Upload
danghanh
View
220
Download
3
Embed Size (px)
Citation preview
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
The Legacy ofHilbert, Godel, Gentzen and Turing
Amılcar Sernadas
Departamento de Matematica - Instituto Superior TecnicoSecurity and Quantum Information Group - Instituto de Telecomunicacoes
TULisbon
LMAC SEMINAROctober 17, 2012
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Abstract
A brief survey of Hilbert’s programme of formalizing mathematics,its initial successes,its failure,its most interesting ramifications,and its ultimate triumph (in a quite unexpected front).
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Plan
Hilbert’s programme
why?how?
Initial successes
first order logic (FOL)complete axiomatization of FOLsymbolic proof of the consistency of FOL
Disaster
arithmetic is not axiomatizableconsistency is not derivable in any rich fragment of arithmetic
Other negative results
halting problem is not decidablemany other undecidability results
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Plan (conc)
Interesting ramifications
independence resultsdecidability of some useful fragments of mathematicspractical applications of formal logicexistence of computable universal functions
Ultimate triumph
advent of the universal computerthe rest is history!
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Why?
Hilbert’s programme: Why?
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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 usingquantifiers.
Bertrand Russell’s paradox (1901),already by Ernst Zermelo (1900).
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Why?
Russel’s paradox
Inconsistency of Frege’s axiomatization of sets
Let R = {x : x 6∈ x}. Then,
R ∈ R ⇔ R 6∈ R.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Why?
Hilbert’s 2nd problem (1900)
David Hilbert asked if mathematics is consistent — free of anyinternal contradictions.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Why?
But above all I wish to designate the following as the most importantamong the numerous questions which can be asked with regard to theaxioms: To prove that they are not contradictory, that is, that a definitenumber of logical steps based upon them can never lead to contradictoryresults.. . .In geometry, the proof of the compatibility of the axioms can be effectedby constructing a suitable field of numbers, such that analogous relationsbetween 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
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
How?
Hilbert’s programme: How?
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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 axiomswere consistent.
The consistency of more complicated systems, such as realanalysis, should be proven in terms of simpler systems.
Ultimately, the consistency of the whole of mathematics should bereduced to the consistency of basic arithmetic.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
How?
Hilbert’s program for mechanizing mathematics (1920)
Formalization (inspired by Gottfried Wilhelm Leibniz): allmathematical statements should be written in a precise formallanguage, and manipulated according to well defined rules.
Completeness: a proof that all true mathematical statementscan be proved in the formalism.
Consistency: a proof that no contradiction can be obtained inthe formalism.
Conservativeness: a proof that any result about real objectsobtained using reasoning about ideal objects (such asuncountable sets) can be proved without using ideal objects.
Decidability: there should be an algorithm for deciding thetruth or falsity of any formal mathematical statement.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
How?
Early criticism of Hilbert’s ideas
Hermann Weyl described Hilbert’s project as replacing contentualmathematics by a meaningless game of formulas.
He noted that Hilbert wanted to secure not truth, but theconsistency of analysis and suggested a criticism that echoes anearlier one by Gottlob Frege:
Why should we take consistency of a formal system ofmathematics as a reason to believe in the truth of the pre-formalmathematics it codifies?
Is Hilbert’s meaningless inventory of formulas not just thebloodless ghost of analysis?
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Initial successes
Hilbert’s programme: Initial successes
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Initial successes
Expressive power of FOL
FOL seemed good enough for the purpose offormally 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.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Initial successes
Completeness of FOL
Kurt Godel’s completeness theorem (1929)
Completeness of the axiomatization of FOL that specifies theproperties of logic connectives and quantifiers and nothing else.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Initial successes
Formal consistency of FOL
Gerhard Gentzen’s consistency theorem (1936)
Proof by purely symbolic means of the consistency of theaxiomatization of FOL (via a sequent calculus).
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Disaster
Hilbert’s programme: Disaster
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Disaster
Godel’s incompleteness theorems (1931)
First incompleteness theorem
An axiomatization of arithmetic (capable of representingcomputable maps) cannot be both consistent and complete.
Therefore, (sufficiently rich) arithmetic is not axiomatizable.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Disaster
Godel’s incompleteness theorems (1931)
Second incompleteness theorem
Self-consistency is not derivable from any sufficiently strongaxiomatization of (a fragment of) arithmetic.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Disaster
Kurt Godel and Albert Einstein in Princeton (1950)
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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 Entscheidungsproblemposed by Hilbert in 1928:
Is there an algorithm capable of deciding if a formula is derivable inFOL from a given (decidable) set of formulas?
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Disaster
By the way...
Undecidability of arithmetic truth
Godel’s first incompleteness problem already provided an exampleof a non-decidable problem:Truth in a sufficiently rich arithmetic cannot be decidable(since it it is not even semidecidable because it cannot beaxiomatized).
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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 numbercan be proven to have Kolmogorov complexity greater than L.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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... <<<
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Interesting ramifications
Hilbert’s programme: Interesting ramifications
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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).
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
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 Godel’s (1940) and Paul Cohen’s (1963) independenceresults, namely concerning Zermelo’s axiom of choice andCantor’s continuum hypothesis. Work goes on...
Development of techniques, namely those coming out of FOLmodel theory, recently used in other areas of mathematics.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Interesting ramifications
The death of formal logic? Not quite...
Practical applications of formal logic insoftware engineering and artificial intelligence
Formal logic is routinely and widely used today for:
reasoning about programs and protocols (analysis andsynthesis);
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 mathematiciansand computer scientists/engineers.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Interesting ramifications
Another outcome of Hilbert’s programme:the universal Turing machine
A machine that can emulate every machine (1936–1937)
Alan Turing conceived a computing machine that could be madeto emulate any computing machine.
∃U ∀M ∃p ∀x U(p, x) = M(x)
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Interesting ramifications
The universal programmable computer
A machine thatcan be programmed to compute any computable function
Long after the programmable analytical engine had been proposedby 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!
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Ultimate triumph
Hilbert’s programme: Ultimate triumph
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Hilbert’s programme
Ultimate triumph
The economic and social success of Hilbert’s programme
The practical impact of Hilbert’s programme
The work on Hilbert’s programme (although not successful per se)made significant contributions tothe advent of the concept of computable function,which led tothe notion of universal computerand, thus, tothe triggering of the latest industrial revolutionon whichour affluent way of life stands.
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Where to learn more
Start learning about the rise of modern logic
Hilbert’s program
http://plato.stanford.edu/entries/hilbert-programhttp://en.wikipedia.org/wiki/Hilbert’s program
Godel’s completeness theorem
http://plato.stanford.edu/entries/goedelhttp://en.wikipedia.org/wiki/Godel’s completeness theorem
Gentzen’s consistency proof
http://plato.stanford.edu/entries/proof-theory-developmenthttp://en.wikipedia.org/wiki/Gentzen’s consistency proof
Legacy of HGGT — A. Sernadas — LMAC October 17, 2012
Where to learn more
Start learning about the rise of modern logic (conc)
Godel’s incompleteness theorems
http://plato.stanford.edu/entries/goedelhttp://en.wikipedia.org/wiki/Godel’s incompleteness theorems
Turing’s contributions
http://plato.stanford.edu/entries/turinghttp://en.wikipedia.org/wiki/Turing
LMAC course in Mathematical Logic, year 2, semester 2
http://wslc.math.ist.utl.pt/teaching.html