incompleteness.of.arithmetic.[sharethefiles.com]

7/28/2019 Incompleteness.of.Arithmetic.[Sharethefiles.com]

1/13

CHAPTERONE

The Incompleteness of Arithmetic

1.1 FormalizationIn elementary logic classes, we are drilled in translating arguments into a formal language,and then we try to construct formal deductions of putative conclusions from given prem-isses. Why bother?

First, why bother with formal languages? Standard answer: because everyday languageis replete with ambiguities and ill-formed sentences which dont have a clear, unambigu-ous sense. So, in assessing arguments, it helps to regiment them into a suitable artificiallanguage which is expressly designed to be free from ambiguities and obscurities, and

where surface form reveals logical structure.Second, why bother with formal deductions? Standard answer: because everyday argu-

ments are replete with suppressed premisses and inferential fallacies. It is only too easy tocheat. Setting out arguments as formal deductions in one style or another enforceshonesty: we have to keep a tally of the premisses we invoke, and of exactly what inferentialmoves we are using. And honesty is the best policy. For suppose things go well with aformal deduction, and we get from the given premisses to the conclusion by small inferencesteps each one of which is obviously valid (there are no suppressed premisses smuggled in,and no suspect inferential moves). This honest toil buys us the right to confidence that ourpremisses really do entail the conclusion.

Granted, outside the logic classroom we almost never go in for presenting deductivearguments in a fully formalized version. No matter. We have glimpsed an ideal: argumentspresented in an entirely perspicuous language with maximal clarity and with everythingopen and above boardwith no room for misunderstanding, and all the argumentscommitments (the premisses and the inferential moves needed to get to the conclusions)systematically and frankly acknowledged.

Old-fashioned presentations of elementary Euclidean geometry illustrate the pursuit of aclosely related idealthe axiomatized theory. Like beginning logic students, schoolstudents used to be drilled in providing fairly rigorous deductions, though not in a formallanguage but in ordinary geometric language. The game was to establish a body of theo-rems about (say) triangles inscribed in circles, by deriving them from simpler results whichhad earlier been derived from still simpler theorems that could ultimately be derivedfrom some bunch of fundamental principles or axioms. And the aim of this enterprise? Bysetting out the derivations of theorems in a laborious, semi-formal, step-by-step style

where each small move is plainly warranted by simple inferences from propositions thathave already been provedwe can develop a body of results that we can be confident musthold if the initial Euclidean axioms are true. Of course, school geometry is not very deepor difficult. Still, tidying things into an axiomatized theory is illuminating about exactlywhat follows from what. And it invites further enquiry into what might happen if we tinkerwith the basic assumptions in various waysfamously leading to investigations of non-Euclidean geometries.

More complex mathematical theories are also frequently presented more or less axiomat-ically. For example, we set down some basic axioms for set theory and explore their conse-quences. Again we will want to see exactly what is guaranteed by the fundamental prin-


2/13

2 Chapter One

ciples embodied in the axioms. And we are also interested in exploring what happens if wechange the axioms and construct alternative set theories (what happens if we drop e.g. theaxiom of choice?).

Theres not only the project of taking some axioms and finding what follows from them,but also the converse project of taking some body of already-accepted propositions andaxiomatizing them, i.e. finding a specifiable set of assumptions from which the given prop-ositions all follow. Ideally, well try to hit on the weakest and most economical set of axiomsthat will do the business. And that way, we may hope to expose the basic commitmentsinvolved in accepting the given body of propositions.

Even mathematics texts are written in an informal mix of ordinary language and novelsymbolism, with proofs rarely spelt out in all their detail. They fall short of the logical ideal.But we might hope that our more informally presented mathematical arguments couldwith a bit of effortbe turned into ideally formalized ones, set out in a strictly regimentedformal language, with absolutely every inferential move made fully explicit and checked asbeing in accord some acknowledged rule of inference. (Thus consider e.g. Michael Pottersdescription of his book Sets: An Introduction as an informal specification on an entirely

formalized text.)The extra effort of spelling out everything in fully formalized detail is usually just not

going to be worth the cost in time and ink. In mathematical practice we use enoughformalization to convince ourselves that our results dont depend on illicit smuggled prem-isses or on dubious inference moves, and leave it at that. Still, putting together the logi-cians aim of perfect clarity and honest inference with the mathematicians notion ofregimentation into an axiomatized theory, we can see the point of the notion of an axioma-tized formal theory, if not as a practical working medium, at least as an ideal.

