CS1001 Lecture 22. Overview Mechanizing Reasoning Mechanizing Reasoning G ö del ’ s Incompleteness Theorem G ö del ’ s Incompleteness Theorem.

CS1001CS1001

Lecture 22Lecture 22

OverviewOverview

Mechanizing ReasoningMechanizing Reasoning GGödel’s Incompleteness Theoremödel’s Incompleteness Theorem

Natural DeductionNatural Deduction

Start with Axioms (fundamental Start with Axioms (fundamental rules) and Factsrules) and Facts

Apply Rules of logicApply Rules of logic Deduce additional factsDeduce additional facts

Can Deduction be Can Deduction be Performed by Performed by Computer?Computer? Assuming all facts about the Assuming all facts about the

natural world were to be described natural world were to be described as facts in a logical system, can all as facts in a logical system, can all other facts be derived using the other facts be derived using the laws of math/logic?laws of math/logic?

Punch line: No! Punch line: No! AnyAny formal system formal system breaks down; there are truths that breaks down; there are truths that can not be derivedcan not be derived

Why?Why?

ParadoxParadox Self ReferenceSelf Reference As shown in the past, paradox and self As shown in the past, paradox and self

reference are fundamental parts of a reference are fundamental parts of a “real world” or generic system. We “real world” or generic system. We must allow these.must allow these.

If we don’t, we have no way of If we don’t, we have no way of reasoning about the infinite case and reasoning about the infinite case and therefore can’t develop generic therefore can’t develop generic algorithmsalgorithms

Mechanical ReasoningMechanical Reasoning Aristotle (~350BC): Aristotle (~350BC): OrganonOrganon

– We can explain logical deduction with We can explain logical deduction with rules of inference (syllogisms)rules of inference (syllogisms)

Every B is AEvery B is A

C is BC is B

C is AC is A

Every human is mortal.Every human is mortal.

Godel is human.Godel is human.

Godel is mortal.Godel is mortal.

More Mechanical More Mechanical ReasoningReasoning Euclid (~300BC): Euclid (~300BC): ElementsElements

– We can reduce geometry to a few We can reduce geometry to a few axioms and derive the rest by axioms and derive the rest by following rulesfollowing rules

Newton (1687): Newton (1687): Philosophiæ Philosophiæ Naturalis Principia MathematicaNaturalis Principia Mathematica – We can reduce the motion of objects We can reduce the motion of objects

(including planets) to following (including planets) to following axioms (laws) mechanicallyaxioms (laws) mechanically

Mechanical ReasoningMechanical Reasoning

Late 1800s – many Late 1800s – many mathematicians working on mathematicians working on codifying “laws of reasoning”codifying “laws of reasoning”– George Boole, George Boole, Laws of ThoughtLaws of Thought– Augustus De MorganAugustus De Morgan– Whitehead and RussellWhitehead and Russell

All All truetrue statements statements about number about number theorytheory

Perfect Axiomatic Perfect Axiomatic SystemSystem

Derives all true statements, and no false

statements starting from a finite number of axioms

and following mechanical inference rules.

IncompleteIncomplete Axiomatic Axiomatic SystemSystem

Derives some, but not all true

statements, and no false statements starting from a

finite number of axioms and following mechanical

inference rules.

incomplete

InconsistentInconsistent Axiomatic Axiomatic SystemSystem

Derives all true

statements, and some false statements starting from a

finite number of axioms and following mechanical

inference rules. some false

statements

Principia MathematicaPrincipia Mathematica Whitehead and Russell (1910– 1913)Whitehead and Russell (1910– 1913)

– Three Volumes, 2000 pagesThree Volumes, 2000 pages Attempted to axiomatize mathematical Attempted to axiomatize mathematical

reasoningreasoning– Define mathematical entities (like numbers) Define mathematical entities (like numbers)

using logicusing logic– Derive mathematical “truths” by following Derive mathematical “truths” by following

mechanical rules of inferencemechanical rules of inference– Claimed to be Claimed to be completecomplete and and consistentconsistent

