Gödel s Theorem Anders O.F. Hendrickson Department of Mathematics and Computer Science Concordia College, Moorhead, MN Math/CS Colloquium, November 15, 2011
Outline 1 A Little History 2 Incompleteness 3 The First Theorem 4 The Second Theorem 5 Implications
My Source:
A search for foundations For much of mathematical history, there could be something a little sketchy about the proofs, and even about the objects being considered. For example, Newton s fluxions; i? 1; Hamilton s quaternions i, j, k; Graves s octonions To what extent do we get to just make stuff up? So in the early 20th century, mathematicians and philosophers worked strenuously to put mathematics on rigorous foundations.
Everything is a set The foundation they built on was set theory. Every mathematical object could be interpreted as a set. A function f : A Ñ B is a certain subset of the set A B. An operation like or is a certain function from R R to R; since a function is a set, so is an operation. The ordered pair pa, bq could be thought of as the set ttau, ta, buu. Even the integers could be modeled recursively with sets: 0 H; 1 t0u; 2 t0, 1u; 3 t0, 1, 2u, etc.
The foundation is shaken Russell s Barber In a certain village, there is one barber who shaves every man except those men who shave themselves.
The foundation is shaken Russell s Barber In a certain village, there is one barber who shaves every man except those men who shave themselves. Question Who shaves the barber?
The foundation is shaken Russell s Barber In a certain village, there is one barber who shaves every man except those men who shave themselves. Question Who shaves the barber? If the barber does not shave himself, then he must be shaved by the barber, i.e., himself. If the barber does shave himself, then the barber (namely, he himself) does not shave him.
The foundation is shaken Russell s Paradox Let S tall sets x : x R xu.
The foundation is shaken Russell s Paradox Let S tall sets x : x R xu. Question Is S P S?
The foundation is shaken Russell s Paradox Let S tall sets x : x R xu. Question Is S P S? If S P S, then by definition, S fails the criterion to belong to S, so S R S. If S R S, then by definition S is one of the elements of S, so S P S.
How to proceed? Two solutions: Outlaw recursion No set x can satisfy x P x, nor can we have x P y P x, etc. Limit construction of sets You cannot construct tx : Ppxqu; you must begin with some set T and then construct tx P T : Ppxqu. In either approach, the paradoxical tx : x R xu is not even a set at all.
Now, how can we be safe? Logicians recovered from that great danger, so they wanted to prove that their foundation could not prove any further paradoxes. There are three goals: Consistency: our system should not prove both P and P. Completeness: If Q is a true mathematical statement, we want our system to prove it. Soundness: our system should not prove anything false.
A Happy Example Basic logic with truth tables is consistent, complete, and sound. A theorem is a statement like P _ P or P ^ Q ùñ P that is true no matter what values of true and false are plugged in for the variables. P P P _ P T F T F T T P Q P ^ Q P ^ Q ñ P T T T T T F F T F T F T F F F T You can test a purported theorem simply by constructing a truth table. If it is true, then your final column is all T s.
A Philosophical Incompleteness Theorem Theorem There exists a statement that is true but unprovable. Proof. Let P denote the sentence this sentence is unprovable. Suppose for the sake of contradiction that P is false. Then P is not unprovable. So P is provable. So P is true, a contradiction. Thus P is true. Since P is true, P is unprovable is true. Thus P is unprovable.
We seem to have proved both P and P, so we re not being consistent. Is logic inconsistent? Is there a paradox at the heart of reality? Perhaps we should outlaw self-referential statements from philosophy. Perhaps we need to define our terms, especially provable. We ll leave the general problem to philosophers, and focus on the mathematical. But wait... Consider this argument: We just proved on the last slide that P is true. Therefore P is provable. Therefore P is not unprovable. Thus P is false.
Stripping proof to its essentials What do we need to prove a theorem? All we really need are Axioms the agreed-upon starting points. Rules of Inference that let you deduce a new fact from already-known facts. Axioms, rules of inference, and the proofs themselves are written in a formal language, an alphabet of symbols such as @, D,, 0, x, 1, P, ñ, ê, etc. The axioms and rules of inference together make up a formal system; we can think of it as an environment for doing proofs, or even as the blueprint for a machine to discover proofs.
Proofs and Theorems Definition Let F be a formal system. A proof in F is a string of statements in F such that each statement either is an axiom of F or can be obtained from earlier statements by one of F s rules of inference. A theorem of F is the last statement of a proof in F. For each formal system F, we can compute all its theorems as follows: 1 List all strings of symbols from F s language. 2 Check whether each string constitutes a proof. 3 Throw away the non-proofs. 4 From the proofs, take their last lines as the theorems.
Warning! A formal system F is just a dumb process, a set of starting points (axioms) and legal moves (rules of inference). Its theorems are just all the legal outcomes. So a theorem of F might be meaningless or even nonsense, depending on whether F s axioms and rules of inference are meaningful or not. A formal system with just one axiom ñ and one inference rule g and h yields gh would only produce as theorems strings like ññññññ, for example. However, some formal systems have axioms we d call true like 0 0 and valid rules of inference like P and P ñ Q yields Q.