An axiomatized formal theory, then, comprises (a) a formalized language, (b) some deduc-tive apparatus, and (c) a set of axioms characterizing the theory. Consider these threeingredients in turn.

(a) Well take it that the basic idea of a formalized language is familiar from elementarylogic, with its syntactic rules that determine which strings of symbols constitute well-formed formulae of some language, and its semantic rules that determine a single unam-biguous interpretation of these expressions.

But note that when doing pure logic, we typically are less interested in individuallanguages than in a language template (e.g. the language of first-order logic) with anindefinite supply of expressions waiting to do duty as names, predicates, function-symbolsetc., in particular applications. A formal theory, by contrast, involves a particular formal-ized language, with a definite fixed vocabulary. For example, the vocabulary of a particulartheory of arithmetic might involve just the function-symbols + and , the numeral 0,and a symbol for the successor (or next integer) functionoften written which thenenables us to construct terms for the integers thus, 0, 0, 0, . In this example,then, there are just four items of non-logical vocabulary to add to the usual logical appa-ratus of connectives, quantifiers, variables and identity.

Now, when we are in the business of formally regimenting (say) some branch of math-ematics, we dont want it to be a disputable matter whether a given string of symbols is awff of the theory. So we want there to be a mechanical procedurean algorithm that acomputer could be programmed to executethat can tell whether a string is a wff or not.In other words, for a properly formalized language, the property of being a wff of thelanguage must be a mechanically decidable one. Likewise, given interpretations for thenames and predicates etc. in the language, it should be a mechanical procedure to workout what the consequent truth-conditions of the complete sentences are. (Note, this


3/13

The Incompleteness of Arithmetic 3

emphatically isnt to say that it can be mechanically decided whether the sentences areactually true, but only that it can be mechanically worked out what the sentences say.)

(b) Next, we need some deductive apparatus, i.e. some sort of formal proof-system. Again,well take it that the basic idea of a formal proof system is familiar, in some form or other,

from elementary logic. The differences between various styles of proof systeme.g. naturaldeduction vs. tableaux (or tree) systemsdont matter. What matters in any system isthat there is a fixed set of rules for what counts as a well-formed proof, and the rules areformal or syntactic in that they take the form given wffs of theseshapes earlier in theproof, you can extend the proof by adding a wff of thisshape.

For a formalized theory, we also require the property of being a well-formed proof in thetheorys system of logic to be a mechanically decidable one. Recall, the whole point offormalizing proofs is to keep a tally of the rules which are in play in an argument, so wecertainly dont want it to be a disputable matter whether a putative proof does or does notcorrectly invoke a certain rule. There should again be an algorithm that a computer couldbe programmed to execute which will test whether an array counts as a well-constructedproof according to the proof-system in question.

But careful again! The claim is that it should be checkable whether something presented

as a proof really is a proof (from certain premisses, in a certain logical system). This is notto say that we can always mechanically determine whether a proof exists to be discovered.The case of propositional logic may mislead: for in that special case thereisa mechanicaltest to decide whether given premisses entail a certain conclusionjust do a truth-tabletest! But thats quite exceptional. Even in familiar first-order logic, for example, it is not ingeneral mechanically decidable whether there is a proof from given premisses to a certainconclusionwhich is a result we will be proving later.

(c) To specify an axiomatized theory, some wffs of the language of the theory need to beselected as axiomsas basic assumptions of the theory. Since the aim of the axiomatizationgame is to see what follows from a bunch of axioms, given a certain underlying logic, wedont want it to be a matter for dispute whether a given proof does or does not appeal onlyto axioms in the chosen set. In other words, given a purported proof of some result, we

should be able to mechanically check not only that each inference move accords to therules, but also that the input premisses are all instances of the official axioms. So the prop-erty of being an axiom of the theory is also to be a mechanically decidable one.

That doesnt rule out theories with infinitely many axioms. We might want to say everywff of such-and-such a form is an axiom (where there is an unlimited number ofinstances). For example, in a formalized arithmetic, we might want every instance of

(((0) & x((x) (x))) x(x))

as an axiom. For these are induction axioms saying, surely correctly, that for each propertyattributed by some expression(), if (i) 0 has it, and (ii) given one number has it, so doesthe next, then (iii) all numbers have it. Plainly, we can mechanically recognize an instanceof this induction schema as such.

