Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen...
Transcript of Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen...
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Perlen der Informatik I
Jan Kretınsky
Technische Universitat MunchenWinter 2020/2021
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Overview 2/65
I language: English/GermanI voluntary courseI lecture on Tuesday, in the slot 10 a.m. – 12 p.m.I https://www7.in.tum.de/˜kretinsk/teaching/perlen.html
I Godel, Escher, Bach: an Eternal Golden Braidby Douglas R. Hofstadter
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Bach 3/65
I Frederick the GreatI Leonhard Euler,. . . , J.S. BachI improvised 6-part fugueI canons
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Bach 3/65
I Frederick the GreatI Leonhard Euler,. . . , J.S. BachI improvised 6-part fugueI canons
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Bach 3/65
I Frederick the GreatI Leonhard Euler,. . . , J.S. BachI improvised 6-part fugueI canons
I copies differing in time, pitch, speed, direction (upside down, crab)I isomorphicI canon endlessly rising in 6 steps – “strange loop”
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Escher 4/65
“Waterfall”6-step endlessly falling loop
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Escher 4/65
“Ascending and Descending”illusion by Roger Penrose
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Escher 4/65
Penrose triangleFaculty of Informatics, Brno
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Escher 4/65
“Drawing hands”his first strange loop
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Escher 4/65
“Metamorphosis”copies of one theme
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”
I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proofI statement about numbers can talk about itself
“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557 is in binary0011001000101011001100100011110100110101 read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proof
I statement about numbers can talk about itself“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557 is in binary0011001000101011001100100011110100110101 read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proofI statement about numbers can talk about itself
“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557 is in binary0011001000101011001100100011110100110101 read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proofI statement about numbers can talk about itself
“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557
is in binary0011001000101011001100100011110100110101 read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proofI statement about numbers can talk about itself
“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557 is in binary0011001000101011001100100011110100110101
read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Godel 5/65
I BrnoI Epimenides paradox: “All Cretans are liars”I mathematical reasoning in exploring mathematical reasoningI Incompleteness theorem:
All consistent axiomatic formulations of number theoryinclude undecidable propositions.
I strange loop in the proofI statement about numbers can talk about itself
“This statement of number theory does not have any proof”
I numberscode↔ statements
215473077557 is in binary0011001000101011001100100011110100110101 read as ASCII2+2=5
I homework:34723379178930453204433293597543819411782291432109326918654063662
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Mathematical logic 6/65
I different geometries, equally validI real world?I proof?I Russel’s paradox
I “ordinary” sets: x < xI “self-swallowing” sets: x ∈ xI R = set of all ordinary sets
I Grelling’s paradoxI self-descriptive adjectives (“pentasyllabic”) vs non-self-descriptiveI what about “non-self-descriptive”?
I self-referencedrawing handsThe following sentence is false. The preceding sentence is true.
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Way out? 7/65
I prohibition (Principia mathematica)I types, metalanguageI “In this lecture, I criticize the theory of types”
cannot discuss the type theoryI David Hilbert: consistency and completeness
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Computers 8/65
I BabbageThe course through which I arrived at it was the mostentangled and perplexed which probably ever occupied thehuman mind.
Ada Lovelace (daughter of Lord Byron)Mechanized intelligence“Eating its own tail” (altering own program)
I axiomatic reasoning, mechanical computation, psycholgy ofintelligence
I Alan Turing ∼ Godel’s counterpart in computation theoryHalting problem is undecidable.Can intelligent behaviour be programmed? Rules for inventing newrules...Strange loops in the core of intelligence
I materialism, de la Metrie: L’homme machine
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Formal system 9/65
Example (over alphabet M,I,U)I initial string (“axiom”):
I MI
I rules (“inference/production rules”) to enlarge your collection (of“theorems”)requirement of formality: not outside the rules
I last letter I⇒ put U at the endI Mx ⇒ Mxx where x can be any stringI replace III by UI drop UU
Homework: Can you produce/derive/prove MU ?
I Which rule to use? That’s the art.
