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CS344 Course Seminar By Amod Jog ( ) Chaitanya Amdekar( ) Aaditya Ramdas ( ) Abhishek Gupta(05d05015) This is not the Title of our Seminar

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Talk Outline & Motivation Consistency and Completeness Consistency and Completeness Historical References Historical References Self-reference. Why is it important? Self-reference. Why is it important? Gödel’s Incompleteness Theorem Gödel’s Incompleteness Theorem Paradoxes and analogies Paradoxes and analogies Basic concepts involved Basic concepts involved Implications of Gödel’s Theorem Implications of Gödel’s Theorem

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Consistency and Completeness An Axiomatic System An Axiomatic System Axioms Axioms Rules Rules Theorems Theorems Examples Examples Consistency Consistency An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. Completeness Completeness An axiomatic system will be called complete if every true statement is derivable as a theorem. An axiomatic system will be called complete if every true statement is derivable as a theorem.

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Historical References Attempts to create a formal number theoretic system which is consistent and complete Attempts to create a formal number theoretic system which is consistent and complete Gottlob Frege – using set theory Gottlob Frege – using set theory Russell’s Paradox due to self-reference Russell’s Paradox due to self-reference Russell and Whitehead - Principia Mathematica, attempt to remove self- reference. Russell and Whitehead - Principia Mathematica, attempt to remove self- reference. Problems encountered in their approach Problems encountered in their approach

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Self-Reference This sentence is false. This sentence is false. The following sentence is true. The following sentence is true. The preceding sentence is false. (Indirect self-reference) The Ouroboros, a dragon that bites its tail, is a symbol for self-reference The Ouroboros, a dragon that bites its tail, is a symbol for self-referenceOuroboros

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Gödel’s Incompleteness Theorem “To every w-consistent recursive class k of formulae there correspond recursive class signs r, such that neither uGenr nor Neg(uGenr) belongs to FLG (k) (where u is the free variable of r)” “To every w-consistent recursive class k of formulae there correspond recursive class signs r, such that neither uGenr nor Neg(uGenr) belongs to FLG (k) (where u is the free variable of r)” “All consistent axiomatic formulations of Number theory include undecidable propositions”

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Paradoxes and analogies Provability is a weaker notion than truth. Robots and self-destructing guitars! Liar’s Paradox Grelling’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?"

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Gödel’s Incompleteness Theorem Roadmap of Basic Concepts involved: Roadmap of Basic Concepts involved: Typographical Number Theory (TNT) Typographical Number Theory (TNT) The killer sentence G The killer sentence G Gödelized version of TNT Gödelized version of TNT ‘Theoremhood’ ‘Theoremhood’ Arithmoquining Arithmoquining

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TNT Typographical Number Theory Typographical Number Theory Axioms Axioms Axiom 1: Aa:~Sa=0 Axiom 2: Aa:(a+0)=a Axiom 3: Aa:Aa':(a+Sa')=S(a+a') Axiom 4: Aa:(a*0)=0 Axiom 5: Aa:Aa':(a*Sa')=((a*a')+a) Rules Rules There are > dozen rules There are > dozen rules Eg: ~~, or ~A = E~ Eg: ~~, or ~A = E~

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The killer sentence G Aim: Construct a sentence that can be used to disprove the assumption Aim: Construct a sentence that can be used to disprove the assumption Idea - “This statement is true but not provable.” Idea - “This statement is true but not provable.” Sentence G: “This statement is not a theorem of TNT.” Sentence G: “This statement is not a theorem of TNT.” We will now formalize TNT We will now formalize TNT

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Gödelized version of TNT TNT statement: ~Ea:a*a=SS0 TNT statement: ~Ea:a*a=SS0 Gödelized: Gödelized: TNT rule: The string ~~ can be deleted wherever it appears in any string. TNT rule: The string ~~ can be deleted wherever it appears in any string. Gödelized: The string can be deleted wherever it appears in any string. Gödelized: The string can be deleted wherever it appears in any string. Nothing has changed! Nothing has changed!

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‘Theoremhood’ We define a new property of natural numbers - "theoremhood" - every number either has it, or doesn't We define a new property of natural numbers - "theoremhood" - every number either has it, or doesn't A number has theoremhood iff it corresponds to a valid theorem of TNT—or, in other words, to a true statement about numbers. A number has theoremhood iff it corresponds to a valid theorem of TNT—or, in other words, to a true statement about numbers. A number has theoremhood if it is possible to create that number from our small set of axiom-numbers, by the application of our small set of function-rules A number has theoremhood if it is possible to create that number from our small set of axiom-numbers, by the application 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 (ie can be derived from axioms). The string 0=0 is a valid TNT theorem (ie can be derived from axioms). The number has the theoremhood property. The number has the theoremhood property.

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‘Theoremhood’ Remember : We assume TNT can express any mathematical statement, no matter how complex, including Remember : We assume TNT can express any mathematical statement, no matter how complex, including " has theoremhood“ " has theoremhood“ The 3 forms again : The 3 forms again : " has theoremhood" is true. " has theoremhood" is true. The TNT string for " has theoremhood" is a valid TNT theorem. The TNT string for " has theoremhood" is a valid TNT theorem. The Gödel number for the TNT string for " has theoremhood," has theoremhood. The Gödel number for the TNT string for " has theoremhood," has theoremhood.