1.2 Decidability

Before saying anything more about theories, wed better pause and spell out just a littlemore what is involved in saying that some question (such as whether a symbol-string isa wff, or whether the wff is an axiom or the array is a well-formed proof) ismechan-ically decidable.

Consider the familiar rules for finding the results of a long division problem, or for find-ing highest common factors, or for avoiding defeat at noughts-and-crosses, These rulescan be executed in a quite mechanical waya dumb computer can be programmed to


4/13

4 Chapter One

follow the rulesand they are guaranteed to deliver a solution to the target problem. Andit is natural to say, more generally, that other questions are mechanically decidable if youcould in a similar way suitably program a computer so that it is guaranteed to settle theissues. A program embodies an algorithm, a series of step-by-step instructions, each oflimited size and clearly specified in the smallest detail: theres no room for doubt as to whatdoes and what does not count as following its instructions; no room is left for imagination;there is no resort to random methods; there is no reliance on outside oracles.

Now, a real-life computer is limited in size. There will be some finite bound on the sizeof the inputs it can handle; there will be a finite bound on the size of the set of instructionsit can store; there will be a finite bound on the size of its working memory. And even if wefeed in inputs and instructions it can handle, it is of little practical use to us if the computerwont finish doing its computation for a thousand years.

But we are going to abstract from all those practical considerations. We will say that aquestion is algorithmically decidable (oriscomputable in principle) if there is a finite set ofstep-by-step instructions which a dumb computer could use, given working space and timeenough, to deliver a decision. Likewise, a function is computable if there is an algorithmfor calculating the value of the function for a given argument.

Theorists call such a function effectively computable. So lets be clear: effective here mostcertainly does not mean that the computation is feasible for us. We count a function ascomputable in this broad, in principle, sense even if it might take more time to computethan we have before the heat death of the universe, and use more bits of storage than thereare particles in the universe so long as theres an algorithm that works in principle.

But then, why bother with these very highly idealized notions of computability anddecidability? Because it turns out that there are problems that it is impossible to decide bya computer even given unlimited time, and unlimited storage, etc. Having a very weak notionof what is required for decidability means that such impossibility results are very strongthey dont depend on mere contingencies about limitations of time or resources.

We have clarified one aspect of our notion of computability/decidability: we are going to

be abstracting from limitations on storage etc. But you might suspect that still leaves muchto be settled. Doesnt the architecture of a computing device affect what it can compute?For example, wont a computer with random access memorythat is to say, a (perhapsunlimited) set of registers where it can store working data to be retrieved immediatelywhen neededbe able to do more than a computer without?

The short answer is no. The central theoretical questions here were the subject ofintensive investigation even before the first electronic computers were built. Thus, in the1930s, Turing famously analysed what it is for a function to be step-by-step computablein terms of what is computable by aTuring machine (an idiot computer of very limitedpower). About the same time, other definitions were offered of computability in terms ofso-called recursive functions. And later we get e.g. definitions of computability in terms ofthe capacities of register machines (ideal computers whose structure is more like that ofa real-world computer with RAM). The details dont matter here and nowthough well

soon be taking a closer look at some of them in later chapters. What does matter is the bigmathematical fact in this areathat these proposed definitions of computability (and allthe other plausible ones that have ever been advanced) do indeed come to exactly the same.That is to say, exactly the same functions are Turing computable as are computable onmachines with different architectures as are recursive, etc.

And this technical fact prompts a philosophical thesisChurchs thesisto the effectthat the mechanically decidable issues in the intuitive sense are those that are recursivelydecidable or (it comes to the same thing) decidable by a Turing machine. After some sixtyyears, no serious counterevidence to this thesis exists. So we can continue to talk infor-mally about computable functions and algorithmically decidable questions, confident thatwe are indeed referring to a determinate kind.


5/13


1.3 Features of Theories

Start with some key definitions. First a general one:

A set of items (e.g. wffs of a theory) can beeffectively enumeratedif there is a mechan-

ical algorithm for listing off the items in some orderfirst, second, third, .suchthat every item in the set eventually appears on the list somewhere. The list may beinfinite, and contain repetitions, so long as every item in the set is correlated withsome natural number.

And next four definitions are specifically to do with theories.

