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Gödel's incompleteness theorem

In mathematical logic, Gödel's incompleteness theoremstwo celebrated theorems proved by Kurt Gödel1930. Somewhat simplified,first theorem states:

In any consistent axiomatic system (formal systemmathematics) sufficiently strongallow onedo basic arithmetic, one can constructstatement about natural numbers that can be neither proved nor disproved within that system.

In this context, an axiomatic systemone withrecursive setaxioms; equivalently,theorems ofsystem can be generated byTuring machine. The statement which cannot be proved nor disproved insystemfurthermore true insense that whatasserts aboutnatural numbersfact holds. Becausesystem failsprovetrue statement, itsaidbe incomplete. In other words, then, Gödel's first incompleteness theorem says that any sufficiently strong formal systemmathematicseither inconsistent or incomplete.

Gödel's second incompleteness theorem, whichproved by formalizing part ofproof offirst withinsystem itself, states:

Any sufficiently strong consistent system cannot prove its own consistency.

(See below fordiscussionwhat "sufficiently strong" meansthis context.) This endedprojectanswering Hilbert's second problem, which set forthchallengeproving that mathematics could be reduced toconsistent setaxioms from which all mathematical truths could be derived -- Gödel proved that there can be no such set. See Hilbert's problemsbackground.

Tablecontents
1 Examplesundecidable statements
2 Discussionimplications
3 Proof sketch forfirst theorem
4 Proof sketch forsecond theorem
5 See also
6 External linksReferences

Examplesundecidable statements

The subsequent combined workGödelPaul Cohen has given concrete examplesundecidable statements (statements which can be neither proven nor disproven): bothaxiomchoice andcontinuum hypothesisundecidable instandard axiomatizationset theory. These results do not requireincompleteness theorem.

In 1973,Whitehead problemgroup theory was shownbe undecidablestandard set theory. In 1977, Kirby, ParisHarrington proved thatstatementcombinatorics,version ofRamsey theorem,undecidable inaxiomatizationarithmetic given byPeano axioms but can be provenbe true inlarger systemset theory. Kruskal's tree theorem, which has applicationscomputer science,also undecidable fromPeano axioms but provableset theory. Goodstein's theorem isrelatively simple statement about natural numbers thatundecidablePeano arithmetic.

Gregory Chaitin produced undecidable statementsalgorithmic information theoryin fact proved his own incompleteness theoremthat setting.

Discussionimplications

The incompleteness results affectphilosophymathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrasefirst theorem as saying that "we can never find an all encompassing axiomatic system whichableprove all mathematical truths, but no falsehoods."
The following rephrasing ofsecond theoremeven more unsettling tofoundationsmathematics:

If an axiomatic system can be provenbe consistent from within itself, then itinconsistent.

Therefore,orderestablishconsistency ofsystem S, one needsutilize some other system T, butproofTnot completely convincing unless T's consistency has already been established without using S. The consistency ofPeano axiomsnatural numbersexample can be provenset theory, but not intheorynatural numbers alone. This providesnegative answerproblem number 2 on David Hilbert's famous listimportant open questionsmathematics.

In principle, Gödel's theorems still leave some hope:might be possibleproducegeneral algorithm that forgiven statement determines whether itundecidable or not, thus allowing mathematiciansbypassundecidable statements altogether. However,negative answer to Entscheidungsproblem shows that no such algorithm exists.

Note that Gödel's theorems only applysufficiently strong axiomatic systems. "Sufficiently strong" means thattheory contains enough arithmeticcarry outcoding constructions needed forproof offirst incompleteness theorem. Essentially, all thatrequiredsome basic facts about additionmultiplication as formalized, e.g.,Robinson arithmetic Q. Thereeven weaker axiomatic systems whichconsistentcomplete,instance Presburger arithmetic which proves every true first-order statement which only involves addition.

The axiomatic system may consistinfinitely many axioms (as first-order Peano arithmetic does), butGödel's theoremapply, there hasbe an effective algorithm whichablecheck proofscorrectness. For instance, one might takesetall first-order sentences whichtrue instandard model ofnatural numbers. This systemcomplete; Gödel's theorem does not apply because thereno effective procedure that decides ifgiven sentencesan axiom. In fact, that thisso isconsequenceGödel's first incompleteness theorem.

Another example ofspecification oftheorywhich Gödel's first theorem does not apply can be constructed as follows: order all possible statements about natural numbers first by lengththen lexicographically, startan axiomatic system initially equal toPeano axioms, go through your liststatements one by one, and, ifcurrent statement cannot be proven nor disproven fromcurrent axiom system, add itthat system. This createssystem whichcomplete, consistent,sufficiently powerful, but not recursively enumerable.

