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Formalism II „ Gödel’S Incompleteness Theorems Hilbert's Programme and Its Collapse „ Curry

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Formalism II „ Gödel’S Incompleteness Theorems Hilbert's Programme and Its Collapse „ Curry Outline Hilbert’s programme Formalism II Gödel’s incompleteness theorems Hilbert's Programme and its collapse Curry 13 April 2005 Joost Cassee Gustavo Lacerda Martijn Pennings 1 2 Previously on PhilMath… Foundations of Mathematics Axiomatic methods Logicism: Principia Mathematica Logicism Foundational crisis Formalism Russell’s paradox Term Formalism Cantor’s ‘inconsistent multitudes’ Game Formalism Law of excluded middle: P or not-P Intuitionism Deductivism = Game Formalism + preservation of truth Consistency paramount Quiet period for Hilbert: 1905 ~ 1920 No intuition needed Rejection of logicism 3 4 1 Birth of Finitism Finitary arithmetic Hermann Weyl: “The new foundational crisis in Basic arithmetic: 2+3=5, 7+7≠10, … mathematics” (1921) Intuitionistic restrictions – crippled mathematics Bounded quantifiers Bounded: 100 < p < 200 and p is prime Hilbert’s defense: finitism Unbounded: p > 100 and p and p+2 are prime ‘Basing the if of if-then-ism’ Core: finitary arithmetic All sentences effectively decidable Construction of all mathematics from core Finite algorithm Epistemologically satisfying 5 6 Finitary arithmetic – ontology Ideal mathematics – the rest Meaningful, independent of logic – Kantian Game-Formalistic rules Correspondance to finitary arithmetic Concrete symbols themselves: |, ||, |||, … … but not too concrete – not physical Restriction: consistency with finitary arithmetic Essential to human thought – not reducible Formal systems described in finitary arithmetic Finitary proof of consistency 7 8 2 What is Mathematics? Gödel's biography Born: 28 April 1906 in Brünn, Austria-Hungary (now Brno, Czech Republic) Inventory of provable formulas happy childhood. Epistemological programme rheumatic fever at age 6 Proof of known body of mathematics completed his doctoral dissertation under Hahn's supervision in 1929 submitting a thesis proving the completeness of the first order functional 1920’s: optimism and elaboration calculus Proof strategies He became a member of the faculty of the University of Vienna in 1930, Technical work: proofs where he belonged to the school of logical positivism until 1938. Until Gödel … 9 10 Gödel’s biography Gödel’s biography In 1931, he published his famous incompleteness results in “Über formal In 1940, he arrived with his wife in America, via Russia and Japan. unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” In America, he was at the Institute for Advanced Study, where he befriended Einstein. Nervous breakdown in 1934. Spent several months in a sanatorium, recovering from depression. Became a US citizen in 1948, despite telling the judge he had found an inconsistency in the US constitution married Adele Porkert in 1938 Gödel and Einstein made each other company, but they were intellectually Gödel was not Jewish, although people often thought he was. isolated. He decided to run away because he thought himself not healthy enough to Near the end of his life Gödel became convinced that he was being serve in the German army, and was afraid they wouldn't believe him. poisoned and, starved himself to death on 14 January 1978. 11 12 3 Incompleteness Incompleteness Hilbert's program (1921): to find a definitive axiomatization of mathematics. To settle, once and for all, on a system for The first incompleteness theorem: doing mathematics: (1) To enumerate all the symbols used in mathematics and logic says that (4) is impossible. (2) To characterize which combinations of symbols are 'meaningful' in classical This has serious consequences to the epistemic goals of Hilbert's program: mathematics Hilbert’s dream system cannot exist. (3) To supply a construction procedure which enables one to construct all the While it shatters the dream of a single system for mathematics, one could formulas which correspond to the 'provable statements' of classical mathematics. This procedure is called 'proving'. say that such a dream system was not at the core of Hilbert’s program. (4) To prove that the method in 3 is complete: i.e. all true formulas will be proven by For any candidate formal system, we can find a sentence that the system this method. does not decide, even though we can see that the sentence is true. Paraphrased from John van Neumann’s 1931 summary of Hilbert's program 13 14 Incompleteness Incompleteness Gödel’s First Incompleteness Theorem, more formally: Goedel's Second Incompleteness Theorem Let T be a formal deductive system that contains a certain amount of arithmetic (PA), which has an effective syntax (1,2), and a proof checker / sound proof search