Godels Theorem

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Godels Theorem
Jesse M. Phillips February 21, 2005
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Outline

• Introduction History
• Why is it Important
• What is Godels Theorem
• Conclusions
• References

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Introduction History
- Kurt Godel - Born 1906 - Rheumatic Fever -
David Hilbert (See reference 2)
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Why Is It Important (1)
- Limits on Logic - Recursive Functions -
Mathematics is not complete (there are statements
that are true that we cannot prove) -
Philosophical Implications (4)
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Godels Theorem (1)
Three Elements 1. Set up axioms with rules of
inference 2. Set up axioms for standard
arithmetic 3. Define a numbering system for
each axiom (Godels Number)
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Notation
I will use different notation on some logic
symbols
For every A There exists a E
Compliment F not F F
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Rules of Inference
Some axioms from predicate calculus 1. Ayi
(F?(G?F) 2. Ayi ((F?(G?H)) ? ((F?G) ?
(F?H))) 3. Ayi ((F?G) ? ((F?G) ? F))) 4. Ayi
(Ax(F?G) ? (F?AxG)) provided F has no free
occurrences of X 5. Ayi ((F?G) ? (Ayi F ? Ayi G
)) 6. Ayi (Ax F(x) ? F(y)) provided that Y is
not quantified when it is substituted
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Rules of Inference
How to read rules of inference 1. Ayi
(F?(G?F) For all possible values in Y, if F is
true, then G implies F. 3. Ayi ((F?G) ?
((F?G) ? F))) If both G and G are implied by
F, then F cannot be true (F must be true).
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Arithmetic Axioms
1. Ax (0 sx) 2. Ax,y (sx sy) ? (xy) 3.
Ax x 0 x 4. Ax,y x sy s(x y) 5.
Ax,y x X sy xy x 6. Ax x X 0 0
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Arithmetic Axioms
Rules 1 and 2 1. Ax (0 sx) 2. Ax,y (sx
sy) ? (xy) Tell us zero is not the
successor of any natural number. If the
successors are equal, the numbers are equal.
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Mathematical Induction
We can logically prove an important rule that is
used for mathematical induction (remember
Discrete Structures?) (P(0)Ax(P(x) ? P(sx))) ?
AxP(x) If P is true for case O (base case) and
if every time it is true for X, it is also true
for x1, then it is always true.
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Godels Theorem
Three Elements 1. Set up axioms with rules of
inference 2. Set up axioms for standard
arithmetic 3. Define a numbering system for
each axiom (Godels Number)
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Assigning A Number
Code Symbol Code Symbol 1 0 9 x2 s 10
13 11 4 X 12 5 13 E6 ( 14
A7 ) 15 ?8 ,
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Multiple Variables
We have to rewrite formulas to fit this
criteria. For formulas with multiple variables,
we use the symbol x followed by a unique
subscript of 1s. Example Substitutions A
B C A x1, B x11, C x111 x1 x11
x111
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Assigning A Number
Code Symbol Code Symbol 1 0 9 x2 s
10 13 11 4 X 12 5 13 E6 ( 14 A
7 ) 15 ?8 ,
Axiom 4 x sy s(x y) Translate x1 sx11
s(x1 x11) 29 310 53 72 119 1310 1710
195 232 296 319 3710 413 439 4710
5310 597
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Heart of Godels Theorem
Proof(x,y,z) x is the Godel number of a proof
X of formula Y (with one free variable and Godel
number y) which has the integer z substituted
into it. Proof X Godel number xFormula
Y Godel number yz is an integer substituted into
formula Y.
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Heart of Godels Theorem
We now look at a special case, in which Y is fed
its own Godel number. Proof(x,y,y) x is the
Godel number of a proof of the formula obtained
by substituting its own Godel number y for its
one free variable. ExProof(x,y,y) Such a
proof does not exist.
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Godels Theorem
Suppose that ExProof(x,g,g) is provable, and p
is the Godel number of this proof P. Then we have
Proof(p,g,g) is this proof with g substituted as
the free variable. This, however, contradicts
ExProof(x,g,g). Therefore, we have proof by
contradiction that no proof exists. Therefore
ExProof(x,g,g) is true and is without proof.
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Heart of Godels Theorem
Statement You cannot prove me. We always fail
to prove this statement. Therefore the statement
is true even though we failed to prove it to be
true (3)
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Conclusions/Review
We are indebted to Godel for his work on
recursive functions, limitations of logic. We may
conclude from his work that the mathematical
system is not complete His studies are more far
reaching than mathematics and even have some
philosophical implications.
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References
1. New Turing Omnibus, Dewdney. Henry Holt and
Company, New York, 1993. 2. Kurt Godel,
OConnor, J.J., Robertson, E.F.
http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Godel.html 3. Kenny's Overview of
Hofstadter's Explanation of Gödel's Theorem,
Felder, K., 1996, http//www.ncsu.edu/felder-publi
c/kenny/papers/godel.html 4. Godels Theorems
and Truth, Graves, D., http//www.evanwiggs.com/a
rticles/GODEL.html