Teoria ModelelorMaster Anul I Semestrul II 2016

Laurentiu LeusteanPagina web httpunibucro~lleustean

In aceasta prezentare sunt folosite partial slideurile Ioanei Leustean din

Semestrul I 20142015

1

Ce este logica

logike tekhne= stiinta rationamentelor logos = cuvantrationament

Aristotel (IV ien)

I httpplatostanfordedu

entriesaristotle-logic

I primul studiu formal al logicii

I a studiat silogismele deductiiformate din doua premize si oconcluzie

BarbaraPremiza Toti oamenii sunt muritoriPremiza Grecii sunt oameniConcluzie Deci grecii sunt muritori

2

Logica si Informatica

rdquo a computing machine isreally a logic machine Its circuitsembody the distilled insights of aremarkable collection of logiciansdeveloped over centuryNowadays as computertechnology advances with suchbreathtaking rapidity as weadmire the truly accomplishmentsof the engineers it is all too easyto overlook the logicians whoseideas made it all possible Thisbook tells their storyrdquo

3

Gottfried Wilhelm Leibniz (1646 -1716)

Visul lui LeibnizI un limbaj matematic universal (lingua characteristica

universalis) ın care toata cunoasterea umana poate fiexprimata si reguli de calcul (calculus ratiocinator) pentru aderiva cu ajutorul masinilor toate relatiile logice

rdquoIf controversies were to arisethere would be no more need ofdisputation between twophilosophers than between twoaccountants For it would sufficeto take their pencils in theirhands and say to each otherCalculemus - Let us calculaterdquo

4

George Boole (1815-1864)

I The Mathematical Analysis of Logic (1847) The Laws ofThought (1854) a initiat analiza rationamentelor logice prinmetode asemanatoare calculului algebric

I Silogismele lui Aristotel sunt despre clase de obiecte care potfi studiate algebric

rdquoThe design of the followingtreatise is to investigate thefundamental laws of theoperations of the mind by whichreasoning is performed to giveexpressions to them in thesymbolic language of calculusand upon this foundation toestablish the science of logic andconstructs its methodsrdquo

5

Gottlob Frege (1848-1925)

Begriffschrift (1847)

I A introdus sintaxa formala obiecte predicate functiiconectori propozitionali cuantificatori

I A inventat logica de ordinul ıntai

I van Heijenoort From Frege to Godel 1967ldquoperhaps the most important single work ever written inlogicrdquo

Exemplu

I Toti oamenii sunt muritori

I Pentru orice x daca x esteom atunci x este muritor

I forallx(Om(x) rarr Muritor(x))

6

Georg Cantor (1848-1925)

I A inventat teoria multimilor

I A definit numere cardinale ordinale

I A dezvoltat o teorie matematica a infinitului

Hilbert

rdquoNo one shall be able to expel usfrom the paradise that Cantorcreated for usldquo

7

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Gottfried Wilhelm Leibniz (1646 -1716)

Visul lui LeibnizI un limbaj matematic universal (lingua characteristica

universalis) ın care toata cunoasterea umana poate fiexprimata si reguli de calcul (calculus ratiocinator) pentru aderiva cu ajutorul masinilor toate relatiile logice

rdquoIf controversies were to arisethere would be no more need ofdisputation between twophilosophers than between twoaccountants For it would sufficeto take their pencils in theirhands and say to each otherCalculemus - Let us calculaterdquo

4

George Boole (1815-1864)

I The Mathematical Analysis of Logic (1847) The Laws ofThought (1854) a initiat analiza rationamentelor logice prinmetode asemanatoare calculului algebric

I Silogismele lui Aristotel sunt despre clase de obiecte care potfi studiate algebric

rdquoThe design of the followingtreatise is to investigate thefundamental laws of theoperations of the mind by whichreasoning is performed to giveexpressions to them in thesymbolic language of calculusand upon this foundation toestablish the science of logic andconstructs its methodsrdquo

5

Gottlob Frege (1848-1925)

Begriffschrift (1847)

I A introdus sintaxa formala obiecte predicate functiiconectori propozitionali cuantificatori

I A inventat logica de ordinul ıntai

I van Heijenoort From Frege to Godel 1967ldquoperhaps the most important single work ever written inlogicrdquo

Exemplu

I Toti oamenii sunt muritori

I Pentru orice x daca x esteom atunci x este muritor

I forallx(Om(x) rarr Muritor(x))

6

Georg Cantor (1848-1925)

I A inventat teoria multimilor

I A definit numere cardinale ordinale

I A dezvoltat o teorie matematica a infinitului

Hilbert

rdquoNo one shall be able to expel usfrom the paradise that Cantorcreated for usldquo

7

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

George Boole (1815-1864)

I The Mathematical Analysis of Logic (1847) The Laws ofThought (1854) a initiat analiza rationamentelor logice prinmetode asemanatoare calculului algebric

