Time to node my homework... This is an essay on Godel's theorem that I wrote to fulfil the requirements of a maths degree a few years back:

# Limitations of the axiomatic method

John von Neumann, the legendary mathematician who mastered calculus by the age of eight, who devised the familiar set-theoretic definition of the ordinal numbers at twenty, whose powers of calculation surpassed those of at least one early electronic computer, and who was described by Polya as "the only student I was ever afraid of", had the following to say regarding a certain episode in mathematical history:

This happened in our lifetime, and I know myself how humiliatingly easily my own values regarding the absolute mathematical truth changed during this episode, and how they changed three times in succession!
The cause of such awe was a short paper published in 1931 by the 25-year-old logician Kurt Godel, entitled Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). The revolutionary (and, to von Neumann and many others, disturbing) implication of the results therein was that any logical system comprehensive enough to describe elementary arithmetic necessarily contains propositions which can neither be proven nor disproven. Moreover, Godel proved that the internal consistency of such a system can never be proven except by employing reasoning which is not expressible within the system itself.

Background

To better understand the impact which Godel's findings must have had on his peers, we should first describe the mathematical climate of the time.

In the nineteenth century it had been discovered, through the work of Riemann, Lobachevsky and others, that coherent models of geometry could be constructed in which Euclid's parallel postulate (that, given a line L and a point P in the plane, exactly one line exists which contains P and is parallel to L) did not hold. This, in itself, was a shock to many mathematicians: for millenia it had been assumed that Euclid's description of geometry, founded as it was on a "self-evident" and minimal set of axioms, was one of the firmest, most trustworthy branches of mathematical knowledge. The existence of non-Euclidean geometries not only challenged mathematicians' geometrical intuition, but also the Platonist view that mathematics consisted of discoveries about eternal, pure forms whose existence was objective and unquestionable. More "monstrosities" such as continuous functions which were nowhere differentiable soon appeared, further fueling the general loss of faith in geometry.

Attempts to re-establish the comfortable certainty of the past, by turning from geometry to set theory as the new foundation of mathematics, also ran aground. Set theory, when pushed too hard, soon yielded such abominations as Russell's "set of all sets which do not include themselves". It proved difficult to construct a theory of sets which outruled such objects without sacrificing one's principles in the process. Logicism, as espoused by Frege, Dedekind and Russell, gave birth to structures so complicated and unwieldy that the stated intention to formalize the intuitive laws of reasoning was obscured. Constructivism, which rejected even the law of trichotomy (that every real number is either greater than, equal to, or less than zero) was deservedly perceived as fanatical.

To sidestep the embarrassing possibility that multiple, equally defensible versions of mathematical truth might exist, mathematicians soon claimed to have never been searching for truth in the first place. The formalists, led by Hilbert, redefined mathematics as consisting of allegedly meaningless symbols which were not "about" anything in particular. The mathematician was recast as a practitioner who merely manipulated these empty signs, attempting to derive theorems (sentences consisting of the aforementioned meaning-free symbols) from axioms without concerning himself with the "truth" of his findings.

Hilbert hoped thus to outmanoeuvre intuition, and, more importantly, to make possible a proof of the consistency of mathematics. The logicists before him had already laid the foundation by developing a formal language in which mathematical statements could be expressed, along with symbolic transformation rules representing steps which could legally be followed in the progression from the beginning to the end of a valid proof. (The climactic, exhaustive chronicle of this endeavour is Russell and Whitehead's Principia Mathematica, page 362 of which finally yields the demonstration that 1 + 1 = 2.) With this framework in place, it should (Hilbert thought) be possible to study the combinatorial properties of the set of all sentences which could legally be derived from the system's axioms, and to prove that no two of them were logical opposites. This would obviously be an assurance that mathematics (or at least the portion modelled by this formal system) was free from internal contradiction: that is, that the axioms could not be used to prove both a theorem and its negation.

As well as proving the impossibility of internal contradiction, it was hoped that the set of "true" sentences (those which could be constructed by applying legal transformations to the axioms) could be proved complete in the sense that, given a sentence, one could be assured that either this sentence or its negation was a member of the set of true sentences. A formal system with this property is said to exhibit "decidability", since one need never be unsure of the truth of a given sentence.