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Working in the system / observing the system 10/65
I human itellingece⇒ notice properties of theoremsI machine can act unobservant, people cannot
Perfect test (“decision procedure”) for theoremsI tree of all theorems?I finite time!
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Another formal system 11/65
I alphabet {p,q,−}I axioms (axiom schema – obvious decision procedure):
xp−qx− for any x composed from hyphens
I production rules:
xpyqz ⇒ xpy−qz− for any x, y, z composed from hyphens
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Decision procedure 12/65
I only lengthening rules⇒ reduce to shorter ones (top-down)⇒ dovetailing longer axioms and rule application (bottom-up)
I hereditary properties of theorems
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Meaning 13/65
IsomorphismI information-preserving transformationI creates meaningI interpretation + correspondence between true statements and
interpreted theoremsI like cracking a codeI meaningless interpretations possibleI “well-formed” strings should produce “gramatical” sentences
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Meaning is passive in formal systems 14/65
I it seems the system cannot avoid taking on meaningI is --p--p--q------ a theorem?I subtractionI does not add new additions, but we learn about nature of additionI (is reality a formal system? is universe deterministic?)
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Is our formal system accurate? 15/65
I 12 × 12: counting vs proofI basic properties to be believed, e.g. commutatitvity and associativityI in reality not always: raindrop, cloud, trinity, languages in IndiaI ideal numbersI counting cannot check Euclid’s Theorem
I reasoningI non-obvious result from obvious stepsI belief in reasoningI overcoming infinity (“all” N)I patterned structure binding statementsI can thinking be achieved by a formal system?
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Escher: Liberation 16/65
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Puzzle 17/65
1 3 7 12 18 26 35 45 56 ?
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Escher: Mosaic II 18/65
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Can we distinguish primes from composites? 19/65
Formal systems ∼ typographical operations:I read, write, copy, erase, and compare symbolsI keep generated theorems
Multiplication:I axiom xt-qx for every hyphen-string xI rule xtyqz ⇒ xty-qzx for hyphen-strings x, y, z
Composites:I rule x-ty-qz ⇒Cz for hyphen-strings x, y, z
Primes:
I rule: Cx is not a theorem⇒ Px for every hyphen-string xI reasoning what cannot be generated is outside of system,
requirement of formality
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Can we distinguish primes from composites? 19/65
Formal systems ∼ typographical operations:I read, write, copy, erase, and compare symbolsI keep generated theorems
Multiplication:I axiom xt-qx for every hyphen-string xI rule xtyqz ⇒ xty-qzx for hyphen-strings x, y, z
Composites:I rule x-ty-qz ⇒Cz for hyphen-strings x, y, z
Primes:
I rule: Cx is not a theorem⇒ Px for every hyphen-string xI reasoning what cannot be generated is outside of system,
requirement of formality
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Can we distinguish primes from composites? 19/65
Formal systems ∼ typographical operations:I read, write, copy, erase, and compare symbolsI keep generated theorems
Multiplication:I axiom xt-qx for every hyphen-string xI rule xtyqz ⇒ xty-qzx for hyphen-strings x, y, z
Composites:I rule x-ty-qz ⇒Cz for hyphen-strings x, y, z
Primes:I rule: Cx is not a theorem⇒ Px for every hyphen-string x
I reasoning what cannot be generated is outside of system,requirement of formality
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Can we distinguish primes from composites? 19/65
Formal systems ∼ typographical operations:I read, write, copy, erase, and compare symbolsI keep generated theorems
Multiplication:I axiom xt-qx for every hyphen-string xI rule xtyqz ⇒ xty-qzx for hyphen-strings x, y, z
Composites:I rule x-ty-qz ⇒Cz for hyphen-strings x, y, z
Primes:I rule: Cx is not a theorem⇒ Px for every hyphen-string xI reasoning what cannot be generated is outside of system,
requirement of formality
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Negative definitions: figure and ground 20/65
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Negative definitions: figure and ground 21/65
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Negative definitions: figure and ground 22/65
SetsI recursive: decision procedureI recursively enumerable (r.e.): can be generatedI non-r.e.
