CS103
HO#42
SlidesGödel, Algorithms
5/9/11
1
In 1920, Hilbert proposed a research project that came to be know as
Hilbert's Program
.
He wanted to achieve:
A formalization of all of mathematics
A proof that mathematics is consistent (no contradictions
can be obtained).
A proof that mathematics is complete (all true statements
can be proved in the formalism).
An algorithm for deciding the truth of falsity of any statement
and thus the correctness of any proof.
Gödel showed that this was not possible.
Now, at last,
The Incompleteness Theorem
We say that a set of sentences T is
formally
complete
if for any sentence S in the language,
either S or ~S is provable from T.
The set of sentences in question are the axioms of Peano arithmetic,
and Gödel showed that Peano arithmetic is not formally complete.
To begin, we need to realize that as well as writing expressions within a
formal system, we can make statements about that system and those
expressions.
For example, the expression
2 + 3 = 5
belongs to mathematics, but the statement
'2 + 3 = 5' is an arithmetical formula
does not express an arithmetical fact and does not belong to the formal
language of arithmetic.
It belongs to
metamathematics
.
The formulas
x = x
0 = 0
belong to mathematics, but the statement
'x' is a variable
belongs to metamathematics, since it characterizes a certain arithmetical
sign as belonging to a specific class of signs, namely, variables.
The following statement also belongs to metamathematics:
The formula '0 = 0' is derivable from the formula 'x = x' by substituting the
numeral '0' for the variable 'x'.
Gödel devised a scheme that let him express metamathematical statements
as purely arithmetical relations.
This provided a way to make statements about the system inside the system
itself.
The starting point is a way to assign a unique number to every sign, formula
(sequence of signs), and proof (sequence of formulas).
We call these
Gödel numbers
.
not
1
or
2
if...then...
3
there exists
4
equals
5
=
zero
6
0
punctuation
10
,
punctuation
9
)
punctuation
8
(
successor of
7
s
Constant
Gödel
Usual
sign
number
meaning
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CS103
HO#42
SlidesGödel, Algorithms
5/9/11
2
0
11
x
s0
13
y
y
17
z
Numerical
Gödel
Possible
variable
number
substitution
There are three kinds of variables.
First are numerical variables like
'x', 'y', and 'z', for which numerals and numerical expressions
can be substituted.
Numerical variables are associated with primes greater than 10.
0 = 0
11
2
p
(
x)(x = sy)
13
2
q
p
q
17
2
r
Next are sentential variables like 'p', 'q', and 'r', for which formulas
(sentences) can be substituted.
Sentential variables are associated with squares of primes greater than 10.
Numerical
Gödel
Possible
variable
number
substitution
Prime
11
3
P
Composite
13
3
Q
Greater than
17
3
R
Next are predicate variables like 'P', 'Q', and 'R', for which predicates
like 'Prime' or 'Greater than' can be substituted.
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 Spring '09
 Algorithms, Formal system, Mathematical logic, Proof theory, Gödel's incompleteness theorems, Godel, Godel number

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