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Self Referencing and Gödel's
Incompleteness Theorem
Group 7
Nisarg Shah : 07005001
Siddharth Dhakad : 07005002
Chirag Sethi : 07005022
Shiv Shankar : 07005026
Dileep Kini : 07005033
Overview
• Self Reference
• Self Referencial Statements and Paradoxes
• Attempts at Resolution
• Truth and Provability
• Completeness, Soundness and Consistency
• Gödel's Incompleteness Theorem
• Douglas Hofstadter's Proof
• Typographical Number Theory (TNT)
• Tarski's Undefinability Theorem
• Some Implications of Gödel's Theorem
• Gödel vs Artificial Intelligence
• References
Self Reference
- Any object that refers to itself or its own referent
• Quines in computing
• Recursion in programming
• Self referential statements in linguistics
Courtesy :http://plato.stanford.edu/entries/self-reference/
Courtesy :http://www.cut-the-knot.org/selfreference/index.shtml
Tupper's Self Referential Formula
The points which satisfy the above
inequality when plotted form the following
graph:
Courtesy : http://en.wikipedia.org
Self Referential Statements and
Paradoxes
• Liar’s Paradox
• "This sentence is false.“
• “The next sentence is false. The previous sentence is true."
• Epimenides Paradox
(7th century BC)
Epimenides (who himself was a Cretan) said :
"All Cretans are liars”
Russell's Paradox for Sets
• S = the set of all sets that are not members of
themselves
• Is S a member of itself?
• If S is an element of S, then S is a member of itself
and should not be in S.
• If S is not an element of S, then S is not a member of
itself, and should be in S.
Attempts at Resolution
• Tarski's hierarchy of languages
o
introduce the constraint that elements of a given
level may only refer to elements of lower levels
• Prior's assumption
every statement includes an implicit assertion of its
own truth.
o S => "This sentence is true and S"
o
• Three valued Logic
o
o
o
o
Dialetheism - Graham Priest
True, False, Both true and false
Identifies Liar's sentence as both true and false
Inherent problems
Truth and Provability
• T-schema by Alfred Tarski to assign truth value of a
statement by inductive definition
o Alfred Tarski, "The Concept of Truth in Formalized
Languages", Corcoran, J. ed. Logic, Semantics
and Metamathematics, 1983
• Semantic theory of truth
• Provability on the other hand is being able to be
derive a sentence using language's axioms and
inference rules
Completeness and Consistency
• Completeness of a formal system:
o each true formula of the language of the system must be
derivable in the system
o Truth => Provability
• Consistency of a formal system:
o for no formula A of the language of the system both A or ¬A
are theorems of the system
• Godel's Incompleteness Theorem: In any consistent
system capable of expressing elementary arithmetic,
provability is a weaker notion of truth.
Gödel's Incompleteness Theorem
Kurt Gödel
Kurt Gödel, born April 28, 1906,
was an Austrian - American
logician , mathematician and
philosopher . One of the most
significant logicians of all time,
Gödel made an immense impact
upon scientific and philosophical
thinking in the 20th century.
Gödel is best known for his two
incompleteness theorems,
published in 1931 when he was 25
years of age, one year after
finishing his doctorate at the
University of Vienna.
Historical Perspective
• In the nineteenth and early twentieth centuries, one of the big
mathematical goals was to reduce all of number theory to a
formal axiomatic system.
• Russell and Whitehead's Principia Mathematica was the most
famous attempt
• Gödel's theorem dashed this hope completely
• Gödel showed that for any formal axiomatic system capable of
expressing elementary arithmetic, there is always a statement
about natural numbers which is true, but which cannot be
proven in the system
• The Gödel sentence combines three elements: the
representation of provability, self-reference and negation.
Douglas Hofstadter's Proof
• Hofstadter, Douglas R., "Gödel, Escher, Bach: An Eternal
Golden Braid”, 1979.
• Proof by Contradiction
• Defines a formal axiomatic system called Typographical
Number Theory (TNT)
• Assumes TNT to be complete and consistent, encapsulates
all of mathematics perfectly
• Leads to contradiction
Typographical Number Theory (TNT)
• TNT is an attempt to represent all of math in an
axiomatic way.
• TNT expresses everything in terms of a few simple
symbols.
Mathematical Symbols : +, *, =
Variables : a, a', a'', a''', ...
