# The Legacy of Hilbert, Gödel, Gentzen and Turing

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1 The Legacy of Hilbert, Gödel, Gentzen and Turing Amílcar Sernadas Departamento de Matemática - Instituto Superior Técnico Security and Quantum Information Group - Instituto de Telecomunicações TULisbon LMAC SEMINAR October 17, 2012

2 Abstract A brief survey of of formalizing mathematics, its initial successes, its failure, its most interesting ramifications, and its ultimate triumph (in a quite unexpected front).

3 Plan why? how? Initial successes first order logic (FOL) complete axiomatization of FOL symbolic proof of the consistency of FOL Disaster arithmetic is not axiomatizable consistency is not derivable in any rich fragment of arithmetic Other negative results halting problem is not decidable many other undecidability results

4 Plan (conc) Interesting ramifications independence results decidability of some useful fragments of mathematics practical applications of formal logic existence of computable universal functions Ultimate triumph advent of the universal computer the rest is history!

5 Why? : Why?

6 Why? Foundations of mathematics (late XIX century) Generalized use of sets in the foundations of mathematics, namely by Georg Cantor. Gottlob Frege s attempt at axiomatizing set theory using quantifiers. Bertrand Russell s paradox (1901), already by Ernst Zermelo (1900).

7 Why? Russel s paradox Inconsistency of Frege s axiomatization of sets Let R = {x : x x}. Then, R R R R.

8 Why? Hilbert s 2nd problem (1900) David Hilbert asked if mathematics is consistent free of any internal contradictions.

9 Why? But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.... In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms.... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. David Hilbert

10 How? : How?

11 How? Hilbert s dream of checking the consistency of mathematics Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. The consistency of more complicated systems, such as real analysis, should be proven in terms of simpler systems. Ultimately, the consistency of the whole of mathematics should be reduced to the consistency of basic arithmetic.

12 How? Hilbert s program for mechanizing mathematics (1920) Formalization (inspired by Gottfried Wilhelm Leibniz): all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules. Completeness: a proof that all true mathematical statements can be proved in the formalism. Consistency: a proof that no contradiction can be obtained in the formalism. Conservativeness: a proof that any result about real objects obtained using reasoning about ideal objects (such as uncountable sets) can be proved without using ideal objects. Decidability: there should be an algorithm for deciding the truth or falsity of any formal mathematical statement.

13 How? Early criticism of Hilbert s ideas Hermann Weyl described Hilbert s project as replacing contentual mathematics by a meaningless game of formulas. He noted that Hilbert wanted to secure not truth, but the consistency of analysis and suggested a criticism that echoes an earlier one by Gottlob Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert s meaningless inventory of formulas not just the bloodless ghost of analysis?

14 Initial successes : Initial successes

15 Initial successes The first step: first-order logic (FOL) Language = terms + formulas (p(0) ( x (p(x) p(x + 1)))) ( x p(x)) Calculus = decidable set of axioms + computable inference rules ( x p(x) p(t))... p(x), (p(x) q(x)) q(x)... Semantics = class of interpretations Interpretation = domain + relations + operations

16 Initial successes Expressive power of FOL FOL seemed good enough for the purpose of formally stating interesting mathematical properties For instance, the induction principle (p(0) ( x (p(x) p(x + 1)))) ( x p(x)) which should be present in any axiomatization of arithmetic.

17 Initial successes Completeness of FOL Kurt Gödel s completeness theorem (1929) Completeness of the axiomatization of FOL that specifies the properties of logic connectives and quantifiers and nothing else.

18 Initial successes Formal consistency of FOL Gerhard Gentzen s consistency theorem (1936) Proof by purely symbolic means of the consistency of the axiomatization of FOL (via a sequent calculus).

19 Disaster : Disaster

20 Disaster Gödel s incompleteness theorems (1931) First incompleteness theorem An axiomatization of arithmetic (capable of representing computable maps) cannot be both consistent and complete. Therefore, (sufficiently rich) arithmetic is not axiomatizable.

21 Disaster Gödel s incompleteness theorems (1931) Second incompleteness theorem Self-consistency is not derivable from any sufficiently strong axiomatization of (a fragment of) arithmetic.

22 Disaster Kurt Gödel and Albert Einstein in Princeton (1950)

23 Disaster Other negative results First undecidable formal problems (1936) Alan Turing: halting problem. Alonzo Church: equivalence of λ-expressions. Both provided a negative answer to the Entscheidungsproblem posed by Hilbert in 1928: Is there an algorithm capable of deciding if a formula is derivable in FOL from a given (decidable) set of formulas?

24 Disaster By the way... Undecidability of arithmetic truth Gödel s first incompleteness problem already provided an example of a non-decidable problem: Truth in a sufficiently rich arithmetic cannot be decidable (since it it is not even semidecidable because it cannot be axiomatized).

25 Disaster Yet another negative result Gregory Chaitin s incompleteness theorem (1987) In a sufficiently strong axiomatization of (a fragment of) arithmetic there is an upper bound L such that no specific number can be proven to have Kolmogorov complexity greater than L.

26 Disaster The death of formal logic? Formal logic rejected by most mathematicians? Unfortunately yes... A great misunderstanding indeed... >>> a mistake that none of you will commit, I hope... <<<

27 Interesting ramifications : Interesting ramifications

28 Interesting ramifications The death of formal logic? Not quite... Useful decidable fragments of mathematics e.g. Mojzesz Presburger s arithmetic (1929); Alfred Tarski s theories of algebraically closed fields (1949), real closed fields (1951).

29 Interesting ramifications The death of formal logic? Not quite... Formal logic remains relevant to mathematics Better understanding of the foundations of mathematics e.g. Kurt Gödel s (1940) and Paul Cohen s (1963) independence results, namely concerning Zermelo s axiom of choice and Cantor s continuum hypothesis. Work goes on... Development of techniques, namely those coming out of FOL model theory, recently used in other areas of mathematics.

30 Interesting ramifications The death of formal logic? Not quite... Practical applications of formal logic in software engineering and artificial intelligence Formal logic is routinely and widely used today for: reasoning about programs and protocols (analysis and synthesis); knowledge representation. Notwithstanding its roots in the foundations of mathematics, formal logic is now (also) a branch of applied mathematics! Thus, also mandatory in the curriculum of applied mathematicians and computer scientists/engineers.

31 Interesting ramifications Another outcome of : the universal Turing machine A machine that can emulate every machine ( ) Alan Turing conceived a computing machine that could be made to emulate any computing machine. U M p x U(p, x) = M(x)

32 Interesting ramifications The universal programmable computer A machine that can be programmed to compute any computable function Long after the programmable analytical engine had been proposed by Charles Babbage (1834), thanks to the theoretical contributions by Alan Turing, the idea of the programmable computer had arrived for good... The rest is history!

33 Ultimate triumph : Ultimate triumph

34 Ultimate triumph The economic and social success of The practical impact of The work on (although not successful per se) made significant contributions to the advent of the concept of computable function, which led to the notion of universal computer and, thus, to the triggering of the latest industrial revolution on which our affluent way of life stands.

36 Where to learn more Start learning about the rise of modern logic Hilbert s program s program Gödel s completeness theorem s completeness theorem Gentzen s consistency proof s consistency proof

37 Where to learn more Start learning about the rise of modern logic (conc) Gödel s incompleteness theorems s incompleteness theorems Turing s contributions LMAC course in Mathematical Logic, year 2, semester 2

38 Where to learn more You can also start having fun now! And if you would like to work in logic, computability or complexity by all means come up to the 5th foor. You will be most welcome.

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