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Arithmoquining A TNT string cannot possibly be big enough to "contain" its own Gödel number. A TNT string cannot possibly be big enough to "contain" its own Gödel number. We resort to “arithmoquining”, the best step! We resort to “arithmoquining”, the best step! It fulfills our aim : Get TNT Sentences About TNT Sentences It fulfills our aim : Get TNT Sentences About TNT Sentences Arithmoquining: Take any TNT sentence which has a free variable say ‘a’, and replace all its occurrences in the sentence with the Gödel number of the sentence. Arithmoquining: Take any TNT sentence which has a free variable say ‘a’, and replace all its occurrences in the sentence with the Gödel number of the sentence. For e.g. For e.g. a=S0 is the TNT sentence then is its Gödel number. Arithmoquining then gives, “ =1”

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Arithmoquining In words, In words, T: a = S0 T: a = S0 A: The Gödel number of Sentence T is 1. A: The Gödel number of Sentence T is 1. A more complicated example A more complicated example T: a = SS0 * a - SSSS0 T: a = SS0 * a - SSSS0 A: Sentence T is 2 times sentence T minus 4. A: Sentence T is 2 times sentence T minus 4. Sentence T is neither true nor false, since a is unspecified. Sentence A, the arithmoquine of Sentence T, is a blatantly false statement about a specific number. Sentence T is neither true nor false, since a is unspecified. Sentence A, the arithmoquine of Sentence 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: Sentence T is not a valid TNT theorem-number. A: Sentence T is not a valid TNT theorem-number.

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Finally – G! Killer-G : Killer-G : T: The arithmoquine of a is not a valid TNT theorem- number. T: The arithmoquine of a is not a valid TNT theorem- number. A: The arithmoquine of Sentence T is not a valid TNT theorem-number. A: The arithmoquine of Sentence T is not a valid TNT theorem-number. G: The arithmoquine of "The arithmoquine of a is not a valid TNT theorem-number" is not a valid TNT theorem-number. G: The arithmoquine of "The arithmoquine of a is not a valid TNT theorem-number" is not a valid TNT theorem-number. So in the end, we have G for the TNT and as discussed earlier it is indeed true but not provable. So in the end, we have G for the TNT and as discussed earlier it is indeed true but not provable. Hence, TNT, is incomplete!!! Hence, TNT, is incomplete!!!

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Conclusion (Implications) Mathematics may have multiple truths, some of which are contradictory. Mathematics may have multiple truths, some of which are contradictory. E.g.: difference between Euclidean and non- Euclidean geometry E.g.: difference between Euclidean and non- Euclidean geometry The fact that we cannot create a formal system which can capture all of mathematical truth casts serious doubt on the objectiveness of such truth The fact that we cannot create a formal system which can capture all of mathematical truth casts serious doubt on the objectiveness of such truth

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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 consistent logic. Each person has his own peculiarly different consistent logic. Given that you have only your own logic to judge itself, how can you tell if your own logic is ‘peculiar’ or not? Given that you have only your own logic to judge itself, how can you tell if your own logic is ‘peculiar’ or not? Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self- fulfilling prophecies, though the process is by no means inevitable. Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self- fulfilling prophecies, though the process is by no means inevitable.

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Conclusion (Implications) There are many people who believe that the human mind, based on neurons and physical principles, is just a very sophisticated formal system. There are many people who believe that the human mind, based on neurons and physical principles, is just a very sophisticated formal system. Does Gödel's theorem imply the existence of facts that must be true, but that our minds can never prove? Or believe? Or conceive? Does Gödel's theorem imply the existence of facts that must be true, but that our minds can never prove? Or believe? Or conceive? Limitation on science, knowledge and mathematics? Limitation on science, knowledge and mathematics? Not likely, knowledge in science is rarely represented in terms of axioms. Not likely, knowledge in science is rarely represented in terms of axioms.

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Conclusion (Implications) Probably the most common fallacy- AI is impossible (rather, machines cannot think) Probably the most common fallacy- AI is impossible (rather, machines cannot think) Axiomatic systems are equivalent to abstract computers (Turing machines) Axiomatic systems are equivalent to abstract computers (Turing machines) Since there are true propositions which cannot be deduced by interesting axiomatic systems, there are results which cannot be obtained by computers, either Since there are true propositions which cannot be deduced by interesting axiomatic systems, there are results which cannot be obtained by computers, either But we can obtain those results, so our thinking cannot be adequately represented by a computer, or an axiomatic system But we can obtain those results, so our thinking cannot be adequately represented by a computer, or an axiomatic system Therefore, we are not computational machines, and none of them could be as intelligent as we are Therefore, we are not computational machines, and none of them could be as intelligent as we are

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References Hofstadter,Douglas Godel, Escher, Bach: an Eternal Golden Braid Hofstadter,Douglas Godel, Escher, Bach: an Eternal Golden Braid bene/godels-theorem.html bene/godels-theorem.html bene/godels-theorem.html bene/godels-theorem.html felder/public/kenny/papers/godel.html felder/public/kenny/papers/godel.html felder/public/kenny/papers/godel.html felder/public/kenny/papers/godel.html

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