CS344 Course SeminarCS344 Course SeminarByBy

Amod Jog (05005009)Amod Jog (05005009)Chaitanya Amdekar(05005018)Chaitanya Amdekar(05005018)

Aaditya Ramdas (05005027)Aaditya Ramdas (05005027)Abhishek Gupta(05d05015)Abhishek Gupta(05d05015)

This is not the Title of our This is not the Title of our SeminarSeminar

Talk Outline & MotivationTalk Outline & Motivation

Consistency and CompletenessConsistency and CompletenessHistorical ReferencesHistorical ReferencesSelf-reference. Why is it important?Self-reference. Why is it important?GGödel’s Incompleteness Theoremödel’s Incompleteness TheoremParadoxes and analogiesParadoxes and analogiesBasic concepts involvedBasic concepts involved Implications of Gödel’s TheoremImplications of Gödel’s Theorem

Consistency and CompletenessConsistency and Completeness

An Axiomatic SystemAn Axiomatic System Axioms Axioms RulesRules Theorems Theorems ExamplesExamples

ConsistencyConsistency An axiomatic system is said to be An axiomatic system is said to be consistentconsistent if it if it

lacks contradiction, i.e. the ability to derive both a lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. statement and its negation from the system's axioms.

CompletenessCompleteness An axiomatic system will be called An axiomatic system will be called completecomplete if every if every

true statement is derivable as a theorem.true statement is derivable as a theorem.

Historical ReferencesHistorical References

Attempts to create a formal number Attempts to create a formal number theoretic system which is consistent and theoretic system which is consistent and completecomplete

Gottlob Frege – using set theory Gottlob Frege – using set theory Russell’s Paradox due to self-referenceRussell’s Paradox due to self-referenceRussell and Whitehead - Principia Russell and Whitehead - Principia

Mathematica, attempt to remove self-Mathematica, attempt to remove self-reference.reference.

Problems encountered in their approachProblems encountered in their approach

Self-ReferenceSelf-ReferenceThis sentence is false.This sentence is false.The following sentence is true.The following sentence is true.

The preceding sentence is false.The preceding sentence is false.

(Indirect self-reference)(Indirect self-reference)The The OuroborosOuroboros, a dragon that bites its tail, , a dragon that bites its tail,

is a symbol for self-reference is a symbol for self-reference

GGödel’s Incompleteness Theoremödel’s Incompleteness Theorem

““To every w-consistent recursive class k of To every w-consistent recursive class k of formulae there correspond recursive class formulae there correspond recursive class signs r, such that neither uGenr nor signs r, such that neither uGenr nor Neg(uGenr) belongs to FLG (k) (where u Neg(uGenr) belongs to FLG (k) (where u is the free variable of r)”is the free variable of r)”

“All consistent axiomatic formulations of Number theory include undecidable propositions”

Paradoxes and analogiesParadoxes and analogies

Provability is a weaker notion than truth.Robots and self-destructing guitars!Liar’s ParadoxGrelling’s Paradox “Is it possible to formally codify the

Universe in such a way that our system of coding will be both complete and consistent?"

GGödel’s Incompleteness Theoremödel’s Incompleteness Theorem

Roadmap of Basic Concepts involved:Roadmap of Basic Concepts involved:Typographical Number Theory (TNT)Typographical Number Theory (TNT)The killer sentence GThe killer sentence G GGödelized version of TNTödelized version of TNT ‘‘Theoremhood’ Theoremhood’ ArithmoquiningArithmoquining

TNTTNT

Typographical Number TheoryTypographical Number Theory AxiomsAxioms

Axiom 1: Aa:~Sa=0 Axiom 1: Aa:~Sa=0 Axiom 2: Aa:(a+0)=a Axiom 2: Aa:(a+0)=a Axiom 3: Aa:Aa':(a+Sa')=S(a+a') Axiom 3: Aa:Aa':(a+Sa')=S(a+a') Axiom 4: Aa:(a*0)=0 Axiom 4: Aa:(a*0)=0 Axiom 5: Aa:Aa':(a*Sa')=((a*a')+a) Axiom 5: Aa:Aa':(a*Sa')=((a*a')+a)

RulesRules There are > dozen rulesThere are > dozen rules Eg: ~~, or ~A = E~Eg: ~~, or ~A = E~

The killer sentence GThe killer sentence G

Aim: Construct a sentence that can be Aim: Construct a sentence that can be used to disprove the assumptionused to disprove the assumption

Idea - “This statement is true but not Idea - “This statement is true but not provable.”provable.”

