The Legacy of Hilbert, Gödel, Gentzen and Turing · Hilbert, G odel, Gentzen and Turing ... Bertrand Russell’s paradox (1901), ... The Legacy of Hilbert, Gödel, Gentzen and Turing

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


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).


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


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!



Why?

Hilbert’s programme: Why?



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).



Why?

Russel’s paradox

Inconsistency of Frege’s axiomatization of sets

Let R = {x : x 6∈ x}. Then,

R ∈ R ⇔ R 6∈ R.



Why?

Hilbert’s 2nd problem (1900)

David Hilbert asked if mathematics is consistent — free of anyinternal contradictions.



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



How?

Hilbert’s programme: How?



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.



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.



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?



Initial successes

Hilbert’s programme: Initial successes



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



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.



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.



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).



Disaster

Hilbert’s programme: Disaster



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.



Disaster

Godel’s incompleteness theorems (1931)

Second incompleteness theorem

Self-consistency is not derivable from any sufficiently strongaxiomatization of (a fragment of) arithmetic.



Disaster

Kurt Godel and Albert Einstein in Princeton (1950)



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?



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).



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.



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... <<<




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).





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.





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.




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)




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!



Ultimate triumph

Hilbert’s programme: Ultimate triumph



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.


Where to learn more

Where to learn more...


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


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


Where to learn more

You can also start having fun now!

And if you would like to work in logic, computability or complexityby all means come up to the 5th foor.

You will be most welcome.

The Legacy of Hilbert, Gödel, Gentzen and Turing · Hilbert, G odel, Gentzen and Turing ... Bertrand Russell’s paradox (1901), ... The Legacy of Hilbert, Gödel, Gentzen and Turing

Documents