Given a proof of the wff from the axioms of the theory using the background logic,we will say that is a theoremof the theory.

A theory is decidableif there is a mechanical procedure for determining, for any wffof the language of theory, whether is a theorem.

A theory isnegation completeif for any sentence of the language of the theory, either or is a theorem. In other words, the theory has the resources to prove, for each

complete claim that can be framed in the language, either that it is true or else thatits negation is.

A theory is inconsistent if it has a pair of theorems of the form, , otherwise atheory is consistent. (Assuming a classical logical framework, where , takentogether imply, for any, we could equally say that a theory is inconsistent if everywff is a theorem.)

Consider a really trivial pair of examplesthe theoriesT1 andT2, whose shared languageconsists of the propositional atoms {p, q, r}and all the wffs that can be constructed out ofthem using the familiar propositional connectives, whose shared underlying logic is a prop-ositional natural deduction system, and whose sets of axioms are respectively {p} and {p,q, r}.T1 andT2 are then both axiomatized theories, and are both consistent. (p q)is a theorem ofT1 but not ofT2; (p&q) is a theorem ofT2 but not ofT1. Both theories are

decidable by using the truth-table test.T1 is negation incomplete, since the axioms dontdecide whether q or q holds, for example. But T2 is negation complete (any wffconstructed fromp, q, r has its truth-value decided).

One quick aside. You will be familiar with the idea of a deductive system being completee.g. a standard natural deduction system for propositional logic is complete with respect tothe standard two-valued semantics because every argument which is semantically validi.e. valid by the truth-table testcan be shown to be valid by a proof in the deductionsystem. But having a complete logic in this sense is one thing, being a (negation) completetheory is something else entirely. For example,T1 has a complete logic, but is not a completetheory. Beware this terminological double-use! (As it happens, the first proof of thecompleteness of a system ofpredicate logicwas due to Gdelthus Gdels Completeness

Theorem. This is not to be confused, then, with Gdels Incompleteness Theorem whichis the topic of this chapter, and which concerns the (negation) incompleteness of certaintheories of arithmetic.)

We can now state and informally prove three simple results:

(a) The set of wffs of an axiomatized theory can be effectively enumerated.

Proof sketch. We can give an algorithm for mechanically listing off in alphabeticalorder all the possiblestrings of symbols from the vocabulary of the theory, e.g. bystarting with all the strings of length 1, followed by those of length 2, followed bythose of length 3 and so on. By the definition of an axiomatized theory, there is a


6/13

6 Chapter One

mechanical recipe for deciding which of these symbol strings form wffs. So, puttingthese recipes together, as we ploddingly generate the possible strings we can throwaway all the non-wffs that turn up, leaving us with a effective enumeration of all thewffs.

(b)The set of wffs of the kind

(v) with one free variable can be effectively enumerated.Proof sketch. Effectively enumerate the wffs again, but this time just keep those whichhave one freeunquantifiedvariable (this selection can be made mechanically).

(c) The set of theorems of an axiomatized theory can be effectively enumerated.

Proof sketch. Just as we can enumerate all the possible wffs, we can enumerate all thepossiblesequences (or other suitable arrays) of wffs in some alphabetical order. Bythe definition of an axiomatized theory, there is an algorithmic recipe for decidingwhich of these sequences are well-formed proofs in the theory (since it is decidablewhether the premisses are axioms, and whether the inference moves are legitimate).So we can mechanically filter out the proof sequences from the other sequences ofwffs, to give us an effective enumeration of all the possible proofs. Now list (not thecomplete proofs, but) just the last wffs in the proofs: this mechanically generated list

now contains all and only the theorems of the theory.

Two comments:

First, that talk about alphabetical order can be cashed out in various waysand ofcourse, any systematic mechanical ordering will do. Heres one simple device. Suppose,to keep things easy, the theory has a basic vocabulary of less than ten symbols (thisis no real restriction). With each of the basic symbols of the theory we correlate adifferent digit from 1, 2, , 9. We will reserve 0 to indicate some punctuation mark,say a comma. So, corresponding to each finite sequence of symbolsand in particu-lar with each wffthere will be a sequence of digits which we can read as expressinga number. For example: suppose we set up a mini-theory in the Predicate Calculususing the symbols

, , (, ), F, x,'.and we associate these symbols with the digits

1, 2, 3, 4, 5, 6, 7.

respectively. Then the wff

(x)(Fx (x')F'x'x)

will be associated with the number

3643562136741576764.