Godel's theorem

Hilbert's dreams of reformulating classical mathematics as a formal axiomatic system equipped with absolute proofs of consistency and completeness were dealt a cruel blow by Godel's findings in 1931.

In his famous paper, Godel proved that it was impossible to find a metamathematical proof of such a system's consistency without employing rules of inference inexpressible within the formal system under consideration. (More precisely, Godel proved his results of any axiomatic system comprehensive enough to contain the whole of arithmetic. Henceforth, when the term "formal system" is used, it should be assumed that we are speaking of a system satisfying the aforementioned requirement. Less powerful systems, such as arithmetic equipped with addition alone or multiplication alone, can in fact be proved decidable and complete, as was shown by Presburger and Skolem in 1930.)

Godel's other main conclusion was that any such formal system is incomplete, and hence that "truth" within the system is undecidable. Specifically, he showed that it possible to construct a sentence such that neither the constructed sentence nor its negation is provable within the system. What is more, even if one were to arbitrarily decide that such a sentence was true and should therefore be added to the system's axioms, there would still exist other equally undecidable sentences within this new system; and no matter how far this process of augmentation is taken, there will always be further truths which elude proof.

Godel numbering

The proofs of Godel's results hinge on the fact that the set of formulas expressible within a symbolic system is countable, and hence each formula may be mapped to a natural number. Therefore, metamathematical statements about these sentences may be construed as statements about natural numbers: meaning that these metamathematical statements are expressible in the system itself. As we will see, this power of the system to codify statements about itself turns out to be an Achille's heel of sorts, allowing Godel's ingenious construction of an undecidable sentence.

Godel considered a formal system containing only seven constant symbols: the left and right parentheses, as well as signs representing "not", "or", "for all", "zero", and "the successor of" (an operator which adds one to an integer, and can therefore be used to express all natural numbers via its repeated application to "zero"). Recall that Godel's aim was to assign a unique integer (usually called the "Godel number") to each sentence expressible within this system; to begin with, the constant symbols described above were allocated distinct natural numbers. Similarly, other primitive signs (such as letters representing sentential variables) are each assigned integers. Since the number of variables which might be needed in a sentence is potentially infinite, Godel was compelled to employ some simple number theory to avoid overlap between the integers associated with different types of variables. As such, a certain class of variables was assigned prime Godel numbers, while another class was allocated from the set of squares of primes, and so on.

A similar trick was used by Godel to calculate a unique integer associated with each sentence. A sentence is just a string of primitive symbols, each of which already has a natural number assigned to it. Obviously, a simple addition of the Godel numbers of symbols in the sentence is inadequate, since it does not guarantee uniqueness over the set of all sentences. Similarly, a weighted sum is out of the question since we do not have an upper bound on the Godel numbers of primitive symbols. (If such a bound existed, say N, then we could simply multiply successive symbols by 1, N+1, (N+1)2, etc., to obtain a unique Godel number for the sentence.) Instead, the Godel number of a sentence containing n symbols with respective Godel numbers G1,...,Gn is defined as the product p1G1* ... *pnGn where pi denotes the ith prime number. This representation allows us to unambiguously (as guaranteed by the fundamental theorem of arithmetic) retrieve a sentence from its Godel number via factorisation. Similarly, a sequence of sentences may be assigned a single Godel number by multiplying successive prime powers, the exponents being the Godel numbers of successive sentences in the sequence.

Outline of Godel's proof

Since every symbol, sentence, and sequence of sentences in the formal system has now been assigned a Godel number, and since the system under discussion is capable of expressing statements about natural numbers, we now have a way of expressing metamathematical statements in the language of the system. For example, the claim that one sentence implies another can be interpreted as asserting a certain numeric relation between the Godel numbers of the two sentences. This relation will obviously be very complex, since it will need to express, in the domain of Godel numbers, all possible legal transformations which may be applied to a sentence in the system. However, since in the end it is merely a statement about integers, it is certainly expressible in the language of the system itself. Similarly, a yet more complex relation between natural numbers m and n exists which expresses the claim "The sequence of sentences with Godel number m is a proof for the sentence with Godel number n".