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Negative definitions: figure and ground 23/65
Characterize false statementsI negative space of theoremsI altered copy of theorems
Impossible!
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Impossibility 24/65
I some negative spaces cannot be positiveI = there are non-recursive r.e. setsI ⇒ there are formal sytems with no decision procedure
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Primes are recursive 25/65
I axiom xyDNDx for hyphen-strings x, yI rules
xDNDy ⇒ xDNDxy--DNDz ⇒ zDF--zDFx and x-DNDz ⇒ zDFx-z-DFz ⇒Pz-
I axiom P--
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Impossibility 26/65
I if a set is generatable in increasing order then so is its complementI lengthening interleaved with shortening causes Godel’s Theorem,
Turing’s Halting Problem etc.
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Diagonalisation: Cantor 27/65
f : S → P(S)
C = {s ∈ S | s < f(s)}
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Diagonalisation: Cantor 27/65
f : S → P(S)
C = {s ∈ S | s < f(s)}
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Diagonalisation: Russell 28/65
R = {x | x < x}
Then R ∈ R ⇐⇒ R < R
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Diagonalisation: Turing 29/65
program e: if f(i,i) == 0 then return 0 else loop forever
I f(e, e) = 0 =⇒ g(e) = 0 =⇒ program e halts on input e=⇒ f(e, e) = 1
I f(e, e) , 0 =⇒ g(e) undef. =⇒ program e doesn’t halt on input e=⇒ f(e, e) = 0
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Diagonalisation: Godel (vague idea) 30/65
”, when preceded by itself in quotes, is unprovable.”, when preceded byitself in quotes, is unprovable.
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Godel and the strange loop 31/65
For any player, there is a record which it cannot play because it will causeits indirect destruction.
Bach – self-reference in the Art of the Fugue
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Story isomorphism 32/65
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Consistency 33/65
Example: pq-system
I Axiom schema II: xp-qx for every hyphen-string xI inconsistent with external world
I reinterpret: ≥I consistency depends on interpretationI consistency = Every theorem, when interpreted, becomes a true
statement.
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Consistency 33/65
Example: pq-system
I Axiom schema II: xp-qx for every hyphen-string xI inconsistent with external worldI reinterpret: ≥I consistency depends on interpretationI consistency = Every theorem, when interpreted, becomes a true
statement.
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Consistency 34/65
Example: non-Euclid geometry
I ElementsI rigorI axiomatic systemI fifth postulate not a consequenceI Saccheri, Lambert, Bolyai, LobachevskiyI elliptical/spherical (no parallel) and hyperbolical (≥ 2 parallels)
geometry (4 geometrical postlates remain, “absolute geometry”included)
I real points and lines vs. explicit definitions vs. implicit propositions
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Consistency 35/65
I internal consistency: theorems mutually compatibleholds in some “imaginable” world
I logical, mathematical, physical, biological etc. consistencyI Is number theory/geometry the same in all conceivable worlds?
I Peano arithmetic ∼ absolute (core) geometryI number theories are the same for practical purposesI Gauss attmepted to measure angles between three mountains
general relativitymore geometries in mathematics and even physics
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Consistency 36/65
Relativity
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Completeness 37/65
Consistency: minimal condition for passive meaning
Completeness: maximal confirmation of passive meanings
“ Every true statement which can be expressed in the notation of thesystem is a theorem”I Example: 2+3+4=9 in pqI Example: pq with Axiom schema II
(1) add rules or (2) tighten the interpretation
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Godel’s 1st incompleteness theorem 38/65
TheoremThere are true arithmetical formulae unprovable in PA (or other consistentformal systems).