We can now list off the wffs constructible from this vocabulary as follows. We exam-ine each integer in turn, from 1 upwards. It will be decidable whether the standardnumeral for that integer corresponds to a sequence of the symbols which forms a

wffsince we are dealing with an axiomatic theory. If the number does correspondto a wff, we enter onto our list of wffs. In this way, we produce mechanically alist of wffswhich obviously must contain all wffs since to any wff corresponds somenumber. Similarly, taking each number in turn, it will be decidable whether thenumber corresponds to a sequence of symbols which forms a seriesof wffs, separatedby commas. If it does, it will also be decidable whether this series of wffs constitutesa derivation of the final wff in the series as a theorem. If the number does thus corre-spond to a proof of theorem, we enter onto our list of theorems. In this way wemechanically produce a list of theorems.

Second, we should quickly note that to say that the theorems of an axiomatic theorycan be mechanically enumerated is not to say that the theory is decidable. For it isone thing to have a method which is bound to turn up any theorem eventually; it is


7/13


another thing to have a method which, starting with an arbitrary wff, can determinewhether it is a theorem or not. (Suppose we set going our method for enumeratingall the theorems. So far hasnt turned up. Maybe thats because isnt a theorem.Maybe thats because it is a theorem, but wont appear on the list until much later.)

However, despite that last general point, we do have the following important result in therather special case of negation-complete theories:

Theorem 1. A consistent, axiomatic, negation-complete theoryT is decidable.

Proof sketch.Letbe any sentence. Since, by hypothesisT is negation-complete, either is a theorem ofT or is. We now set going the procedure for mechanicallyenumerating the theorems ofT. Eventuallywithin a finite number of stepseither or will be produced. If is produced, this means that it is a theorem. If on theother hand is produced, this means that is not a theorem, for the theory isassumed to be consistent. Thus there is an effective procedure for deciding whether is a theorem.

Note here we are relying on the very weak notion of decidability-in-principle we explained

in 1.2. We might have to wait an immense time for one of or to turn up. Still, themethod is guaranteed to produce a result eventually, in an entirely mechanical waysocounts as an effectively computable procedure in our official, weak sense.

1.4 Formalized arithmetic

Lets turn from generalities to a specific case, and consider what might seem to be a verymodest aim, to give a formal axiomatic theory of arithmetic (more exactly, of the arithmeticof the non-negative integers).

Take a sample theory, one well get rather familiar with over the coming chapters, so-calledfirst-order Peano arithmetic, or PA1 for short. (The theory is less helpfully labelled Zby Boolos and Jeffrey. The label first-order indicates that the background logic is first order

logic: later well meet second-order Peano arithmetic, PA2.)The special vocabulary ofPA1 comprises the two-place function symbols + and , the

one-place successor function symbol the , and the numeral 0. As noted, the logicalapparatus will be some standard proof system for first-order logic. And the axioms are asfollows.

First we adopt the following seven general axioms

xy(x =y x =y) having equal successors implies being equalx(0 x) 0 is not the successor of any numberx(x 0 y(x =y) every number other than 0 is a successorx(x + 0 =x) adding 0 has no effectxy(x +y = (x +y)) i.e. x + (y + 1) = (x +y) + 1x(x 0 = 0) multiplying by 0 gives 0

xy(xy = (xy) + x) i.e. x(y + 1) =xy +x;with the domain of quantification the natural numbers (i.e. 0, together with the positiveintegers). We then add all instances of the induction schema

(((0) & x((x) (x))) x(x))

where(x) is a wff with x its one free variable, and (0) is the result of replacing theoccurrences of that free variable with 0.

A little investigation and experimentation reveals that PA1 has the resources to establishall the familiar basic truths about the addition and multiplication of the natural numbers.Still, you might say, even ifPA1 can establish lots of truths about addition and multiplica-tion, thats not yet very impressive. After all, there are plenty of arithmetical facts that


8/13

8 Chapter One

arent, or at least arent immediately, about these two functions. For example, there arefacts about the less than relation, facts about being prime, and so forth.