Sentence G: “This statement is not a Sentence G: “This statement is not a theorem of TNT.”theorem of TNT.”

We will now formalize TNTWe will now formalize TNT

GGödelized version of TNTödelized version of TNT

TNT statement: ~Ea:a*a=SS0 TNT statement: ~Ea:a*a=SS0 Gödelized: Gödelized:

223333262636262236262111123123666223333262636262236262111123123666TNT rule: The string ~~ can be deleted TNT rule: The string ~~ can be deleted

wherever it appears in any string. wherever it appears in any string. Gödelized: The string 223223 can be Gödelized: The string 223223 can be

deleted wherever it appears in any string. deleted wherever it appears in any string. Nothing has changedNothing has changed!!

‘‘Theoremhood’Theoremhood’ We define a new property of natural numbers - We define a new property of natural numbers -

"theoremhood" - every number either has it, or doesn't"theoremhood" - every number either has it, or doesn't A number has A number has theoremhoodtheoremhood iff it corresponds to a valid iff it corresponds to a valid

theorem of TNT—or, in other words, to a true statement theorem of TNT—or, in other words, to a true statement about numbers. about numbers.

A number has A number has theoremhoodtheoremhood if it is possible to create that if it is possible to create that number from our small set of axiom-numbers, by the number from our small set of axiom-numbers, by the application of our small set of function-rulesapplication of our small set of function-rules

The 3 forms :The 3 forms : "Zero equals zero" is true. "Zero equals zero" is true. The string 0=0 is a valid TNT theorem (The string 0=0 is a valid TNT theorem ( ieie can be derived from can be derived from

axioms). axioms). The number 666111666 has the theoremhood property.The number 666111666 has the theoremhood property.

‘‘Theoremhood’Theoremhood’

Remember : We assume TNT can express Remember : We assume TNT can express any any mathematical statement, no matter how mathematical statement, no matter how complexcomplex, including , including

"666111666 has theoremhood“ "666111666 has theoremhood“ The 3 forms again :The 3 forms again :

"666111666 has theoremhood" is true. "666111666 has theoremhood" is true. The TNT string for "666111666 has theoremhood" The TNT string for "666111666 has theoremhood"

is a valid TNT theorem. is a valid TNT theorem. The Gödel number for the TNT string for The Gödel number for the TNT string for

"666111666 has theoremhood," has theoremhood. "666111666 has theoremhood," has theoremhood.

ArithmoquiningArithmoquining A TNT string cannot possibly be big enough to "contain" A TNT string cannot possibly be big enough to "contain"

its own Gödel number.its own Gödel number. We resort to “arithmoquining”, the best step!We resort to “arithmoquining”, the best step! It fulfills our aim : It fulfills our aim : Get TNT Sentences About TNT Get TNT Sentences About TNT

SentencesSentences ArithmoquiningArithmoquining: Take any TNT sentence which has a : Take any TNT sentence which has a

free variable say ‘a’, and replace all its occurrences in free variable say ‘a’, and replace all its occurrences in the sentence with the Gthe sentence with the Gödel number of the sentence.ödel number of the sentence.

For e.g. For e.g. a=S0 is the TNT sentence then 262111123666 is its Ga=S0 is the TNT sentence then 262111123666 is its Gödel ödel number. Arithmoquining then gives,number. Arithmoquining then gives,““262111123666=1”262111123666=1”

ArithmoquiningArithmoquining In words,In words,

T: a = S0T: a = S0 A: A: The Gödel number of Sentence TThe Gödel number of Sentence T is 1. is 1.

A more complicated exampleA more complicated example T: a = SS0 * a - SSSS0 T: a = SS0 * a - SSSS0 A: A: Sentence TSentence T is 2 times is 2 times sentence T sentence T minus 4. minus 4.

Sentence T is neither true nor false, since a is Sentence T is neither true nor false, since a is unspecified. Sentence A, the arithmoquine of Sentence unspecified. Sentence A, the arithmoquine of Sentence T, is a blatantly false statement about a specific number. T, is a blatantly false statement about a specific number.