Gödel himself only provedtechnically slightly weaker version ofabove theorems;first proof forversions stated above was given by Rosser1936.

In essence,proof offirst theorem consistsconstructingstatement

p = "This statement cannot be proven"

withinformal axiomatic system. As such,can be seen asmodern variant ofLiar paradox.

Ifaxiomatic systemconsistent, Gödel's proof shows that p (and its negation) cannot be proven insystem. Therefore ptrue (p claims notbe provable,it isn't) yetcannot be formally proved insystem. Note that adding p toaxioms ofsystem would not solveproblem: there would be another Gödel sentence forenlarged theory.

Roger Penrose claims that this (alleged) difference between "what can be mechanically proven""what can be seenbe true by humans" shows that human intelligencenot mechanicalnature. This viewnot widely accepted, because as stated by Marvin Minsky, human intelligencecapableerror andunderstanding statements whichin fact inconsistent or false. However, Marvin Minsky has reported that Kurt Gödel told him personally that he believed that human beings had an intuitive, not just computational, wayarriving at truththat therefore his theorem did not limit what can be knownbe true by humans.

The position thattheorem shows humanshave an ability that transcends formal logic can also be criticized as follows: We do not know whethersentence ptrue or not, because we do not (and can not) know whethersystemconsistent. Sofact we do not know any truth outside ofsystem. All we know isfollowing statement:

Either punprovable withinsystem, orsysteminconsistent.

This statementeasily proved withinsystem. In fact, suchproof will now be given.

Proof sketch forfirst theorem

The main problemfleshing outabove mentioned proof idea isfollowing:orderconstructstatement p thatequivalent"p cannot be proved", p would havesomehow containreferencep, which could easily endan infinite regress. Gödel's ingenious trick, which was later used by Alan TuringsolveEntscheidungsproblem, will be described below.

To begin with, every formula or statement that can be formulatedour system getsunique number, called its Gödel number. Thisdonesuchway that iteasymechanically convert backforth between formulasGödel numbers. Because our systemstrong enoughreason about numbers, itnow also possiblereason about formulas.

A formula F(x) that contains exactly one free variable x is calledstatement form. As soon as xreplaced byspecific number,statement form turns intobona fide statement,it theneither provable insystem, or not. Statement forms themselvesnot statementstherefore cannot be proved or disproved. But every statement form F(x) hasGödel number which we will denote by G(F). The choice offree variable used inform F(x)not relevant toassignment ofGödel number G(F).

By carefully analyzingaxiomsrules ofsystem, one can then write downstatement form P(x) which embodiesidea that x isGödel number ofstatement which can be provedour system. Formally: P(x) can be proved if x isGödel number ofprovable statement,its negation ~P(x) can be proved ifisn't. (While thisgood enoughthis proof sketch, ittechnically not completely accurate. See Gödel's paper forproblemRosser's paper forresolution. The key word"omega-consistency".)

Now comestrick:statement form F(x)called self-unprovable ifform F, appliedits own Gödel number,not provable. This concept can be defined formally,we can constructstatement form SU(z) whose interpretationthat z isGödel number ofself-unprovable statement form. Formally, SU(z)defined as: z = G(F)some particular form F(x),y isGödel number ofstatement F(G(F)),~P(y). Nowdesired statement p that was mentioned above can be defined as:

p = SU(G(SU)).

Intuitively, when asking whether ptrue, we ask: "Ispropertybeing self-unprovable itself self-unprovable?" Thisvery reminiscent ofBarber paradox aboutbarber who shaves precisely those people who don't shave themselves: does he shave himself?

We will now assume that our axiomatic systemconsistent.

If p were provable, then SU(G(SU)) would be true,by definitionSU, z = G(SU) would beGödel number ofself-unprovable statement form. Hence SU would be self-unprovable, which by definitionself-unprovable means that SU(G(SU))not provable, but this was our p: pnot provable. This contradiction shows that p cannot be provable.

Ifnegationp= SU(G(SU)) were provable, then by definitionSU this would mean that z = G(SU)notGödel number ofself-unprovable form, which implies that SUnot self-unprovable. By definitionself-unprovable, we conclude that SU(G(SU))provable, hence pprovable. Againcontradiction. This one shows thatnegationp cannot be provable either.

Sostatement p can neither be proved nor disproved within our system.

Proof sketch forsecond theorem

Let p stand forundecidable sentence constructed above,let's assume thatconsistency ofsystem can be proven from withinsystem itself. We have seen above that ifsystemconsistent, then pnot provable. The proofthis implication can be formalized insystem itself,thereforestatement "pnot provable", or "not P(p)" can be proven insystem.

But this last statementequivalentp itself (and this equivalence can be proven insystem), so p can be proven insystem. This contradiction shows thatsystem must be inconsistent.

See also

External linksReferences


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