However, a moments reflection shows that we can also easily construct wffs that in aclear sense capture facts about many other such arithmetic relations. For example, ifm isthe numeral in the language ofPA

1

denoting the number m (i.e. m is 0 followed by moccurences of the accent symbol ), andn is the numeral denoting the number n,thenthe sentence

x(m+x =n)

is true just when m


9/13


facts, well surely want there to be a wff(x)constructible inTsuch that expressesP andsuch that, for any n,

ifnhas property P, the wff(n) is a theorem ofTifndoesnt have property P, the wff(n) is a theorem ofT.

When this obtains we say that P is representableinT. And finally, when every computableproperty and relation can be represented inT, well say that theoryT issufficiently strong.

The idea, to repeat, is that a sufficiently strong theory of arithmetic enables us to derivewffs corresponding to every computable fact about numbersi.e. we can formally prove inthe theory what we can informally calculate. Being sufficiently strong looks a highly plau-sible desideratum for any axiomatic formalization of arithmetic. AndPA1 meets this desid-eratumor at least, it certainly does if we buy Churchs thesis. For then the questionwhether PA1 captures all computable properties can be sharpened into the questionwhether PA1 captures all the properties computable by some Turing machine. And in thissharpened form, the question is a purely technical, mathematical one to which theanswer is yes.

We certainly arent going to prove this here and now. The basic proof idea is fairlysimple, though. A Turing computable function is one that is computed by a certain recipe;so what we want in order to express that function is a wff ofPA1 which encodes that recipe.But the devil is in the details.

A quick final point. It is a surprising fact that, even if we delete all the induction axiomsfrom PA1 and just use only the first seven axioms, the resulting cut-down theory is stillsufficiently strong. This cut-down theory (known as Robinson arithmetic) is fairly stand-ardly designated Q.

1.5 The Undecidabilility of Arithmetic

Weve suggested that we should want any attractive axiomatic theory of arithmeticT to be

sufficiently strong, proving all the truths that can be established to be so by some informalmechanical procedure. Obviously we dont wantT to be able to prove all those truths invirtue of being inconsistent and so being able to prove anything. In other words, we wanta consistent, sufficiently strong axiomatic theoryT.

It would be very nice if our theoryTwere decidable as welli.e. it would be nice if therewere an algorithmic test for theoremhood inT. Then we would have a mechanical way ofsettling every arithmetic question. We are familiar from school arithmetic with mechanicalalgorithms for settling various arithmetic questions (like what is the square of 666 orwhat is the highest common factor of 28 and 42?). Couldnt there be a master algo-rithmso to speakwhich always does the trick for every arithmetical question that canbe expressed in the langauge ofT?

No. For we have the following key theorem:

Theorem 2. Any consistent, sufficiently strong, axiomatic theory of arithmeticT isundecidable.

Proof sketch.Well proceed by reductio. Well supposeT is a consistent and sufficientlystrong yet also decidable theory of arithmetic. And well derive a contradiction.

Let1(x),2(x),3(x) be an effective enumeration of the wffs ofTwith one freevariable (there is such an enumeration by 1.3.3b). We now make the following defi-nition:

n has property D iff the wffn(n) is a theorem ofT.

Note the link between the subscript and the numeral inserted inton()! Well see in laterchapters why Dmight be said to be a diagonal property, of a kind that will in fact play a


10/13

10 Chapter One

major role in what is to come. But here, of course, before discussing any other diagonalproperties, the definition ofDmust look rather artificial and unmotivated. Thats the pricewe pay at this point if we are to come straight to the substantiveTheorem 2 without fillingin lots more background details first. So please bear with the sense of artificiality for now;the construction should later come to see a very natural one. The proof sketch continues:

Now, on the supposition thatT is decidable, the numerical property D is a computableone. For given any number n, it will be a mechanical matter to enumerate the wffswith one free variable until thenth one,n(x), is produced; and then it is a mechan-ical matter then to substitute the appropriate term to getn(n), and prefix a negationsign. Now we just apply the supposed mechanical decision procedure forT to decidewhether the resulting wffn(n) is a theorem.

So the property D is computable. Since by hypothesis the theoryT is sufficientlystrong, it is capable of representing all computable numerical properties. So, in partic-ular, D is representable. It will be represented (like any other property) by a wff withone free variableand this wff must occur somewherein our enumeration of such wff.Lets suppose thed-th wff in the enumeration does the trick. That is to say, propertyD is represented byd(x).