If T is consistent, then one cannot derive "T is consistent" in T. algorithm (3). Then Goedel showed that: there is a sentence G in the language of T such that: (1) if T is consistent, then G is not a theorem of T, and Thus the Gödel sentence from the 1st Theorem is not just an exception: the consistency of T itself can’t be proven in T. (2) if T is omega-consistent (slightly stronger), then ~G is not a theorem of T. This "Goedel sentence" G is roughly a formalization of "G is not provable in T". Note This is a serious attack on Hilbert's program. the self-reference! If PA is a formalization of (ideal) arithmetic, then Hilbert requires a finitary If T is consistent, then G is true but not provable. This is because if G were false, T proof of the consistency of PA. But if PA is consistent, then "PA is would be inconsistent! (it proves a false statement, namely G) consistent" cannot be proved in PA, let alone in its finitary fragment. So if we assume that all well-formed sentences are either true or false in T, then G must be true in T. 15 16 4 Impact on Philosophy Impact on Mathematics Bernays-Gödel Conclusion: Finitary arithmetic is not enough to prove the consistency of classical mathematics. Gödel's Incompleteness Theorems have had almost no impact on the working mathematician. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than "interesting statements about number theory are much more likely to that envisaged by Hilbert. resemble "2 + 2 = 4" than Gödel's vexing "P." " - Jordan Ellenberg An “alternative” conclusion :"finitary arithmetic is inherently informal" - Detlefsen (most purported consequences are unjustified) 17 18 Fallacious Conclusions Fallacious Conclusions - "If we cannot prove that T is consistent, then no theorem proven in T can Some people will conclude that "it's best to leave things unformalized", but be considered as definitely proven. In fact, T could be inconsistent." that's like closing your eyes to the problem. A theorem of a theory T *never* has more guarantee than its axioms, and the fact that we can't prove axioms has nothing to do with Gödel. And *even* if “Absolute Knowledge is Impossibe” T could prove its own consistency, that would be just another theorem of T. [From W. Carnielli: "OS TEOREMAS DE GÖDEL, E O QUE ELES NÃO SIGNIFICAM"] 19 20 5 Fallacious Conclusions Some people mistakenly interpret Gödel's theorem as saying that mathematics cannot be formalized. Haskell B. Curry BUT: computer formalizations of mathematics in Mizar, Coq, Isabelle, etc. In fact, Gödel's incompleteness theorem itself has been formalized in most * 12 Sept 1900 Millis, Massachusetts of these systems. † 1 Sept 1982 State College, Pennsylvania Theorem PAIncomplete : exists f : Formula, (forall v : nat, ~ In v (freeVarFormula LNT f)) /\ ~ (SysPrf PA f \/ SysPrf PA (notH f)). [Thanks to Russell O’ Connor] 21 22 Biographical Sketch Biographical Sketch 1916-1920 Harvard College: from medicine to 1929 Appointed to State College, Pennsylvania (roughly until 1966). mathematics (A.B. degree). Daughter (1930) and son (1934). 1924 Harvard University: Master’s degree physics. Founded The Association for Symbolic Logic (1936). Important work in combinatory logic. Doctoral studies on differential equations 1945 Involved with ENIAC. 1928 Married Mary Virginia Wheatly before going to: 1966-1970 Prof. of Logic, History of Logic & Phil. of Science 1929 Göttingen: Doctoral dissertation on logic, officially Amsterdam. supervised by Hilbert (but supported by Bernays). 1970 Returns to State College till death do them part. Programming languages Haskell, Curry, process of currying. 23 24 6 Biographical Sketch Formal System (definition) “there are always many […] less formal gatherings [at the Primitive Frame (of slightly restricted type of system): Currys], and we conjecture that Virginia’s cooking has Terms (recursively defined): also played a role in the growth of interest in List of primitive terms combinatory logic” – Anonymous? Operations Rules of formation Elementary Propositions: List of elementary predicates with numbers and kinds of arguments. Elementary Theorems (recursively defined): Axioms (or axiom schemes) Rules of procedure 25 26 About Formal Systems Meta-theory A formal system is not a game, but a body of We study the formal system as datum by any means at propositions. our command. All symbols are to be taken as designating the entities This leads to meta-propositions, which give rise to a talked about (term formalism). meta-theoretic method, studying a language consisting A formal system is not an entity existing independently of of ordinary discourse and symbols from formal systems. representation (anti-realism in ontology). Examples are metatheorems…: combining elementary propositions, The ontology of a formal system
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