Godel’s proof (sketch)
I it is possible to construct a PA formula ρ such that
PA ` ρ ⇐⇒ ¬Provable([ρ])
i.e. “ρ says “I’m not provable”” is provable in PAI by consistency of PA this is true in arithmeticsI if ¬ρ then Provable([ρ]), a contradiction
if ρ then ¬Provable([ρ]) hence PA 0 ρ �
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Godel’s 1st incompleteness theorem 39/65
Recall:I Accept := {i | Mi accepts i}I Accept is r.e., but not recursiveI Accept is not r.e.
Alternative proof:
I Provable is r.e. (for PA and similar)I Provable ⊆ Valid by consistencyI we prove Valid is not r.e., hence (
I construct a program transforming n ∈ N into a formula ϕ:
ϕ ∈ Valid iff n ∈ Accept
it computes the formula “Mn does not accept n”I computation is a sequence of configurations (numbers)I one can encode that a configuration c follows a given configuration dI every finite sequence can be encoded by a formula β:
For every n1, . . . , nk there are a, b ∈ N such that
β(a, b , i, x) iff x = ni
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Godel’s 1st incompleteness theorem 40/65
Let β(a, b , i, x) be true iff x = a mod (1 + b(1 + i))I expressible in simple arithmetics:
a ≥ 0∧b ≥ 0∧∃k(k ≥ 0∧k ∗c ≤ a∧(k +1)∗c > a∧x = a−(k ∗b)
)where c is a shortcut for (1 + b ∗ (1 + i))
I for every a, b the predicate β induces a unique sequence,where the ith element is a mod (1 + b(1 + i))
I every finite sequence can be encoded by β for some a, b:
TheoremFor every n1, . . . , nk there are a, b ∈ N such that
β(a, b , i, x) iff x = ni
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Godel’s 1st incompleteness theorem 41/65
β(a, b , i, x) iff x = a mod (1 + b(1 + i))
TheoremFor every n1, . . . , nk there are a, b ∈ N such that
β(a, b , i, x) iff x = ni
Proof.
I b := (max{k , n1, . . . , nk })!
I pi := 1 + b(1 + i) is ≥ ni and mutually incommensurableI ci :=
∏j,i pj
I ∃! 0 ≤ di ≤ pi : ci · di mod pi = 1I a :=
∑ki=1 ci · di · ni
I hence ni = a mod pi
�
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Recursion 42/65
ExamplesI recursive defintions
I in terms of simpler versions of itselfI some part avoids self-reference (vs. circular definitions)
I pushdown systemsI music: tonic and pseudo-tonicI language: verb at the endI indirect recursion in EpimenidesI Fib(n)=Fib(n-1)+Fib(n-2)I computer programsI fractalsI Cantor set
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Mandelbrot set 43/65
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Mandelbrot set 44/65
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Further examples 45/65
energies of electrons in a crystal in a magnetic field
Cantor set
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Sidenote: Location of meaning 46/65
I is meaning of a message an inherent property of the message?I meaning is part of an object to the extent that it acts upon
intelligence in a predictable wayI levels of informtion
I frame message: “this bears information”I outer message: “this is in Japanese”I inner message: “this says ...”