All true theorems could be derivedAll true theorems could be derived No falsehoods could be derivedNo falsehoods could be derived

Russell’s ParadoxRussell’s Paradox

Some sets are not members of Some sets are not members of themselvesthemselves– In a certain town in Spain, there lives an excellent barber In a certain town in Spain, there lives an excellent barber

who shaves all the men who do not shave themselves. Who who shaves all the men who do not shave themselves. Who

shaves the barber?shaves the barber? Some sets are members of themselvesSome sets are members of themselves Call the set of all sets that are not Call the set of all sets that are not

members of themselvesmembers of themselves S S Is Is SS a member of itself? a member of itself?

Russell’s ParadoxRussell’s Paradox

SS: set of all sets that are not : set of all sets that are not members of themselvesmembers of themselves

Is Is SS a member of itself? a member of itself?– If If SS is an element of is an element of SS, then , then SS is a is a

member of itself and should not be in member of itself and should not be in SS..

– If If S S is not an element of is not an element of SS, then , then SS is is not a member of itself, and should be not a member of itself, and should be in in SS..

Ban Self-Reference?Ban Self-Reference?

Principia MathematicaPrincipia Mathematica attempted to attempted to resolve this paragraph by banning resolve this paragraph by banning self-referenceself-reference

Every set has a typeEvery set has a type– The lowest type of set can contain only The lowest type of set can contain only

“objects”, not “sets”“objects”, not “sets”– The next type of set can contain The next type of set can contain

objects and sets of objects, but not objects and sets of objects, but not sets of setssets of sets

Russell’s Resolution?Russell’s Resolution?

Set ::= SetSet ::= Setnn

SetSet00 ::= { ::= { xx | | xx is an is an ObjectObject } }

SetSetnn ::= { ::= { xx | | x x is an is an Object Object or a or a SetSetn n - 1 - 1 }}

SS: Set: Setnn

Is Is SS a member of itself? a member of itself?No, it is a Setn so, it can’t be a member of a Setn

Epimenides ParadoxEpimenides Paradox

Epidenides (a Cretan): Epidenides (a Cretan):

““All Cretans are liars.”All Cretans are liars.”

Equivalently:Equivalently:

““This statement is false.”This statement is false.”

Russell’s types can help with the set paradox, but not with this one.

GGödel’s Solutionödel’s Solution

All consistent axiomatic All consistent axiomatic formulations of number theory formulations of number theory include include undecidableundecidable propositions. propositions.

(GEB, p. 17)(GEB, p. 17)

undecidableundecidable – cannot be proven – cannot be proven either true or false inside the either true or false inside the system.system.

Kurt GKurt Gödelödel

Born 1906 in Brno (now Born 1906 in Brno (now Czech Republic, then Czech Republic, then Austria-Hungary)Austria-Hungary)

1931: publishes 1931: publishes Über Über formal unentscheidbare formal unentscheidbare Sätze der Principia Sätze der Principia Mathematica und Mathematica und verwandter Systemeverwandter Systeme ((On Formally Undecidable On Formally Undecidable Propositions of Principia Mathematica Propositions of Principia Mathematica and Related Systemsand Related Systems))

1939: flees 1939: flees ViennaVienna

Institute for Institute for Advanced Study, Advanced Study, PrincetonPrinceton

Died in 1978 – Died in 1978 – convinced convinced everything was everything was poisoned and poisoned and refused to eatrefused to eat

GGödel’s Theoremödel’s Theorem

In theIn the Principia Mathematica Principia Mathematica system, system, there are statements there are statements that cannot be proven either that cannot be proven either true or false.true or false.

GGödel’s Theoremödel’s Theorem

In In any interesting rigid any interesting rigid systemsystem, there are statements , there are statements that cannot be proven either that cannot be proven either true or false.true or false.

GGödel’s Theoremödel’s Theorem

All logical systems of any All logical systems of any complexity are incomplete: complexity are incomplete: there are statements that are there are statements that are truetrue that cannot be proven that cannot be proven within the system.within the system.

