about the existence of undecidablenon-arithmetical or non-mathematical statementsThe incompleteness theorem does not apply incontexts where there’s no formal system like, say,the Bible, or the Constitution, analogies andmetaphors rarely sufﬁce
- 19. Mind vs. ComputerSome people speculate that the incompletenesstheorem implies the mind surpasses the machinebecause it can see the truth of formulas themachine won’t be able to demonstrateThis argument is invalid, in general we have noidea about whether the Gödel sentence of asystem is true, what we know is that it is true iffthe system is consistent, and this much is provablein the system itself
- 20. Incompleteness and the TOESome people question the feasibility of a Theoryof Everything in physics because of the theoremA TOE would probably be a formal system whereGödel’s theorem applies, and in such case thetheorem tells us there would be an incompletenessin its arithmetical component, but whether or notthe basic equations of physics are completeconsidered as a description of the physical world,and what completeness might mean in such acase, is not something that the incompletenesstheorem tells us anything about
- 21. Truth“Truth” is not a mathematical concept, and so itis normally avoided by mathematicians, a proof inlogic is a syntactic derivation and has nothing todo with the “truth” or “falsehood” of a sentenceWhen a mathematician says that the twin primeconjecture may be “true” he means that theremay be inﬁnitely many primes p such that p+2 isalso prime, no more no less
- 22. ZFCThere’s nothing that supports in that arithmeticalstatements like the twin prime conjecture could beactually undecidable in ZFC, in practice there’sno need to feel at all worried by the possibility ofnatural mathematical problem being unsolvableSome problems in Set Theory are known to beunsolvable in ZFC, like Cantor’s continuumhypothesis
- 23. Some Later DevelopmentsThere has been considerable work seekingundecidable statements close to “ordinarymathematics”In arithmetic the ﬁrst result of this kind was theParis-Harrington theorem (1976) which establishes theunprovability of a combinatorial statementGödel himself suggested a way to extend thearithmetical component of ZFC with axioms ofinﬁnity that imply some undecidable statements
- 24. Another line of development—followed byGregory Chaitin in the 1960s—relatesincompleteness to Kolmogorov complexity
- 25. The SecondIncompleteness Theorem
- 26. Second Incompleteness Theorem Second incompleteness theorem (Gödel). For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved in S itself
- 27. ObservationsThe “certain amount of arithmetic” in the secondtheorem is not the same we ask for in the ﬁrst one“S is consistent” can be expressed in said systemsthanks to a technique called arithmetization of syntaxthat uses a way of representing syntactical objectssuch as sentences and proofs as numbers calledGödel numbering
- 28. The consistency of such systems is sometimesproved in other systems, for instance Gentzenshowed that PA is consistent using a theory T thatextends PA, but that yields an inﬁnite regress sincewe need now to investigate the consistency of T
- 29. DoubtsSome people think the second incompletenesstheorem raises doubts about the consistency offormal systems used in mathematics, we can’tprove it in an absolute manner, so there’s a logicalpossibility of mathematics being inconsistentFrom a metamathematical view, though, it wouldbe of little use to prove the consistency of ZFCwithin ZFC, for if we doubt about its consistencyin the ﬁrst place how can we trust the theorem?
- 30. In fact, if ZFC was inconsistent it would indeedproof its consistency, since every sentence is atheorem in an inconsistent system, so even if ZFCcould prove its own consistency little would followas far as our conﬁdence in the system goThe second incompleteness is mainly of interestto logicians, whose object of study are logics, thesame way an algebraist studies, say, rings and isinterested in their properties
- 31. So, Are Mathematics At Risk?No one doubts of the consistency of the axiomson which ordinary mathematics are based, andtherefore of the validity of the theorems such asthat there are inﬁnitely many primesThat belief comes from consensus, commonsense, experience, tradition, etc., not from logic
- 32. Hilbert’s ProgramIt is often said that the second incompletenesstheorem demolished Hilbert’s program, whosegoal was to prove the consistency of mathematicsby ﬁnitistic reasoningThis was not the view of Gödel himself. Rather,the theorem showed that the means by whichacceptable consistency proofs could be carried outhad to be extended and Gödel gave one way inhis “Dialectica interpretation,” published in 1958
- 33. Computability, Formal Systems and Incompleteness
- 34. Alternative ProofsOver time logicians have developed techniques todemonstrate both incompleteness theorems usingother approaches than the original proofs
- 35. StringsBy a string is meant any ﬁnite sequence of symbolsThe length of a string is deﬁned to be the numberof occurrences of its symbolsFor instance numerals in decimal positionalnotation are strings of the alphabet of digits
- 36. Computably Enumerable SetsNumerals can be generated in lexicographic orderby a computer program, such a set of strings iscalled computably enumerableIn that deﬁnition we allow repetitions, and if theset is ﬁnite we allow the program not to halt, aslong as it eventually exhausts the set of stringsTuring machines or some other model ofcomputation are used to formalize these concepts,but we informal descriptions will do for us
- 37. Computably Decidable SetsThere is an algorithm that decides whether agiven string will ever appear in the enumeration ofnumerals mentioned in the previous slide—“13”does, “007” does not, neither does “foo”—bydeﬁnition this property makes the set of numeralscomputable, or computably decidable, or just decidableA set that is not decidable is said to be (computably)undecidable
- 38. We’ve Got Two “Decidable”sWe have by now a deﬁnition of decidable thatapplies to sentences of formal systems, an anotherone that applies to sets of stringsThey are different, but there’s a connectionbetween them that we will see in this section
- 39. Enumerability & DecidabilityEvery computably decidable set is computablyenumerableA set E is computably decidable iff both E and itscomplement are computably enumerableUndecidability theorem (Turing, Church).There are computably enumerable sets which arenot computably decidable
- 40. Formal Systems and Decidability The set of sentences of the language of a formal system is assumed to be a decidable set of strings Axioms and inference rules need to be deﬁned in such a way that the set of theorems is computably enumerable If the set of theorems is indeed decidable the formal system is said to be decidable
- 41. A complete formal system is always decidable, foreither it is inconsistent, or else we determine if asentence A is a theorem enumerating its theoremsuntil either A or ¬A is foundThere are also theories which are decidable andincomplete
- 42. Computability & Incompleteness There is a proof of the incompleteness theorem that does not use any arithmetical formalization of self-referential sentences (so they are not essential) It is based on computability theory and uses the Matiyasevich-Robinson-Davis-Putman theorem The proof shows that there are inﬁnitely many equations D(x1, ..., xn) = 0 for which is undecidable in S whether or not they have solution
- 43. The Completeness Theorem
- 44. The Completeness TheoremGödel himself proved that ﬁrst-order theories arecomplete, in that theorem “complete” has anothermeaning thoughA model of a ﬁrst-order system is a mathematicalstructure that satisﬁes its axioms given a certaininterpretationGiven a ﬁrst-order system S, the completenesstheorem establishes that a sentence of S is atheorem of S iff it holds in all models of S
- 45. Nonstandard ModelsSometimes it is said that the ﬁrst incompletenesstheorem implies that in any theory T of a certaindegree of complexity there are sentences that aretrue in all models of T yet undecidable in TThe ﬁrst incompleteness theorem neither statesnor implies such a thing, on the contrary, thecompleteness theorem implies that if a sentence istrue in all models of a ﬁrst-order theory T (suchas PA or ZFC) then it is decidable in T, for it isindeed a theorem of T

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