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

1

A Little History

2

Incompleteness

3

The First Theorem

4

The Second Theorem

5

Implications

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, bq could be thought of as the set tta✉, ta, b✉✉. Even the integers could be modeled recursively with sets: 0 ✏ ❍; 1 ✏ t0✉; 2 ✏ t0, 1✉; 3 ✏ t0, 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.

A Little History Incompleteness The First Theorem The Second Theorem Implications

The foundation is shaken

Russell’s Paradox Let S ✏ tall sets x : x ❘ x✉. P P ❘ ❘ P

A Little History Incompleteness The First Theorem The Second Theorem Implications

The foundation is shaken

Russell’s Paradox Let S ✏ tall sets x : x ❘ x✉. Question Is S P S? P ❘ ❘ P

A Little History Incompleteness The First Theorem The Second Theorem Implications

The foundation is shaken

Russell’s Paradox Let S ✏ tall 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 tx : P♣xq✉; you must begin with some set T and then construct tx P T : P♣xq✉. In either approach, the paradoxical tx : 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 ✥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 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

• n 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,

• r 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:

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.

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, a2 of length 2, a3 of length 3, etc. Thus there are only countably many formal systems, and we could list all formal systems as F1, F2, F3, . . .. For each Fi, some of its “theorems” might be positive

• integers. Let Si be the set of all positive integers “proved”

by Fi.

A Little History Incompleteness The First Theorem The Second Theorem Implications

An Important Set

Let F be a formal system. Now some of the theorems of F might just happen to look like “k ❘ Sk.” Define D ✏ tk P N : F proves ‘k ❘ Sk’✉ 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 Fn for some n. Thus D ✏ Sn.

A Little History Incompleteness The First Theorem The Second Theorem Implications

An incompleteness theorem

Sn ✏ tk P N : F proves ‘k ❘ Sk’✉ Is n P Sn? n ❘ Sn ð ñ F does not prove ‘n ❘ Sn.’ Let P denote the statement ‘n ❘ Sn.’ Thus P ð ñ “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.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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, 232 for Unicode. this corresponds to this string s N♣sq P N length of s rlogb♣N♣sqqs concatenation s✦t N♣sq ✝ blen♣tq N♣tq All string operations, like breaking a text into lines, substituting for a variable, etc., can be done with arithmetic in N.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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

• r follows from preceding lines by an inference rule of F.

G can test whether the last line of the proof is t.

A Little History Incompleteness The First Theorem The Second Theorem Implications

Most importantly for us right now, Lemma G can prove every true statement of the form ‘x P Sk.’ Proof. Suppose ‘x P Sk’ is true. Thus formal system Fk proves x. Thus there exists a proof p of theorem x in system Fk. G can verify that p is a proof of x in Fk. Thus G can prove that “Fk proves x.” Thus G can prove that x P Sk.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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.

A Little History Incompleteness The First Theorem The Second Theorem Implications

Proof. G is itself a formal system, so by our argument from before, there exists an n such that Sn ✏ tk P N : G proves k✉. We also had a statement P ✏ “n ❘ Sn.” Recall that P ð ñ “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 ❘ Sn. On the other hand, since P is false, n P Sn. But G proves all true statements of the form x P Sk, so G proves n P Sn. Thus G is inconsistent, a contradiction.

Thus P is true, so G does not prove P, so G is incomplete.

A Little History Incompleteness The First Theorem The Second Theorem Implications

Okay, but honestly, so what? So you can’t prove some obscure true statement “n ❘ Sn.” Who cares?

A Little History Incompleteness The First Theorem The Second Theorem Implications

A more concerning blow

Note that we can express in the language of G: The sentence Con♣Gq ✏ “G is consistent.” The sentence P ✏ “n ❘ Sn” 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 ❘ Sn. Also, P ✏ “n ❘ Sn,” so n P Sn. Thus G can prove n P Sn. Thus G is inconsistent.

Thus Con♣Gq ù ñ P. So if G could prove Con♣Gq, it could also prove P. In that case, we would know that G is inconsistent by the First Theorem.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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.

A Little History Incompleteness The First Theorem The Second Theorem Implications

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

A Little History Incompleteness The First Theorem The Second Theorem Implications

Questions?