Possible Attempt at G:Possible Attempt at G: T: a is not a valid TNT theorem-number. T: a is not a valid TNT theorem-number. A: A: Sentence TSentence T is not a valid TNT theorem-number. is not a valid TNT theorem-number.

Finally – G!Finally – G! Killer-G :Killer-G :

T: The arithmoquine of a is not a valid TNT theorem-T: The arithmoquine of a is not a valid TNT theorem-number.number.

A: The arithmoquine of A: The arithmoquine of Sentence T Sentence T is not a valid TNT is not a valid TNT theorem-number. theorem-number.

G: The arithmoquine of "The arithmoquine of a is not G: The arithmoquine of "The arithmoquine of a is not a valid TNT theorem-number" is not a valid TNT a valid TNT theorem-number" is not a valid TNT theorem-number. theorem-number.

So in the end, we have G for the TNT and as So in the end, we have G for the TNT and as discussed earlier it is indeed true but not discussed earlier it is indeed true but not provable.provable.

Hence, TNT, is incomplete!!!Hence, TNT, is incomplete!!!

Conclusion (Implications)Conclusion (Implications)

Mathematics may have multiple truths, Mathematics may have multiple truths, some of which are contradictory.some of which are contradictory.E.g.: difference between Euclidean and non-E.g.: difference between Euclidean and non-

Euclidean geometryEuclidean geometryThe fact that we cannot create a formal The fact that we cannot create a formal

system which can capture all of system which can capture all of mathematical truth casts serious doubt on mathematical truth casts serious doubt on the objectiveness of such truththe objectiveness of such truth

Conclusion (Implications)Conclusion (Implications)

How can you figure out if you are sane? How can you figure out if you are sane? Each person has his own peculiarly different Each person has his own peculiarly different

consistent logic.consistent logic. Given that you have only your own logic to judge Given that you have only your own logic to judge

itself, how can you tell if your own logic is itself, how can you tell if your own logic is ‘peculiar’ or not?‘peculiar’ or not?

Once you begin to question your own sanity, you Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no fulfilling prophecies, though the process is by no means inevitable. means inevitable.

Conclusion (Implications)Conclusion (Implications)

There are many people who believe that the There are many people who believe that the human mind, based on neurons and physical human mind, based on neurons and physical principles, is just a very sophisticated formal principles, is just a very sophisticated formal system. system.

Does Gödel's theorem imply the existence of Does Gödel's theorem imply the existence of facts that must be true, but that our minds can facts that must be true, but that our minds can never prove? Or believe? Or conceive?never prove? Or believe? Or conceive?

Limitation on science, knowledge and Limitation on science, knowledge and mathematics? mathematics? Not likely, knowledge in science is rarely represented Not likely, knowledge in science is rarely represented

in terms of axioms.in terms of axioms.

Conclusion (Implications)Conclusion (Implications)

Probably the most common fallacy- AI is Probably the most common fallacy- AI is impossible (rather, machines cannot think)impossible (rather, machines cannot think) Axiomatic systems are equivalent to abstract Axiomatic systems are equivalent to abstract

computers (Turing machines)computers (Turing machines) Since there are true propositions which cannot be Since there are true propositions which cannot be

deduced by interesting axiomatic systems, there are deduced by interesting axiomatic systems, there are results which cannot be obtained by computers, eitherresults which cannot be obtained by computers, either

But But wewe can obtain those results, so our thinking can obtain those results, so our thinking cannot be adequately represented by a computer, or cannot be adequately represented by a computer, or an axiomatic systeman axiomatic system

Therefore, we are not computational machines, and Therefore, we are not computational machines, and none of them could be as intelligent as we arenone of them could be as intelligent as we are

ReferencesReferences

Hofstadter,Hofstadter,Douglas Godel, Escher, Bach: Douglas Godel, Escher, Bach: an Eternal Golden Braidan Eternal Golden Braid

http://www.cscs.umich.edu/~crshalizi/notahttp://www.cscs.umich.edu/~crshalizi/notabene/godels-theorem.htmlbene/godels-theorem.html

http://www4.ncsu.edu/unity/lockers/users/f/http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/godel.htmlfelder/public/kenny/papers/godel.html

http://www.miskatonic.org/godel.htmlhttp://www.miskatonic.org/godel.html