To prove that an undecidable sentence existed, Godel needed to find a formula G which, somewhat like Epimenides (the Cretan who claimed "All Cretans are liars"), expressed the assertion that no proof of G exists. More precisely, this claim could be expressed in the language of the system as

There does not exist a natural number m such that m is the Godel number of a sequence of sentences forming a proof for the sentence with Godel number g.
where g is actually the Godel number of the sentence just quoted. The sentence can therefore be construed as making a claim about itself, namely that it is unprovable.

A little thought should show that constructing such a sentence is somewhat difficult. To calculate the Godel number of the above sentence, one follows the process described above of splitting it into primitive symbols, whose Godel numbers are encoded as exponents of successive primes. However, the result of this calculation, g, appears in the sentence itself, and therefore affects the calculation! It would appear at first that we need to be "lucky" by stumbling upon a number g with the property that, when substituted literally into this sentence, brings about the coincidence that the Godel number of the resultant sentence is also g.

Luck, of course, plays no part. Godel conceived of a complex but elegant construction which, through a process of iteration, shows how to find such a number in a finite number of steps. The details of this process, while readily understandable, are somewhat tedious and will not be described here. The end result is the important point: for a very general class of formal sysems, we have an explicit method for constructing a sentence, G, which asserts its own unprovability. Further, Godel showed that if the axioms of the system are consistent (meaning that it is impossible to derive two contradictory sentences from them) then G is indeed unprovable: since if a proof for G existed, then it would also be possible to prove its negation, making the system inconsistent. The converse also holds: discovery of a proof for G's negation would imply the existence of a proof for G. In other words, if the axioms are consistent, then G is formally undecidable.

Godel further noted that, although unprovable within the formal system itself, the sentence G can in fact be proved true via metamathematical reasoning. In fact, the immediately preceding discussion shows this: since we have established that no proof for G can exist, and since this is exactly the assertion made by G about itself, G is a true statement. Thus the system not only contains an undecidable sentence, but: since it contains a true, unprovable sentence: the system is also incomplete. (The term "completeness", applied to a formal system, implies that all true statements in the system are derivable from its axioms.) What is more, simply adding G to the axioms would not suffice to make the system complete, since exactly the same process could be applied to this augmented system to obtain another, similarly undecidable, sentence. Godel thus shattered all hope of ever constructing a consistent, complete formal system.

The final blow landed by Godel's paper was a demonstration of the impossibility of proving a formal system's consistency via a proof expressible within the system itself. A brief description of how he obtained this result follows. Above we saw how, from the assumption that the system's axioms were consistent, Godel proved that it contained a true, undecidable sentence and was thus incomplete. It turns out that the proof of this fact:

If this system is consistent, then it is incomplete.
is achievable within the system itself. To see how, note that the sentence G, which asserts its own unprovability, is equivalent to the statement "This system is incomplete", since it gives an explicit example of a true, undecidable sentence. Thus the statement above is equivalent to:
(This system is consistent) implies that G is true
Next, let A be the statement "There exists a sentence which is unprovable". This claim is in fact equivalent to asserting the system's consistency, since if the system were inconsistent, then every sentence would be provable. (This is closely related to the fact that, if we have a false statement p in any logical system, then the sentence "p implies q" is true for any sentence q.) Hence the above statement may be expressed within the formal system as simply "A implies G". Godel showed that this latter sentence was formally provable within the system. Now, assume that a proof for A, i.e., a proof of the system's consistency, also existed. Then since we have proofs for both A and "A implies G", we have a proof of G. But G was previously proven unprovable. Therefore no proof of A can exist: the system cannot prove its own consistency.

Consequences of Godel's proof

Godel's findings were the catalyst for many philosophical controversies which continue even to the present day. The Oxford philosopher J.R. Lucas has made the claim that Godel's theorem precludes the existence of artificial intelligence, since any calculating machine is isomorphic to a formal system to which Godel's theorem applies. Others, notably Douglas Hofstadter, dismiss this view as "a transient moment of anthropocentric glory" and claim that Godel's proof may even offer insights about the workings of human intelligence which will be useful in the creation of AI.

