Ch5 - Gцdel's Theorem Limits on Logic

1
Ch5 - Gödel's TheoremLimits on Logic

• Jason HandUber

2

• there have been within the experience of people
  now living at least three serious crisesThere
  have been two such crises in physicsThe third
  crisis was in mathematics. It was a very serious
  conceptual crisis, dealing with rigor and the
  proper way to carry out a correct mathematical
  proof. In view of the earlier notions of the
  absolute rigor of mathematics, it is surprising
  that such a thing could have happened, and even
  more surprising that it could have happened in
  these latter days when miracles are not supposed
  to take place. Yet it did happen.

3
Overview

• Who is Gödel?

• History and such

• Necessary Definitions

• Gödel Numbering

• Gödels Theorem

• Effects Abuses of Gödels Theorem

• References, Discussion, Questions

4
Gödel who?

• Czech-born
  mathematician Kurt
  Gödel proposed his
  remarkable theorem
  in 1931 at the age of 25.
• John von Neuman (1963) is quoted (concerning
  Gödels theorem)

5
Before 1931

• In the early 1900s there was a quest within the
  mathematical community to reduce all of number
  theory to a formal axiomatic system. Such a
  system would start off with a few simple axioms
  and some mechanical way of deriving theorems from
  such axioms.
• Such a proof would yield a system that would
  represent every statement one could possibly make
  concerning natural numbers (i.e. a Universal
  Truth Machine).

6
Before (part deux)

• Gödel attempted his proof in the early 1930s,
  attempting to show that Principia Mathematica
  and related systems were complete.
• He proved that Principia Mathematica was either
  incomplete or inconsistent. Since it had been
  proven consistent, Gödel had shown it to be
  incomplete. Such failure led to Gödels theorem.

7
1931

• Gödels Theorem states (simplified) All
  axiomatic systems, including the whole of number
  theory, are incomplete.
• In other words, for any formal (valid) axiomatic
  system, Gödel proved that their exists a
  statement which is true but cannot be proven
  using the conventions of the given system.

8
After 1931

• Hermann Weyl (1946)
• We are less certain than ever about the ultimate
  foundation of (logic and) mathematics Outwardly
  it does not seem to hamper our daily work, and
  yet I for one confess that it has had a
  considerable practical influence on my
  mathematical life it directed my interests to
  fields I considered relatively safe, and has
  been a constant drain on the enthusiasm and
  determination with which I pursued my research
  and work. This experience is probably shared by
  other mathematicians who are not indifferent to
  what their scientific endeavors mean in the
  context of mans who caring and knowing,
  suffering and creative existence in the world.

9
Definitions
Axiom

• 1) A self-evident or universally recognized
  truth a maxim It is an economic axiom as old
  as the hills that goods and services can only be
  paid for only with goods and services (Albert
  Jay Nock).

• 2) An established rule, principle, or law. A
  self-evident principle or one that is accepted as
  true without proof as the basis for argument a
  postulate.

10
Definitions

• Axiomatic System - A logical system which
  possesses an explicitly stated set of axioms from
  which theorems can be derived.

11
Definitions

• Completeness - Can a system solve all problems
  that can be posed of it, or, in particular, can
  it be used to reason about all properties of its
  own members?

• For example, can a Turing machine be built that
  can decide which Turing machines halt on all
  input?

12
Definitions

• Gödel Numbering An elaborately clever system
  devised by Gödel that enables one to uniquely
  identify all possible statements created from
  Gödels building blocks.

• For instance

13
Gödel Numbering

• Given the following tables
• Constant Sign Gödel Number Meaning
• 1 not
• ?
  2 or
• ?
  3 implies
• ?
  4 there exists
• 
  5 equals
• 0
  6 zero
• s
  7 immediate successor
• (
  8 punctuation mark
• )
  9 punctuation mark
• ,
  10 punctuation mark
• 
  11 plus
• ?
  12 times

14
Godel Numbering

• Numerical Variable Gödel Number
  Possible Substitution
• x 13 0
• y 17 s0
• z
  19 y
• All Gödel Numbers are associated with prime
  numbers gt 12.
• Sentential Variable Gödel Number
  Possible Substitution
• p
  132 0 0
• q
  172 (?x) (x sy)
• r
  192 p ? q
• Predicate Variable Gödel Number
  Possible Substitution
• P 133 x sy
• Q
  173 (x ss0
  y)
• R
  193 (?z) (x y
  sz)

15
Gödel Numbering

• ( ? x ) ( x s y )
  

? ? ? ? ? ? ? ? ? ?
8 4 13 9 8 13 5 7 17 9
In order to use this system to its full advantage
we must assign a unique number to every possible
formula.
16
Gödel Numbering

• 28 34 513 79 118 1313 175 197
  2317 295

m
Suppose we have another (arbitrary) formula, lets
call it n. We can now conveniently denote a
particular sequence of formulas, lets call this
sequence k, defined as k 2m 3n
We now have the power to define an entire
SEQUENCE of formulas and represent such as one
(albeit large) number.
17
Gödels Theorem

• While Gödels Theorem has a direct impact on the
  Theory of Computability, the theorem in all its
  glory is best left to an advanced Number Theory
  class.
• I will present an alternate and much simpler yet
  less formal proof using the Paradox of the
  Liar.