I Silogismele lui Aristotel sunt despre clase de obiecte care potfi studiate algebric

rdquoThe design of the followingtreatise is to investigate thefundamental laws of theoperations of the mind by whichreasoning is performed to giveexpressions to them in thesymbolic language of calculusand upon this foundation toestablish the science of logic andconstructs its methodsrdquo

5

Gottlob Frege (1848-1925)

Begriffschrift (1847)

I A introdus sintaxa formala obiecte predicate functiiconectori propozitionali cuantificatori

I A inventat logica de ordinul ıntai

I van Heijenoort From Frege to Godel 1967ldquoperhaps the most important single work ever written inlogicrdquo

Exemplu

I Toti oamenii sunt muritori

I Pentru orice x daca x esteom atunci x este muritor

I forallx(Om(x) rarr Muritor(x))

6

Georg Cantor (1848-1925)

I A inventat teoria multimilor

I A definit numere cardinale ordinale

I A dezvoltat o teorie matematica a infinitului

Hilbert

rdquoNo one shall be able to expel usfrom the paradise that Cantorcreated for usldquo

7

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Gottlob Frege (1848-1925)

Begriffschrift (1847)

I A introdus sintaxa formala obiecte predicate functiiconectori propozitionali cuantificatori

I A inventat logica de ordinul ıntai

I van Heijenoort From Frege to Godel 1967ldquoperhaps the most important single work ever written inlogicrdquo

Exemplu

I Toti oamenii sunt muritori

I Pentru orice x daca x esteom atunci x este muritor

I forallx(Om(x) rarr Muritor(x))

6

Georg Cantor (1848-1925)

I A inventat teoria multimilor

I A definit numere cardinale ordinale

I A dezvoltat o teorie matematica a infinitului

Hilbert

rdquoNo one shall be able to expel usfrom the paradise that Cantorcreated for usldquo

7

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Georg Cantor (1848-1925)

I A inventat teoria multimilor

I A definit numere cardinale ordinale

I A dezvoltat o teorie matematica a infinitului

Hilbert

rdquoNo one shall be able to expel usfrom the paradise that Cantorcreated for usldquo

7

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Georg Cantor (1848-1925)

I Aristotel ldquoInfinitum Actu Non Daturrdquo - nu exista infinitactual

I Leibniz ldquoI am so in favor of the actual infinite that instead ofadmitting that Nature abhors it I hold that Nature makesfrequent use of it everywhererdquo

I Gauss ldquoI protest above all the use of an infinite quantity as acompleted one which in mathematics is never allowedldquo

I Frege rdquoFor the infinite will eventually refuse to be excludedfrom arithmetics Thus we can foresee that this issue willprovide for a momentous and decisive battleldquo

I Poincare rdquograve disease infecting mathematicsrdquoI Kronecker despre Cantor ldquoscientific charlatanrdquo ldquocorrupter of

youthrdquoI Wittgenstein ldquoutter nonsenserdquoI Mittag-Leffler despre lucrarile lui Cantor ldquoabout one hundred

years too soonrdquo8

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Criza fundamentelor matematicii

Scrisoarea lui Bertrand Russell catre Frege (16 iunie 1902)

ldquoI find myself in agreement with you in all essentials I find inyour work discussions distinctions and definitions that one seeksin vain in the work of other logicians There is just one pointwhere I have encountered a difficultyrdquo

Frege apendix la The Fundamental Laws of Arithmetic Vol 2

ldquoThere is nothing worse that can happen to a scientist than tohave the foundation collapse just as the work is finished I havebeen placed in this position by a letter from Mr Bertrand Russellrdquo

9

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Criza fundamentelor matematicii

Conform teoriei naive a multimilor orice colectie definibila estemultime Fie U multimea tuturor multimilor

Paradoxul lui Russel (1902)

Fie R = A isin U | A isin A Atunci R este multime deci R isin UObtinem ca R isin R hArr R isin R

Criza fundamentelor matematiciiI Paradoxul lui Russel rArr Sistemul logic al lui Frege inconsistent

I a declansat criza fundamentelor matematicii (rdquofoundations ofmathematicsrdquo)

I s-a dezvoltat teoria axiomatica a multimilor Zermelo-Fraenkel(ZF) ZFC ZF+ Axioma alegerii (Axiom of Choice)

10

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

David Hilbert (1862-1943)

I unul dintre matematicieniide varf ai generatiei sale

I unul dintre fondatorii teorieidemonstratiei si logiciimatematice

I lista sa de 23 problemedeschise (1902) a influentatfoarte mult matematicasecolului XX

11

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Programul lui Hilbert

Programul lui Hilbert (1921)

Sa se formalizeze matematica si sa se stabileasca urmatoarele

I Matematica este consistenta un enunt matematic si negatiasa nu pot fi demonstrate simultan

I Matematica este completa toate enunturile matematiceadevarate pot fi demonstrate

I Matematica este decidabila exista o regula mecanica pentru adetermina daca un enunt matematic dat este adevarat sau fals