Proof – General IdeaProof – General Idea

Theorem: In the Principia Theorem: In the Principia Mathematica system, there Mathematica system, there are statements that cannot are statements that cannot be proven either true or be proven either true or false.false.

Proof: Find such a statementProof: Find such a statement

GGödel’s Statementödel’s Statement

GG: This statement of number : This statement of number theory does not have any theory does not have any proof in the system of proof in the system of Principia MathematicaPrincipia Mathematica..

GG is unprovable, but true! is unprovable, but true!

GGödel’s Proofödel’s Proof

GG: This statement of number theory : This statement of number theory does not have any proof in the does not have any proof in the system of system of PMPM..

If If GG were provable, then PM would be were provable, then PM would be inconsistent.inconsistent.

If If GG is unprovable, then PM would be is unprovable, then PM would be incomplete.incomplete.

PM cannot be complete and PM cannot be complete and consistent!consistent!

Finishing The ProofFinishing The Proof

Turn Turn GG into a statement in the into a statement in the Principia Mathematica Principia Mathematica systemsystem

Is Is PMPM powerful enough to powerful enough to express “express “This statement of This statement of number theory does not number theory does not have any proof in the have any proof in the system of system of PMPM.”?.”?

How to express How to express ““does not does not have any proof in the have any proof in the system of system of PMPM”” What does it mean to have a proof of What does it mean to have a proof of SS in PM? in PM?

– There is a sequence of steps that follow the There is a sequence of steps that follow the inference rules that starts with the initial axioms inference rules that starts with the initial axioms and ends with and ends with SS

What does it mean to What does it mean to notnot have have anyany proof of proof of SS in PM?in PM?– There is There is nono sequence of steps that follow the sequence of steps that follow the

inference rules that starts with the initial axioms inference rules that starts with the initial axioms and ends with and ends with SS

Can PM express Can PM express unprovability?unprovability?

There is There is nono sequence of steps that sequence of steps that follow the inference rules that follow the inference rules that starts with the initial axioms and starts with the initial axioms and ends with ends with SS

Can we express “This Can we express “This statement of number statement of number

theory”theory” We can write turn every We can write turn every

statement into a number, so statement into a number, so we can turn “This statement we can turn “This statement of number theory does not of number theory does not have any proof in the system have any proof in the system of of PMPM” into a number” into a number

GGödel’s Proofödel’s Proof

GG: This statement of number theory : This statement of number theory does not have any proof in the does not have any proof in the system of system of PMPM..

If If GG were provable, then PM would be were provable, then PM would be inconsistent.inconsistent.

If If GG is unprovable, then PM would be is unprovable, then PM would be incomplete.incomplete.

PM cannot be complete and PM cannot be complete and consistent!consistent!

GeneralizationGeneralization

All logical systems of any All logical systems of any complexity are incomplete: complexity are incomplete: there are statements that are there are statements that are truetrue that cannot be proven that cannot be proven within the system.within the system.

Practical ImplicationsPractical Implications

Mathematicians will Mathematicians will nevernever be be completely replaced by computerscompletely replaced by computers– There are mathematical truths that There are mathematical truths that

cannot be determined mechanicallycannot be determined mechanically– We can build a computer that will prove We can build a computer that will prove

only true theorems about number only true theorems about number theory, but if it cannot prove something theory, but if it cannot prove something we do not know that that is not a true we do not know that that is not a true theorem.theorem.

Russell’s DoctrineRussell’s Doctrine

““I wish to propose for the reader's I wish to propose for the reader's favourable consideration a doctrine favourable consideration a doctrine which may, I fear, appear wildly which may, I fear, appear wildly paradoxical and subversive. The doctrine paradoxical and subversive. The doctrine in question is this: that it is undesirable in question is this: that it is undesirable to believe a proposition when there is no to believe a proposition when there is no ground whatever for supposing it true.”ground whatever for supposing it true.”(Russell, (Russell, Introduction: On the Value of Introduction: On the Value of ScepticismScepticism, 1928), 1928)