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CS103A
HO #57
Gödel II
3/12/08
1
Gödel's Incompleteness Theorem
Kurt Gödel (1906 – 1978)
(
∃
x
)
(
x
=
s
y
)
8
4
11
9
8
11
5
7
13
9
2
±·3
±·5
±·7
±·11
±·13
±·17
±·19
±·23±·29
This scheme allows us to represent every formula with a unique
number.
Given a number, we can determine whether it is a Gödel
number, and if so, we can recover the formula, since every number
has a unique prime factorization.
Gödel Numbering
(
∃
x
)
(
x
=
s
y
)
8
4
11
9
8
11
5
7
13
9
2
±·3
±·5
±·7
±·11
±·13
±·17
±·19
±·23±·29
S:
Syntactic remark:
S begins with '('
Arithmetic statement:
The Gödel number is S is divisible by 2
8
but not by 2
9
Correspondence between syntactic properties of sentences
and arithmetic properties of Gödel numbers
Statement A:
(p
∨
p)
⊃
p
Statement B:
(p
∨
p)
Syntactic remark:
B is an initial part of A
Arithmetic statement:
The Gödel number of A is divisible by
the Gödel number of B
Correspondence between syntactic properties of sentences
and arithmetic properties of Gödel numbers
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View Full DocumentCS103A
HO #57
Gödel II
3/12/08
2
Now consider the statement
'The sequence of formulas with the Gödel number x is a proof of the formula
with Gödel number z.'
This statement is represented by a definite formula in the arithmetic calculus
that expresses a purely arithmetic relation between x and z.
So the truth or falsity of the statement hangs on whether x and z bear the
proper arithmetic relationship.
We will denote this relationship with the name 'Dem', so we will write the
sentence above as
Dem(x,z)
Note that even though the statement at the top is a metamathematical
statement, there is a (complicated) formula for Dem in the arithmetic calculus.
We need one more notation, concerning substitution for a variable:
Suppose m is the Gödel number of the formula (
∃
x)(x = sy)
Subsitute in this formula for the variable with Gödel number 13 (i.e., y)
the numeral for m.
We get the formula (
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 Winter '07
 Plummer,R
 Computer Science

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