And now it is routine to get out a contradiction. For note that by the definition ofrepresentability (applied to D andd) we have: for any n

ifnhas the property D, the wffd(n) is a theorem ofTifn doesnt have property D, the wffd(n) is a theorem ofT

Taking in particular the casen =d, we have

(i) if dhas the property D, the wffd(d) is a theorem ofT(ii) if ddoesnt have property D, the wffd(d) is a theorem ofT

But note also that from the initial general definition of the property D, we have inparticular:

(iii) dhas the property D iff the wffd(d) is a theorem ofT

From (ii) and (iii), it follows that whether dhas property D or not, the wffd(d) isa theorem either way. So by (iii) again, ddoes have property Dand now by (i) thewffd(d) must be a theorem too. So both a wff and its negation are theorems ofT.And soT is inconsistent, contrary to our initial assumption thatT is consistent.

So givenT is a consistent and sufficiently strong axiomatized formal theory ofarithmetic, it cant be decidable. QED.

This result will immediately yield Gdels first incompleteness theorem. But it is alreadydeeply interesting in its own right. And lets briefly pause to note the following importantcorollary:

Theorem 3.There is no mechanical test for deciding whether a wff of first-order logicis a theorem.

Proof sketch. We noted (though didnt prove) that the cut-down Robinson arithmetic,with its seven first-order axioms, is sufficiently strong. It makes no odds if we conjoinits seven axioms into a single sentence. Then in any standard deduction system forlogic, is provable in Robinson arithmetic just so long asfollows from, i.e. justso long as the conditional () is a theorem provable from no premisses.

Hence, ifwe could always mechanically decide theoremhood for first-order logic,then we could mechanically decide whether follows from. But since we cant dothe latter (a sufficiently strong arithmetic is not decidable), we cant do the former.QED.


11/13


1.6 Gdels First Incompleteness Theorem

Now lets put together the first two Theorems:

Theorem 1. Any consistent, axiomatic, negation-complete theoryTisdecidable.

Theorem 2. Any consistent, sufficiently strong, axiomatic theory of arithmeticT isundecidable.

These immediately entail

Gdels First Incompleteness Theorem(one version). Any consistent, sufficiently strong,axiomatic theory of arithmetic cannot also be negation complete.

That is to say, there will always be some pair of sentences and, neither of which aretheorems. But one and must be true. So, for any theoryT which is consistent,axiomatized, and sufficiently strong, there are arithmetical truths not provable inT.

Note the terrific strength of this result. We have not shown merely that some particularaxiomatic system of arithmetic is incomplete; rather we have shown, in effect, that suffi-ciently strong axiomatic theories are in principle incompletablein the sense that theirincompleteness cant be cured by adding further axioms (except at the price of inconsist-ency).

For supposeT is a consistent, sufficiently strong, axiomatic theory of arithmetic (say, itcontains the Robinson arithmetic Q and as many more true axioms as take your fancy)and is a wff which is unprovable but true on interpretation. Suppose we seek tocompleteTby adding the wffas a new axiom, to give us a new theoryT*. We will nowtrivially be able to provein the resulting new theoryT*. But this new theory is axioma-tized, still consistent (all its axioms are true on interpretation, and well assume the logicis sound), and still sufficiently strong, since it contains Q. Soby the incompleteness theo-rem, the new theoryT*muststill contain another wff*which is again true-on-interpre-tation but unprovable.

And so it goesany patching-up process, trying to complete the theory, will be never-ending. This is an astonishing result. It puts paid to an idea that goes back at least to Leib-

niz, namely thatif only we could see how to do itwe should be able to capture all math-ematical truth, or at least all arithmetic truth, in a simple axiomatic systemi.e. weshould be able to do for arithmetic what Euclid (supposedly) did for geometry. But we cant.

Lets say, for short, that an axiomatized system of arithmetic is respectableif it is consistent,is sufficiently strong, and its axioms are true on the intended interpretation (hence all thetheorems are true too). The label will, I hope, not seem tendentious by this stage. We wantrespectability in our formal theories of arithmetic.