I if all juke-boxes would play the same song on “A-5”, it wouldn’t bejust a trigger but a meaning of “A-5”
I mass is intrinsic, weight is not; or yes, but at the cost of geocentricityI
·
·
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Propositional calculus: Definition 47/65
I purely typographicI alphabet: < > P Q R ′ ∧ ∨ ⊃ ∼ [ ]
I well-formed strings:I atoms: P, Q, R + adding primesI formation rules: if x and y are wel-formed then so are∼ x, < x ∧ y >, < x ∨ y >, < x ⊃ y >
I rulesI joining: x and y ⇒ < x ∧ y >I separation:< x ∧ y >⇒ x and yI double-tilde: ∼∼ can be deleted or insertedI contrapositive: < x ⊃ y > and <∼ y ⊃∼ x interchangableI De Morgan: ∼< x ∨ y > and <∼ x∧ ∼ y interchangableI Switcheroo: < x ∨ y > and <∼ x ⊃ y interchangableI no axioms
I fantasy rule (Deduction Theorem): y derived from x ⇒< x ⊃ y >I carry-over theorems into fantasyI detachment (Modus Ponens): x and < x ⊃ y >⇒ y
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Propositional calculus: Definition 47/65
I purely typographicI alphabet: < > P Q R ′ ∧ ∨ ⊃ ∼ [ ]
I well-formed strings:I atoms: P, Q, R + adding primesI formation rules: if x and y are wel-formed then so are∼ x, < x ∧ y >, < x ∨ y >, < x ⊃ y >
I rulesI joining: x and y ⇒ < x ∧ y >I separation:< x ∧ y >⇒ x and yI double-tilde: ∼∼ can be deleted or insertedI contrapositive: < x ⊃ y > and <∼ y ⊃∼ x interchangableI De Morgan: ∼< x ∨ y > and <∼ x∧ ∼ y interchangableI Switcheroo: < x ∨ y > and <∼ x ⊃ y interchangableI no axiomsI fantasy rule (Deduction Theorem): y derived from x ⇒< x ⊃ y >I carry-over theorems into fantasyI detachment (Modus Ponens): x and < x ⊃ y >⇒ y
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Propositional calculus: Properties 48/65
I decision procedure:
truth tables
I simplicity, precisionI other versions (axiom schemata + detachment)
extensions (valid propositional inferences,incompleteness/inconsistency only due to embedding system)
InformalI proof: normal thoughtI simplicity: sounds rightI complexity: human language
FormalI derivation: artificial, explicitI simplicity: trivialI astronomical size
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Propositional calculus: Properties 48/65
I decision procedure: truth tables
I simplicity, precisionI other versions (axiom schemata + detachment)
extensions (valid propositional inferences,incompleteness/inconsistency only due to embedding system)
InformalI proof: normal thoughtI simplicity: sounds rightI complexity: human language
FormalI derivation: artificial, explicitI simplicity: trivialI astronomical size
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Propositional calculus: Properties 48/65
I decision procedure: truth tables
I simplicity, precisionI other versions (axiom schemata + detachment)
extensions (valid propositional inferences,incompleteness/inconsistency only due to embedding system)
InformalI proof: normal thoughtI simplicity: sounds rightI complexity: human language
FormalI derivation: artificial, explicitI simplicity: trivialI astronomical size
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Propositional calculus: Contradictions 49/65
I < <P ∧ ∼P> ⊃ Q >
I infection vs. mental break-downI 1 − 1 + 1 − 1 + 1 · · ·I relevant implication
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Typographical number theory: Syntax 50/65
I natural-numbers theory N→ TNT1. 2 is not a square.2. 5 is a prime.3. There are infinitely many primes.
I primitives: for all numbers, there exists a number, equals, greaterthan, times, plus, 0, 1, 2, . . .
I variables: a, b , a′
terms: (a · b), (a + b), 0,S0,SS0atoms: S0 + S0 = SS0quantifiers: ∃b : (b + S0) = SS0, similarly ∀
Puzzle: encode the followingI b is a power of 2I b is a power of 10
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Typographical number theory: Syntax 50/65
I natural-numbers theory N→ TNT1. 2 is not a square.2. 5 is a prime.3. There are infinitely many primes.
I primitives: for all numbers, there exists a number, equals, greaterthan, times, plus, 0, 1, 2, . . .
I variables: a, b , a′
terms: (a · b), (a + b), 0,S0,SS0atoms: S0 + S0 = SS0quantifiers: ∃b : (b + S0) = SS0, similarly ∀
Puzzle: encode the followingI b is a power of 2I b is a power of 10
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Typographical number theory: Syntax 50/65
I natural-numbers theory N→ TNT1. 2 is not a square.2. 5 is a prime.3. There are infinitely many primes.
I primitives: for all numbers, there exists a number, equals, greaterthan, times, plus, 0, 1, 2, . . .