12

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Programul lui Hilbert

Hilbert a fost convins ca aceste obiective pot fi atinse

rdquoEvery mathematical problem must necessarily be susceptible to anexact statement either in the form of an actual answer to thequestion asked or by the proof of the impossibility of its solutionrdquo

rdquoOnce a logical formalism is established one can expect that asystematic so-to-say computational treatment of logic formulas ispossible which would somewhat correspond to the theory ofequations in algebrardquo

13

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Kurt Godel (1906-1978)

Teoremele de incompletitudine ale lui Godel (1931-33)

I Incompletitudinea aritmeticii obisnuite

I Imposibilitatea de a demonstra consistenta teoriei multimilor

I Au marcat esecul programului lui Hilbert

I Este considerat cel mai mare logician alsecolului XX

I A introdus functiile calculabile

I A demonstrat teorema de completitudinea logicii de ordinul l

I A demonstrat ca Axioma Alegerii siIpoteza Continuumului sunt consistentecu axiomele teoriei multimilor

14

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Kurt Godel (1906-1978)

John von Neumann

ldquoKurt Godelrsquos achievement in modern logic is singular andmonumental - indeed it is more than a monument it is a landmarkwhich will remain visible far in space and time The subject oflogic has certainly completely changed its nature and possibilitieswith Godelrsquos achievementrdquo

Revista TIME (19 martie 1999)

Godel a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

15

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Problema de decizie (Entscheidungsproblem)

I Hilbert si Ackermann (1928) Exista un algoritm pentru averifica daca o anumita formula din logica de ordinul ıntai esteadevarata

I Cu alte cuvinte Este logica de ordinul ıntai decidabila

16

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Alan Turing(1912-1954)

Turing On computable numbers with an application to theEntscheidungsproblem Proc London Math Soc 42 (1936)

I a demonstrat ca logica de ordinul ıntai este nedecidabila(rezultat obtinut independent de Church (1936))

I a introdus masina Turing (universala) pentru a formalizanotiunea de algoritm

I parintele informaticii siinteligentei artificiale

I masina Turing universalaeste model al calculatoareloractuale

17

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Alan Turing(1912-1954)

Revista TIME (19 martie 1999)

Turing a fost inclus in lista cu cei mai importanti 20 oameni destiinta si ganditori ai secolului XX

ldquoVirtually all computers today from 10 million supercomputers tothe tiny chips that power cell phones and Furbies have one thingin common they are all rdquovon Neumann machinesldquo variations onthe basic computer architecture that John von Neumann buildingon the work of Alan Turing laid out in the 1940rsquos

Premiul Turing

I httpamturingacmorg

I decernat anual de catre Association for Computing Machinery(ACM) pentru contributii ın informatica

I este considerat un Premiu Nobel pentru Informatica18

Logica si Informatica

E W Dijkstra The next fifty years (EWD1243a) EW DijkstraArchive Center for American History University of Texas atAustin

rdquoComputing and Computing Science unavoidably emerge as anexercise in formal mathematics or if you wish an acronym asexercise in VLSAL (Very Large Scale Application of Logic)ldquo

Aaron R Bradley Zohar Manna The Calculus of ComputationDecision Procedures with Applications to Verification Springer2007

rdquoLogic is the calculus of computationrdquo

Georg Gottlob Logic and Artificial Intelligence VSL 2014

ldquoComputer science is the continuation of logic by other meansrdquo19

Logica si Informatica

Aplicatii ale logicii ın informatica

I calculabilitate si complexitate

I arhitectura calculatoarelor (circuite logice)

I software engineering (verificare model checking)

I limbaje de programare (semantica programare logicaprogramare functionala)

I baze de date (algebre de relatii teoria modelelor finite)

I inteligenta artificiala

I criptografie si securitate

J Y Halpern R Harper N Immerman PGKolaitis MY VardiVVianu On the Unusual Effectiveness of Logic in ComputerScience Bulletin of Symbolic Logic 7(2001)

20

Logica si Informatica ın Romania

Grigore C Moisil (1906-1973)

Computer Pioneer Award of IEEE Computer Society

S Marcus Grigore C Moisil A life becominga myth 2006

rdquoAs a professor of the Bucharest University hewas the first to teach there mathematicallogic Articulating logic and automata Moisilwas well prepared to organize the Romaniandevelopment in the emergent field ofComputer Sciencewe can say that 1957 isthe date of birth of Romanian ComputerScience under the guidance of ProfessorMoisil and with the collaboration of engineersand mathematiciansrdquo

21

Logica si Informatica ın Romania

Grigore C Moisil (1906-1973)

Computer Pioneer Award of IEEE Computer Society

S Marcus Grigore C Moisil A life becominga myth 2006

rdquoAs a professor of the Bucharest University hewas the first to teach there mathematicallogic Articulating logic and automata Moisilwas well prepared to organize the Romaniandevelopment in the emergent field ofComputer Sciencewe can say that 1957 isthe date of birth of Romanian ComputerScience under the guidance of ProfessorMoisil and with the collaboration of engineersand mathematiciansrdquo

21