But we have now shown that there is some true-but-unprovablesentence in any respect-able system of arithmetic. However, we havent given any idea of what such a sentencemight look like. For many philosophical purposes, you dont need to know. But Gdel infact established his devastating result by finding a way of actually producing a true-but-

unprovable wff for a given respectable theory T. In later chapters, well be seeing how hedid itbut the basic idea is actually quite easily grasped.Suppose for a moment that a respectable theoryT could have a sentenceGwhose inter-

pretation is I amnot provable in T. Now, ifGwereprovable, then Twould have a theoremwhose interpretation is a false statementnamelyG itselfcontradicting respectability: soG is not provable. In which caseG will be truebut will not be a theorem. So clearly, oneway of showing the incompleteness of a theoryT is by finding a wff whose interpretationis likeGs. Such a wff we will call Gdelian. But how can we do this with a formal arith-metic whose wffs are to be interpreted as statements about numbers, and notas statementsabout the theory itself?

Well, the essential trick is one we have used already, in 1.3: associate the symbols ofthe theoryT with digits, and then each wff will have a corresponding numberwe will


12/13

12 Chapter One

call this its Gdel-number, or g.n. for short. And so corresponding to claims about astring of wffs ofTbeing a proof there will be claims about the g.n. of a string of wffs havinga certain arithmetic property. And this property is a decidable one and so representable ina sufficiently strong theoryT. So we can construct an arithmetic wff inT which can beinterpreted, via the g.n. encoding, to be about proofs inT. And with a bit of cunning, wecan construct a Gdelian wff G which, via the encoding, says I am not provable. Moreaccurately G says There is no number which is the code for a proof inT of the wff withcode number g where the wff with code number g is G itself!

But further details will have to wait (as well as the answer to the natural query, well, ifthats Gdels first incompleteness theorem, whats the second one?). We first need to saymore about the various key notions like decidability that weve been using in a prettyrough and ready way


13/13


Summary.Gdel in a page

SupposeT is a respectabletheory of arithmetic, i.e. one which has true axioms and isaxiomatic (you can mechanically decide whats a wff, whats an axiom, etc.)consistent (of course!)sufficiently strong (any numerical facts you can informally calculate, you can formallyprove inTso ifP is a computable property of numbers, there is a wffA(x)ofTsuch that

ifnhas property P, the wff(n) is a theorem ofTifndoesnt have property P, the wff(n) is a theorem ofT

wheren is the numeral for n.)

Then (Theorem 1) if T were negation complete it would be decidable. For letbe any sentence.Then either or would be a theorem ofT. Now just start churning out (in alphabeticalorder) all the theorems ofT, by enumerating the possible proofs. Eventually either or is going to be produced. If is produced, it is a theorem. If is produced, this meansthat is not a theorem, for the theory is assumed to be consistent. Thus there would bean effective procedure for deciding whether is a theorem.

But we also have (Theorem 2)T is not decidable. Let1(x),2(x) be an effective enumer-ation of the wffs ofT with one free variable. We define:

n has the property D iff the wffn(n) is a theorem ofT.

Now, assume thatT is decidable. Then D is a computable property. For any n, it is amechanical matter to find thenth wff with one free variablen(x); a mechanical matterto formn(n); and by hypothesis mechanical to compute whether n(n) is a theorem.SinceT is sufficiently strong, it is capable of representing a computable property likeD. Sothere must be some wffd(x) in the enumeration which does the tricki.e. for any n

ifnhas the property D, the wffd(n) is a theorem ofTifn doesnt have property D, the wffd(n) is a theorem ofT

Taking in particular the casen =d, we have

(i) if dhas the property D, the wff(d) is a theorem ofT(ii) if ddoesnt have property D, the wff(d) is a theorem ofT

But from the definition ofD, we have in particular:

(iii) dhas the property D iff the wff(d) is a theorem ofT

Simple logic now shows that the three results (i), (ii) and (iii) taken together implyT isinconsistent. So, givenT is respectable and so consistent, it cant be decidable.

Putting these theorems together we get Gdels First Incompleteness Theorem: any respecta-ble theory of arithmetic cannot also be negation complete. So there are truths unprovablein the theory. (Adding some in as new axioms will just give us a new theory, which is still

incomplete.)

Gdel himself established this result by finding a way of constructing for a givenT asentenceGwhich in effect means I am unprovable inT. The trick for doing this is to codefacts about the formal systemT into facts about numbers. So that Gsays in effect there isno number which is the code for a proof inTof the wff with code number g where the wffwith code number g is G itself! Cunning, but quite legitimate.

And thisdoesnt all show that there is something suspect about the idea of a theorys beingsufficiently strong. Even the very simple Robinson Arithmetic Qhas that property.

incompleteness.of.arithmetic.[sharethefiles.com]

Documents