Gdels Theorem Anders O.F . Hendrickson Department of Mathematics - PowerPoint PPT Presentation

A Little History Incompleteness The First Theorem The Second Theorem Implications Gödel’s Theorem Anders O.F . Hendrickson Department of Mathematics and Computer Science Concordia College, Moorhead, MN Math/CS Colloquium, November 15, 2011

A Little History Incompleteness The First Theorem The Second Theorem Implications Outline A Little History 1 Incompleteness 2 The First Theorem 3 The Second Theorem 4 Implications 5

A Little History Incompleteness The First Theorem The Second Theorem Implications My Source:

A Little History Incompleteness The First Theorem The Second Theorem Implications 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 ♣ a , b q could be thought of as the set tt a ✉ , t a , b ✉✉ . Even the integers could be modeled recursively with sets: 0 ✏ ❍ ; 1 ✏ t 0 ✉ ; 2 ✏ t 0 , 1 ✉ ; 3 ✏ t 0 , 1 , 2 ✉ , etc.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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?

A Little History Incompleteness The First Theorem The Second Theorem Implications 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.

P ❘ ❘ P P A Little History Incompleteness The First Theorem The Second Theorem Implications The foundation is shaken Russell’s Paradox Let S ✏ t all sets x : x ❘ x ✉ .

P ❘ ❘ P A Little History Incompleteness The First Theorem The Second Theorem Implications The foundation is shaken Russell’s Paradox Let S ✏ t all sets x : x ❘ x ✉ . Question Is S P S ?

A Little History Incompleteness The First Theorem The Second Theorem Implications The foundation is shaken Russell’s Paradox Let S ✏ t all sets x : x ❘ x ✉ . Question Is S P S ? If S P S , then by definition, S fails the criterion to belong to S , so S ❘ S . If S ❘ S , then by definition S is one of the elements of S , so S P S .

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 t x : P ♣ x q✉ ; you must begin with some set T and then construct t x P T : P ♣ x q✉ . In either approach, the paradoxical t x : x ❘ x ✉ is not even a set at all.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 Little History Incompleteness The First Theorem The Second Theorem Implications A Happy Example Basic logic with truth tables is consistent, complete, and sound. A theorem is a statement like “ P ❴ ✥ P ” or “ P ❫ Q ù ñ P ” that is true no matter what values of true and false are plugged in for the variables. P Q P ❫ Q P ❫ Q ñ P P ✥ P P ❴ ✥ P T T T T T F T T F F T F T 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 Little History Incompleteness The First Theorem The Second Theorem Implications 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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. 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 ❅ , ❉ , ✏ , 0, x , ✶ , 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications 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: List all strings of symbols from F ’s language. 1 Check whether each string constitutes a proof. 2 Throw away the non-proofs. 3 From the proofs, take their last lines as the theorems. 4

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 .”

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 .

A Little History Incompleteness The First Theorem The Second Theorem Implications 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 Little History Incompleteness The First Theorem The Second Theorem Implications 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 .