Logical Symbols : A, E, v,^, ~
o Numbers : 0, S0, SS0, SSS0, ...
o
o
o
TNT
• Statements
S0 + SS0 = SSS0
• Axioms
o Aa:~Sa=0
o Aa:(a+0)=a
o Aa:Aa':(a+Sa')=S(a+a')
o Aa:(a*0)=0
o Aa:Aa':(a*Sa')=((a*a')+a)
• Rules
o ~~ can be removed from any statement
o Au:~ and ~Eu: are interchangeable anywhere
inside a string.
o ... (Refer Appendix A for all rules)
o
TNT
• TNT-Theorem: TNT-Statement which can be inferred
from TNT-Axioms and TNT-Rules
• Example:
• S0+S0=SS0
• Aa:Aa':(a+a')=(a'+a) (Commutativity)
• ~Ea:a*a=SS0 (No root of 2)
• ~(Ea':Ea'':SSa'*SSa'' = SSSSS0) (Primality of 5)
• Can even express exponent or in general any
statement about numbers
• Assumption: Any true statement can be derived in
TNT
Gödel's Statement in TNT
• G: This statement is not a theorem of TNT
• If G is false, then it is a theorem of TNT. Then we
have a valid theorem which is false, this is not
possible.
• Hence G is true and it is not a theorem of TNT
• Hence there is a true statement in TNT which is not
provable in TNT proving the theorem.
• Question: TNT has statements about numbers. How
do we write G in TNT?
Gödel Numbering
•
•
•
•
•
•
Replace every symbol with three digit unique number
0 - 666
a - 262
S - 123
= - 111
and so on ...
• Every TNT-Statement thus has a unique Gödel number
• Rules become functions on natural numbers
[Kurt Gödel, "On formally undecidable propositions of Principia
Mathematica and related systems", Monatshefte für Mathematik
und Physik, 1931]
Gödel Numbering
• Every TNT-Statement thus has a unique Gödel number
o
Example
~ E a : a * a = SS0
223 333 262 636 262 236 262 111 123 123 666
• Rules become functions on natural numbers
o
Example
Dummy Rule: Whenever a string ends in the symbol "000",
you can replace that symbol with "005".
The same fake rule, written differently: Whenever a number is
a multiple of 1000, you can add 5 to it.
Theoremhood
• Theoremhood of a number: Being derivable by applying
certain functions (Rules) on certain initial numbers (Axioms)
• Theoremhood thus becomes a mathematical function and
hence can be represented by TNT by our assumption that
TNT can express all of mathematics
• Benefit of Gödel Numbering: A statement can talk about a
statement of TNT by using its Gödel number
• Difficulty: Gödel number of a statement is longer than the
statement itself => Intelligent way of referring
Arithmoquining
• Mechanism for writing a TNT sentence about another TNT
sentence
• Take a sentence with one free variable and replace all
occurrences of the free variable with the Gödel number of
the sentence.
• Example:
• T: a=S0
A: Sentence T is 1
where “Sentence T” is the Gödel number of T
• Arithmoquining is thus a mathematical function taking as
input Gödel number and giving Gödel number of its
arithmoquine and can be represented in TNT.
Finally G is here!!!
• T: The arithmoquine of a is not a valid TNT theorem-
number
• G: The arithmoquine of Sentence T is not a valid TNT
theorem-number
• G uses "arithmoquine of Sentence T" to intelligently
refer to itself
• G in words: "This statement is not a valid TNT
theorem"
• G is true and not a theorem of TNT as shown before
Some Implications of
Gödel’s Theorem
• No consistent system of axioms whose theorems
can be listed by a computer program is capable of
proving all facts about the natural numbers
• No sufficiently strong (and useful) reasoning
system can derive all truths
• Gödel's second incompleteness theorem
(assumption of consistency within the system leads
to inconsistency)
• Hilbert's second problem cannot be solved within
arithmetic
Notion of truth inside Formal system
• The statement of the Gödel's theorem invokes the
notion of truth.
• There is a precise characterization of what it
means for a statement in a formal language to be
true within a given structure for that language
• This characterization is independent of any syntactic
considerations, and in the case of formalized
arithmetic it is not expressible in the theory itself
(unlike the notion of theoremhood)
- John W Dawson Jr
Tarski's Undefinability Theorem
• The theorem says that given some formal arithmetic, the
concept of truth in that arithmetic is not definable using the
expressive means of that arithmetic
• It is possible to define a formula True(x) whose extension is
set of all true statements in the theory, only by drawing on a
meta-language whose expressive power goes beyond that
of L, say second-order arithmetic.