Whilst the dream of establishing secure foundations for mathematics has never recovered from Godel's attack, his findings have not been construed as a reason to abandon all hope of extracting meaning from mathematical inquiry. Godel himself seemed to hold the view that Platonic realism provided the clearest definition of mathematical truth: of mathematical concepts, he said "It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence". According to Davis and Hersh, most modern mathematicians also secretly subscribe to Platonism: "like an underground religion, it is observed in private and rarely mentioned in public".

Godel's methods also sparked various fruitful lines of investigation which had far-reaching consequences. Since the publication of his paper, the first naturally-arising example of an undecidable set-theoretic statement has been found. Known as the continuum hypothesis, it is the statement that no set has a cardinality greater than that of the natural numbers but less than that of the reals. Godel himself showed in 1937 that this hypothesis cannot be proved from the axioms of set theory; Paul J. Cohen demonstrated in 1964 that neither can it be disproved.

A fascinating variant of Godel's theorem was discovered in 1970, when it was proved that no general algorithm for solving all Diophantine equations (polynomial equations with integer coefficients and roots) can be formulated. Loosely, it can be shown that in any formal number theory, a Diophantine equation exists which is in some sense equivalent to Godel's self-denying sentence G. Such an equation can be interpreted as stating of itself that it has no solutions; in fact, if a solution were found, one could construct from it the Godel number of a proof that the equation had no solutions. It seems unlikely that we have come close to exhausting the list of surprises derived from Godel's work. Perhaps von Neumann may be allowed the last word on Godel's significance:

Kurt Godel's achievement in modern logic is singular and monumental: indeed it is more than a monument, it is a landmark which will remain visible far in space and time... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement.

Bibliography

Davis and Hersh , The Mathematical Experience, 1980
Ferris, Timothy (ed.), The World Treasury of Physics, Astronomy and Mathematics , 1991
Hofstadter, Douglas, Gödel, Escher, Bach: An Eternal Golden Braid, 1979
Nagel and Newman, Godel's Proof, 1958
van Heijenoort, Jean (ed.), From Frege to Godel: a Sourcebook in Mathematical Logic ,1967

Essentially, the theorem is proved by first proving that in any sufficiently advanced system, it is always possible to make meta-mathematical statements (this is done through Godel numbers). And if you can do that, you can, within the system, formulate the statement "This system is inconsistent", which is basically an Epimenides paradox.

Here's the simple (i.e. less than rigorous) version. It applies to a elementary number theory (ENT). I'll skip the formal definitions, but ENT is basically the theory of the nonnegative integers.

ENT is built up from a number of axioms. If an expression follows from an axiom or a theorem, then it is said to be a theorem of ENT, and the list of expressions it follows from are its proof. To differentiate mathematical expressions from ENT expressions, I will write ENT expressions in bold type.

Now, we can write any expression or proof in ENT as a number, called the Godel number of the expression. I won't explain how to do this, but it is not difficult.

Since we can do this we can define a relation P(m,n) which is true for all numbers m and n, where m is the Godel number of an expression A and n is the Godel number of the proof of A. (Still with me?) I won't explain how, but we can write an expression P(x1,x2) such that P is provable in ENT if P(x1,x2) is true and not P is provable in ENT if P(x1,x2) is false.

Got that? Then let r be the Godel number of the expression for all x2, not P. This expression will be provable in ENT if the expression x1 is not provable in ENT. Note that r is a Godel number. Now, here's the clever bit: let E be the expression for all x2, not P(r|x1). (There is nothing mysterious about P(r|x1). It just means P with x1 replaced by r.)

Why is this clever? Think about what E means. Well, E means "E has no proof in ENT". Is this true? If E is false, then E has a proof, so E is true. But that is ridiculous, so E must be true.

This might seem like a paradox because we have just proven E. This is not a paradox however, because we have proven E in English, but E just has to be unprovable in ENT. What this means, though, is that there are expressions in ENT which cannot be proven to be either true or false (in ENT). Thus ENT is incomplete. (This is the definition of incomplete).

The same line of proof applies to any sufficiently complex theory, including 'all of mathematics' (in a very loose sense), so it basically shows that there are true statements about maths that maths can't prove.

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