18
Gödels Theorem

• 1) Assume we have a machine that represents a
  valid system of logic (axiomatic), lets call this
  system LoG (for logic)
• 2) Let G LoG will never say G is true
• 3) Ask LoG whether or not G is true
• 4) If LoG(G) is true, then G is false. If LoG(G)
  is false, then G is true. So, if LoG(G) returns
  true, then G is in fact false, and LoG has made a
  false statement. So, LoG will never say G is
  true, since LoG never makes false statements.
• 5) We know that G is true ? LoG cannot exist.

19
Gödels Theorem

• 1) Assume we have a machine that represents a
  valid (axiomatic) system of logic, lets call this
  system LoG (for logic)
• 2) Let G LoG will never say G is true
• 3) Ask LoG whether or not G is true
• 4) If LoG(G) returns true, then G is false. If
  LoG(G) is false, then G is true. So, if LoG says
  G is true, then G is in fact false, and LoG has
  made a false statement. So, LoG will never say G
  is true, since LoG never makes false statements.
• 5) We know that G is true ? LoG cannot exist!

20
Effects of Godels Theorem

• An axiomatic approach to number theory cannot
  fully characterize the nature of
  number-theoretical truth.
• Information-theoretic approaches to Godels
  theorem is actively being researched in the realm
  of Theoretical Physics.

21
Information-theoretic approaches

• Suggests that the incompleteness phenomenon
  discovered by Godel is more natural and
  widespread rather than pathological and unusual.
• Basic Conclusion if a theorem contains more
  information than a given set of axioms, then it
  is impossible for the theorem to be derived from
  the axioms.

22
Gödels effects on C.S.

• Significant effects on Artificial Intelligence.
• Dismisses the possibility of creating a
  Universal Truth Machine (based on an axiomatic
  scheme) or similar machines that could, given all
  axioms and valid rules (within some universe),
  deduce all truths (in such universe).

23
Gödel and Artificial Intelligence

• Gödels conclusions bear on the question whether
  a calculating machine can be constructed that
  would match the human brain in mathematical
  intelligence.
• Any machine which reasons solely on a fixed
  axiomatic method is incapable of answering
  innumerable problems in elementary number theory,
  no matter how complex or fast the machine may be.
• Consequently, there is no immediate prospect of
  replacing the human mind by robots.

24
Common Abuses of Gödels Theorem

• This theorem has been applied everywhere from
  philosophy to biology.
• We must remember that the domain of Gödels
  theorem is axiomatic systems and axiomatic
  systems alone.

25
References

• Nagel, Ernest. Newman, James. Gödel's Proof. New
  York University Press. 2001.
• - UCF Call QA9.65.N34 2001
• Gödels Incompleteness Theorem http//www.miskaton
  ic.org/godel.html
• Gödels Theorem http//www.cscs.umich.edu/crshali
  zi/notebooks/godels-theorem.html
• Gödels Theorem and Information.
  http//www.auckland.ac.nz/CDMTCS/chaitin/georgia.h
  tml
• Kennys Overview of Hofstadters Explanation of
  Gödels Theorem http//www2.ncsu.edu/unity/lockers
  /users/f/felder/public/kenny/papers/godel.html
• Mathematical Limitations to Software Estimation
  http//www.idiom.com/zilla/Work/Softestim/softest
  im.html
• Special thanks to Dr. Hughes

26
Additional Post-Speech Slides

• Another parallel approach to the Paradox of the
  liar / proof of Gödels theorem
• 1) Constructing a formula G in some arithmetic
  system (lets call this system PM for Principia
  Mathematica). This formula, G, represents the
  meta-mathematical statement The formula G is
  not demonstrable using the rules of PM. This
  formula thus says of itself that it is not
  demonstrable.
• 2) The formula G is then associated with its
  Gödel number, g, and G is thus constructed to say
  The formula that has Gödel number g is not
  demonstrable.
• 3) BUT, Gödel showed that G is demonstrable
  if and only if its formal negation, G, is also
  demonstrable. However, if both a formula and its
  complement are demonstrable, then the system of
  logic being used is NOT consistent (i.e. it
  contradicts itself). Otherwise (if we cant show
  that G and G are demonstrable), the formula (and
  thus system) is determined to be formally
  undecidable.

27
Gödel Numbering

• Gödel Numbering uses the s or immediate
  successor of function in order to represent
  numbers greater than 0. Recall that this system
  only represents natural numbers.
• Thus, to represent the number 4 in Gödel
  Numbering one would have to do the following
• 1) Replace the number with its appropriate
  symbol(s)
• 2) Replace those symbols with their corresponding
  Gödel number ? Raise that Gödel number to its
  corresponding prime
• Ex 4 ? ssss0 ? 7 7 7 7 6 ? 27 37 57
  77 96