I variables: a, b , a′
terms: (a · b), (a + b), 0,S0,SS0atoms: S0 + S0 = SS0quantifiers: ∃b : (b + S0) = SS0, similarly ∀
Puzzle: encode the followingI b is a power of 2I b is a power of 10
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Propositional calculus: Examples 51/65
I ∼∀c : ∃b : (SS0 · b) = cI ∀c : ∼∃b : (SS0 · b) = cI ∀c : ∃b : ∼(SS0 · b) = cI ∼∃b : ∀c : (SS0 · b) = cI ∃b : ∼∀c : (SS0 · b) = cI ∃b : ∀c : ∼(SS0 · b) = c
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Propositional calculus: Derivations 52/65
Axioms:1. ∀a : ∼Sa = 02. ∀a : (a + 0) = a3. ∀a : ∀b : (a + Sb) = S(a + b)4. ∀a : (a · 0) = 05. ∀a : ∀b : (a · Sb) = ((a · b) + a)
Rules:1. specification: ∀u : x ⇒ x[u′/u] for any term u′
2. generalization: x ⇒ ∀u : x for a free variable u3. interchange: ∀u :∼ and ∼ ∃u : are interchangeable4. existence: x[u′/u]⇒ ∃u : x5. symmetry: r = s ⇒ s = r6. transitivity: r = s and s = t ⇒ r = t7. successorship: r = t ⇔ Sr = St
Example: S0 + S0 = SS0
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Propositional calculus: Derivations 52/65
Axioms:1. ∀a : ∼Sa = 02. ∀a : (a + 0) = a3. ∀a : ∀b : (a + Sb) = S(a + b)4. ∀a : (a · 0) = 05. ∀a : ∀b : (a · Sb) = ((a · b) + a)
Rules:1. specification: ∀u : x ⇒ x[u′/u] for any term u′
2. generalization: x ⇒ ∀u : x for a free variable u3. interchange: ∀u :∼ and ∼ ∃u : are interchangeable4. existence: x[u′/u]⇒ ∃u : x5. symmetry: r = s ⇒ s = r6. transitivity: r = s and s = t ⇒ r = t7. successorship: r = t ⇔ Sr = St
Example: S0 + S0 = SS0
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Propositional calculus: Derivations 52/65
Axioms:1. ∀a : ∼Sa = 02. ∀a : (a + 0) = a3. ∀a : ∀b : (a + Sb) = S(a + b)4. ∀a : (a · 0) = 05. ∀a : ∀b : (a · Sb) = ((a · b) + a)
Rules:1. specification: ∀u : x ⇒ x[u′/u] for any term u′
2. generalization: x ⇒ ∀u : x for a free variable u3. interchange: ∀u :∼ and ∼ ∃u : are interchangeable4. existence: x[u′/u]⇒ ∃u : x5. symmetry: r = s ⇒ s = r6. transitivity: r = s and s = t ⇒ r = t7. successorship: r = t ⇔ Sr = St
Example: S0 + S0 = SS0
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Typographical number theory: Induction 53/65
I can deriveI (0 + 0) = 0I (0 + S0) = S0I (0 + SS0) = SS0
I
...
I can derive ∀a : (0 + a) = a ?
I nor its negationundecidable in TNT (like Euclid’s 5th postulate in absolute geometry)
I rule of induction: u variable, X{u} well-formed formula with u free,X{0/u}, ∀u :< X{u} ⊃ X{Su/u} > ⇒ ∀u : X{u}
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Typographical number theory: Induction 53/65
I can deriveI (0 + 0) = 0I (0 + S0) = S0I (0 + SS0) = SS0
I
...
I can derive ∀a : (0 + a) = a ?I nor its negation
undecidable in TNT (like Euclid’s 5th postulate in absolute geometry)I rule of induction: u variable, X{u} well-formed formula with u free,
X{0/u}, ∀u :< X{u} ⊃ X{Su/u} > ⇒ ∀u : X{u}
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Typographical number theory: Consistency 54/65
I we believe in each ruleI are natural numbers a coherent construct??