• No sufficiently powerful language is strongly-semantically-
self-representational
• This implies a major limitation on the scope of "self-
representation"
Gödel vs Artificial Intelligence
• Gödel's theorem has been taken to imply that we humans
have an innate ability to recognize a valid proof when we
see one.
• "Mathematical thinking (and hence conscious thinking
generally) is something that cannot be encapsulated within
any purely computational model of thought" - Penrose
• Reason is creative and original.
• Although any bit of reasoning can be codified into a set of
rules, there will always be further exercises of reason.
Gödel vs Artificial Intelligence
• Human mathematician cannot be accurately
represented by an algorithmic automaton.
• For any such automaton, there would be some
mathematical formula which it could not prove, but
which the human mathematician could both see, and
show, to be true.
o
Minds, Machines and Gödel - J. R. Lucas
• This argument is based on the inherent assumption
that human mind can escape Gödel's result
!!! Guarantee the last line of Lucas's argument !!!
Gödel vs Artificial Intelligence
• Gödel's theorem shows that if human reasoning was
computable, then it had to either be unsound, or it had to
be impossible for human to know both what human's
own reasoning powers were and to also know that they
were sound
• Gödel's theorem places limits on all 'sufficiently'
expressive reasoning systems (including human mind)
• This Gödelian limitation is not due to lack of human
intelligence, but is inherent in any reasoning system that
is capable of reasoning about itself
-McCulloug,"Can Humans Escape Gödel?"
Godel vs Artificial Intelligence
• One can never entirely understand oneself, since the
mind, can be sure of what it knows about itself only
by relying on what it knows about itself
• All the limitative theorems of mathematics suggest
that once the ability to represent your own structure
has reached a certain critical point it guarantees that
you can never represent yourself totally.
- Hofstader, "Godel,Escher, Bach"
• So when we ask "Is Artificial Intelligence possible?"
Gödel does not rule out AI.
Gödel vs AI
As a concluding remark to the above argument
regarding Godel's theorem and mind we quote an
ancient philosopher and poet :But as for certain truth, no man has known it,
Nor shall he know it, neither of the gods
Nor yet of all the things of which I speak.
For even if by chance he were to utter
The final truth, he would himself not know it:
Xenophanes
References
• Kurt Gödel, "On formally undecidable propositions of
Principia Mathematica and related systems", Monatshefte
für Mathematik und Physik, 1931.
• Hofstadter, Douglas R., "Gödel, Escher, Bach: An Eternal
Golden Braid”, 1979.
• Alfred Tarski, "The Concept of Truth in Formalized
Languages", Corcoran, J. ed. Logic, Semantics and
Metamathematics, 1983
• J. R. Lucas, "Minds, Machines and Gödel", Oxford
Philosophical Society, 1959
References
• D. McCullough, "Can Humans Escape Gödel?", 1995
• http://en.wikipedia.org/wiki/Godel's_incompleteness_theore
•
•
•
•
•
ms
http://www.cut-the-knot.org
http://en.wikipedia.org/wiki/Liar_paradox
http://www.sdsc.edu/~jeff/Godel_vs_AI.html
http://plato.stanford.edu/entries/self-reference/
http://www.cscs.umich.edu/~crshalizi/notabene/godelstheorem.html
Thank You!
Appendix A : Rules of TNT
•
All the rules for the Propositional Calculus are also rules of TNT
•
Rule of Specification:
If ∀u: x{u} is a theorem, then so is x{u} and so is x{v/u} where v is any term
Restriction: v cannot be quantified within x{u}.
If it's true for all, it's true for one.
•
Rule of Generalization:
If x{u} is a theorem, then so is ∀u: x{u}
Restriction: u must be free
If something is true for something without restrictions, then it's true for all.
•
Rule of Interchange:
The strings ∀u:~ and ~∃u: are interchangeable
If it isn't true for all, then there doesn't exist any for which it's true
.
Appendix A : Rules of TNT
•
Rule of Existence:
Any instance of a term may be replaced by u, if ∃u: is placed in front.
Restriction: The term cannot contain variables that aren't free.
•
Rules of Equality:
Symmetry: If r = s is a theorem, then so is s = r
Transitivity: If r = s and s = t are theorems, then so is s = t
•
Rule of Successorship:
Add S: If r = t is a theorem, then so is Sr = St
Drop S: If Sr = St is a theorem, then so is r = t
•
Rule of Induction:
If ∀u: <x{u} ⊃ x{Su/u}> is a theorem
and
x{0/u}>
is a theorem
then
∀u: x{u}
is a theorem
Restrictions: x{u} is well-formed; u is free