One-Slide Summary Godel and Computability A proof of X in a formal - PDF document

One-Slide Summary Godel and Computability • A proof of X in a formal system is a sequence of steps starting with axioms. Each step must use a valid rule of inference and the final step must be X. • All interesting logical systems are incomplete : there are true statements that cannot be proven within the system. • An algorithm is a (mechanizable) procedure that always terminates. • A problem is decidable if there exists an algorithm to solve it. A problem is undecidable if it is not possible for an algorithm to exists that solves it. • The halting problem is undecidable. Halting Problems Hockey Team #2 Outline Epimenides Paradox • Gödel's Proof Epimenides (a Cretan): • Unprovability “All Cretans are liars.” • Algorithms • Computability Equivalently: • The Halting Problem “This statement is false.” Russell’s types can help with the set paradox, but not with these. #3 #4 Gödel’s Solution Kurt Gödel • Born 1906 in Brno (now Czech Republic, then All consistent axiomatic formulations of Austria-Hungary) number theory include undecidable • 1931: publishes Über propositions. formal unentscheidbare (GEB, p. 17) Sätze der Principia Mathematica und undecidable – cannot be proven either verwandter Systeme ( On true or false inside the system. Formally Undecidable Propositions of Principia Mathematica and Related Systems ) #5 #6

Gödel’s Theorem • 1939: flees Vienna • Institute for Advanced Study, In the Principia Mathematica Princeton system, there are statements that • Died in 1978 – cannot be proven either true or convinced false. everything was poisoned and refused to eat #7 #8 Gödel’s Theorem Gödel’s Theorem All logical systems of any In any interesting rigid system, complexity are incomplete : there are statements that cannot there are statements that are true that cannot be proven be proven either true or false. within the system. #9 #10 Proof – General Idea Gödel’s Statement •Theorem: In the Principia G : This statement does not have any proof in the Mathematica system, there system of Principia are statements that cannot be Mathematica . proven either true or false. •Proof: Find such a statement! G is unprovable, but true! Why? #11 #12

Gödel’s Statement Gödel’s Proof Idea G : This statement does not have any G : This statement does not have proof in the system of PM . any proof in the system. Possibilities: If G is provable, PM would be inconsistent. 1. G is true ⇒ G has no proof If G is unprovable, PM would be incomplete. System is incomplete 2. G is false ⇒ G has a proof Thus, PM cannot be complete and consistent! System is inconsistent #13 #14 Liberal Arts Trivia: Liberal Arts Trivia: Philosophy Women's Studies • This American-invented contact sport involves • In philosophy, this is a hypothetical being that two teams roller skating around an oval track. cannot be distinguished from a normal human It became popular in 1935 during the Great except that it lacks conscious experience, Depression and continued to grow in the '50s, qualia or sentience. That is, it does not feel '60s and '70s. Teams score points when the pain, but will react appropriately when poked jammer passes an opposing blocker or pivot . with a sharp stick. They are typically invoked The sport is strongly associated with third- in thought experiments in the philosophy of wave feminism. mind to argue against physicalist stances such as materialism or behaviorism, such as those of David Chalmers in The Conscious Mind . #15 #16 How to express “ does not have Finishing The Proof any proof in the system of PM ” • Turn G into a statement in the • What does “ have a proof of S in PM” mean? Principia Mathematica system – There is a sequence of steps that follow the inference rules that starts with the initial • Is PM powerful enough to express axioms and ends with S “ This statement does not have • What does it mean to “ not have any proof any proof in the PM system.”? of S in PM”? – There is no sequence of steps that follow the inference rules that starts with the initial axioms and ends with S #17 #18

Can we express Can PM express unprovability? “This statement” ? • There is no sequence of steps that • Yes! follows the inference rules that starts with the initial axioms and ends with S – Optional Reading: the TNT Chapter in GEB • Sequence of steps: • We can write turn every statement T 0 , T 1 , T 2 , ..., T N into a number, so we can turn “This statement does not have any proof T 0 must be the axioms T N must include S in the system” into a number Every step must follow from the previous using an inference rule #19 #20 Gödel’s Proof Generalization G : This statement does not have any All logical systems of any proof in the system of PM . complexity are incomplete: there are statements that are If G is provable, PM would be inconsistent. true that cannot be proven If G is unprovable, PM would be incomplete. within the system. PM can express G . Thus, PM cannot be complete and consistent! #21 #22 Practical Implications What does it mean for an axiomatic system to be complete and consistent? • Mathematicians will never be completely replaced by computers – There are mathematical truths that cannot Derives all true be determined mechanically statements, and no false – We can build a computer that will prove only statements starting from a true theorems about number theory, but if it finite number of axioms cannot prove something we do not know that and following mechanical that is not a true theorem. inference rules. #23 #24

Pick one: What does it mean for an axiomatic some false incomplete statements system to be complete and consistent? Derives Derives all true some, but not all true statements, and some false statements, and no false statements starting from a It means the axiomatic system is weak. statements starting from a finite number of axioms finite number of axioms and following mechanical and following mechanical Indeed, it is so weak, it cannot express: inference rules. inference rules. “This statement has no proof.” Incomplete Inconsistent Axiomatic System Axiomatic System #25 #26 Inconsistent Axiomatic System Liberal Arts Trivia: Chemistry • Also known as the rapture of the deep, this is Derives a completely reversible alteration in all true consciousness that occurs while scuba diving statements, and some false at depth. The state is quite similar to alcohol statements starting from a intoxication, and usually occurs at depths finite number of axioms beyond 100 feet. It is caused by breathing and following mechanical gasses that dissolve into nerve membranes inference rules. and disrupt transmission: apart from helium, some false statements all breathable gasses have a narcotic effect, Once you can prove one false statement, everything can be proven! false ⇒ anything which is greater as lipid solubility increases. #27 #28 Liberal Arts Trivia: History Liberal Arts Trivia: Music • Between 1945 and 1946, the political and • This is the name given to a chord consisting of military leadership of Nazi Germany, such as only the root note of the chord and the fifth, Hermann Göring, were tried in military usually played on an electric guitar through tribunals in this location. The trials had a an amplification process with distortion. They lasting legacy on international criminal law, are a key element of many styles of rock including the later Geneva Conventions. music. In these, the ratio between the frequencies of the root and fifth is simply 3:2, leading to a coherent sound and harmonics closely related to the original two notes when played through distortion. #29 #30

Computability Algorithms • Is there an algorithm that solves a problem? • What’s an algorithm ? • Computable ( decidable ) problems: A procedure that always terminates. – There is an algorithm that solves the problem. • What’s a procedure ? – Make a photomosaic, sorting, drug discovery, A precise (mechanizable) description of winning chess (it doesn’t mean we know the algorithm, but there is one) a process. • Uncomputable ( undecidable ) problems: – There is no algorithm that solves the problem. – There might be a procedure, but it doesn’t always terminate. #31 #32 The Halting Problem Input: a specification of a procedure P Are there any uncomputable problems? Output: If evaluating an application of P halts, output true. Otherwise, output false. #33 #34 Alan Turing (1912-1954) Halting Problem • Codebreaker at Bletchley Park Define a procedure halts? that takes a – Broke Enigma Cipher procedure specification and evaluates to – Perhaps more important than Lorenz #t if evaluating an application of the • Published On Computable Numbers … (1936) procedure would terminate, and to #f if – Introduced the Halting Problem evaluating an application of the would – Formal model of computation not terminate. (now known as “Turing Machine”) • After the war: convicted of homosexuality (then a crime in Britain), committed (define ( halts? proc) … ) suicide eating cyanide apple 5 years after Gödel’s proof! #35 #36