Peano’s axioms:1. zero is a number2. every number has a successor (which is a number)3. zero is not a successor of any number4. different numbers have different successors5. if zero has X and every number relays X to its successor, then all
numbers have X
I want to convince of consistency of TNT using a weaker systemI Godel’s 2nd Theorem: Any system that is strong enough to prove
TNT’s consistency is at least as strong as TNT itself.
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Typographical number theory: Consistency 54/65
I we believe in each ruleI are natural numbers a coherent construct??
Peano’s axioms:1. zero is a number2. every number has a successor (which is a number)3. zero is not a successor of any number4. different numbers have different successors5. if zero has X and every number relays X to its successor, then all
numbers have X
I want to convince of consistency of TNT using a weaker systemI Godel’s 2nd Theorem: Any system that is strong enough to prove
TNT’s consistency is at least as strong as TNT itself.
![Page 80: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/80.jpg)
Typographical number theory: Consistency 54/65
I we believe in each ruleI are natural numbers a coherent construct??
Peano’s axioms:1. zero is a number2. every number has a successor (which is a number)3. zero is not a successor of any number4. different numbers have different successors5. if zero has X and every number relays X to its successor, then all
numbers have X
I want to convince of consistency of TNT using a weaker system
I Godel’s 2nd Theorem: Any system that is strong enough to proveTNT’s consistency is at least as strong as TNT itself.
![Page 81: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/81.jpg)
Typographical number theory: Consistency 54/65
I we believe in each ruleI are natural numbers a coherent construct??
Peano’s axioms:1. zero is a number2. every number has a successor (which is a number)3. zero is not a successor of any number4. different numbers have different successors5. if zero has X and every number relays X to its successor, then all
numbers have X
I want to convince of consistency of TNT using a weaker systemI Godel’s 2nd Theorem: Any system that is strong enough to prove
TNT’s consistency is at least as strong as TNT itself.
![Page 82: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/82.jpg)
Recall: MIU system 55/65
I alphabet M,I,UI initial string (“axiom”):
I MI
I rules1. xI⇒ xIU2. Mx ⇒ Mxx3. xIIIy ⇒ xUy4. xUUy ⇒ xy
I Can you produce MU ?
I No:I I-count starts at 1 (not multiple of 3)I I-count is a multiple of 3 only it was before applying the most recent
rule
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Recall: MIU system 55/65
I alphabet M,I,UI initial string (“axiom”):
I MI
I rules1. xI⇒ xIU2. Mx ⇒ Mxx3. xIIIy ⇒ xUy4. xUUy ⇒ xy
I Can you produce MU ?I No:
I I-count starts at 1 (not multiple of 3)I I-count is a multiple of 3 only it was before applying the most recent
rule
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Godel numbering 56/65
I All problems about any formal system can be encoded into numbertheory!
I define arithmetization on symbols (Godel number):I M↔ 3I I↔ 1I U↔ 0
I extend it to all strings1. MI↔ 312. MIU↔ 310
Example: Rule 11. xI⇒ xIU2. x1⇒ x103. x ⇒ 10 · x for any x mod 10 = 1
Typographical rules on numerals are actually arithmetical rules onnumbers.I Is MU a theorem of the MIU-system?I Is 30 a MIU-number?
1. “MU is a theorem” into number theory2. number theory into TNT
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Godel numbering 56/65
I All problems about any formal system can be encoded into numbertheory!
I define arithmetization on symbols (Godel number):I M↔ 3I I↔ 1I U↔ 0
I extend it to all strings1. MI↔ 312. MIU↔ 310
Example: Rule 11. xI⇒ xIU2. x1⇒ x103. x ⇒ 10 · x for any x mod 10 = 1
Typographical rules on numerals are actually arithmetical rules onnumbers.I Is MU a theorem of the MIU-system?I Is 30 a MIU-number?