Definition We call a formal system F sound if all of its theorems are true. Definition We call a formal system F consistent if it never proves both σ and σ. Definition We call a formal system F complete if it proves every true statement σ.
So our informal word provability has been formalized as provable in the formal system F. Our next goal Let s construct a mathematical version of the statement this statement is unprovable. We want a mathematical equivalent of P P is not a theorem of F. (How do we do this without an infinite regress?)
A List of Lists Every formal system has finitely many axioms and rules of inference, which we can list as one long string of symbols from some alphabet of length a. There are only countably many strings of symbols: a of length 1, a 2 of length 2, a 3 of length 3, etc. Thus there are only countably many formal systems, and we could list all formal systems as F 1, F 2, F 3,.... For each F i, some of its theorems might be positive integers. Let S i be the set of all positive integers proved by F i.
An Important Set Let F be a formal system. Now some of the theorems of F might just happen to look like k R S k. Define D tk P N : F proves k R S k u Now it is possible (but tedious) to build a formal system that produces as theorems the numbers in D and no other numbers. That formal system was somewhere in our list of all formal systems, so it is F n for some n. Thus D S n.
An incompleteness theorem S n tk P N : F proves k R S k u Is n P S n? n R S n ðñ F does not prove n R S n. Let P denote the statement n R S n. Thus P ðñ P is not a theorem of F. What can we prove now? If P is true, then P is true but unprovable in F. If P is false, then F proves P even though it s false. Theorem Every formal system F is either unsound or incomplete.
Theorem Every formal system F is either unsound or incomplete. So either F proves some false things or it doesn t prove some true things. This is impressive, perhaps, but the trouble is there s no mathematical symbol for truth. Gödel s Theorem replaces unsound with inconsistent. It also brings it close to home with basic arithmetic.
Encoding in numbers Every formal system is a manipulation of strings of symbols. Let s turn these strings into positive integers say, by writing out their ASCII (or unicode) symbols. Let b be the number of symbols available e.g., 16 for hexadecimal, 256 for ASCII characters, 2 32 for Unicode. this corresponds to this string s Npsq P N length of s rlog b pnpsqqs concatenation s!t Npsq b lenpt q Nptq All string operations, like breaking a text into lines, substituting for a variable, etc., can be done with arithmetic in N.
Let G be a formal system that encodes arithmetic. Then since every formal system F can be encoded into arithmetic, G can encode F and run its proofs! That is, G can test whether, in the formal system F, a string p is a proof of theorem t. G can break up p into lines. G can check whether each line is an axiom of F or follows from preceding lines by an inference rule of F. G can test whether the last line of the proof is t.
Most importantly for us right now, Lemma G can prove every true statement of the form x P S k. Proof. Suppose x P S k is true. Thus formal system F k proves x. Thus there exists a proof p of theorem x in system F k. G can verify that p is a proof of x in F k. Thus G can prove that F k proves x. Thus G can prove that x P S k.
The First Incompleteness Theorem Theorem (Gödel) If a formal system G is strong enough to encode arithmetic in N, then G is either inconsistent or incomplete. Note that any set of axioms and rules of logic you choose for doing mathematics is a candidate for G! Theorem (An Alternate Formulation) Let G be a formal system encoding arithmetic in N. If G is consistent, then there is some true statement about positive integers that G cannot prove.
Proof. G is itself a formal system, so by our argument from before, there exists an n such that S n tk P N : G proves ku. We also had a statement P n R S n. Recall that P ðñ G does not prove P. Suppose G is consistent; we ll prove G is incomplete. Suppose P is false. Then G does prove P. Thus G proves n R S n. On the other hand, since P is false, n P S n. But G proves all true statements of the form x P S k, so G proves n P S n. Thus G is inconsistent, a contradiction. Thus P is true, so G does not prove P, so G is incomplete.
Okay, but honestly, so what? So you can t prove some obscure true statement n R S n. Who cares?
A more concerning blow Note that we can express in the language of G: The sentence ConpGq G is consistent. The sentence P n R S n Thus G can prove the following logic: P is equivalent to G does not prove P. Suppose P is false. Then G does prove P, so G proves n R S n. Also, P n R S n, so n P S n. Thus G can prove n P S n. Thus G is inconsistent. Thus ConpGq ùñ P. So if G could prove ConpGq, it could also prove P. In that case, we would know that G is inconsistent by the First Theorem.
The Second Incompleteness Theorem Theorem (Gödel) Every formal system G strong enough to encode arithmetic in N is either inconsistent or cannot prove its own consistency.
Gödel s Theorems have some mathematical and philosophical implications. The old mathematical program of putting math on secure logical foundations has its limits: you never can be sure it won t be undermined. Some philosophers (e.g., John Lucas) argue that Gödel s Theorem proves minds are not the same as machines, and that computers will never achieve an artificial intelligence equal to the human mind. There has been a wealth of new true-but-unprovable statements discovered. recognizing whether certain groups are isomorphic recognizing whether certain manifolds are homeomorphic proving that a computer file is incompressible
Questions?