1. “MU is a theorem” into number theory2. number theory into TNT
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Self-swallowing TNT 57/65
I Godel-number TNTI S0 = 0 is a theorem of TNT −→ 123, 666, 111, 666 is a TNT-number
I for any formalization of number theory, its metalanguage isembedded in it
I find a string G that says “G is not a theorem”I thoerem =⇒ truth =⇒ not a theorem =⇒ truth, but unprovableI summary:
There is a string of TNT expressing a statement about numbers(interpretable as “I am not a theorem (of TNT)”). By reasoningoutside of the system, we can show it is true. But still it is not atheorem of TNT (TNT says neither true nor false).
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Self-swallowing TNT 57/65
I Godel-number TNTI S0 = 0 is a theorem of TNT −→ 123, 666, 111, 666 is a TNT-numberI for any formalization of number theory, its metalanguage is
embedded in it
I find a string G that says “G is not a theorem”I thoerem =⇒ truth =⇒ not a theorem =⇒ truth, but unprovableI summary:
There is a string of TNT expressing a statement about numbers(interpretable as “I am not a theorem (of TNT)”). By reasoningoutside of the system, we can show it is true. But still it is not atheorem of TNT (TNT says neither true nor false).
![Page 88: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/88.jpg)
Self-swallowing TNT 57/65
I Godel-number TNTI S0 = 0 is a theorem of TNT −→ 123, 666, 111, 666 is a TNT-numberI for any formalization of number theory, its metalanguage is
embedded in itI find a string G that says “G is not a theorem”I thoerem =⇒ truth =⇒ not a theorem =⇒ truth, but unprovable
I summary:There is a string of TNT expressing a statement about numbers(interpretable as “I am not a theorem (of TNT)”). By reasoningoutside of the system, we can show it is true. But still it is not atheorem of TNT (TNT says neither true nor false).
![Page 89: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/89.jpg)
Self-swallowing TNT 57/65
I Godel-number TNTI S0 = 0 is a theorem of TNT −→ 123, 666, 111, 666 is a TNT-numberI for any formalization of number theory, its metalanguage is
embedded in itI find a string G that says “G is not a theorem”I thoerem =⇒ truth =⇒ not a theorem =⇒ truth, but unprovableI summary:
There is a string of TNT expressing a statement about numbers(interpretable as “I am not a theorem (of TNT)”). By reasoningoutside of the system, we can show it is true. But still it is not atheorem of TNT (TNT says neither true nor false).
![Page 90: Perlen der Informatik I - Theoretical Computer Sciencekretinsk/teaching/perlen/slides.pdf · Perlen der Informatik I Jan Kˇret ´ınsk y´ Technische Universitat M¨ unchen¨ Winter](https://reader033.fdocuments.us/reader033/viewer/2022060521/604f7b98d4f32c4564170111/html5/thumbnails/90.jpg)
Last words
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Free will, consciousness 59/65
I computer vs. human?I self-design, choosing one’s wants?
I Do words and thoughts follow formal rules?
I rules on the lowest level, e.g. neuronsI software rules change, harware cannot
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Free will, consciousness 59/65
I computer vs. human?I self-design, choosing one’s wants?
I Do words and thoughts follow formal rules?I rules on the lowest level, e.g. neuronsI software rules change, harware cannot
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Strange loops, tangled hierarchies: Examples 60/65
I self-modifying gameI Escher’s handsI symbols in brain (on neuronal substrate)I ? washing hands, dialogueI language, Klein bottleI Watergate
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Klein bottle 61/65
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Dualism 62/65
Subject vs ObjectI old scienceI prelude to modern phase: quantum mechanics, metamathematics,
science methodology, AI
Use vs MentionI symbols vs just beI John Cage: Imaginary Landscape No.4I Rene Magritte: Common Sense, The Two Mysteries
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Magritte: Common Sense 63/65
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Magritte: The Two Mysteries 64/65
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Self 65/65
I evidence, meta-evidence... → built-in hardwareI limitative theorems (Godel, Church, Turing, Tarski,. . . )
I imagine your own non-existenceI cannot be done fully, TNT does not contain its full meta-theory
I “self” necessary for free willI strange loops necessaryI not non-determinism, but choice-maker: identification with a
high-level description of the process when program is runningI Godel, Escher, Bach