Transcript Gödel's Incompleteness Theorem Dr. Philip Cannata Gödel's Incompleteness Theorems – see Delong pages, 165 - 180 Gödel showed that any system rich.

Gödel's Incompleteness Theorem
Dr. Philip Cannata
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Gödel's Incompleteness Theorems – see Delong pages, 165 - 180
Gödel showed that any system rich enough to express primitive recursive
arithmetic (i.e., contains primitive recursive arithmetic as a subset of
itself) either proves sentences which are false or it leaves unproved
sentences which are true … in very rough outline – this is the reasoning
and statement of Gödel's first incompleteness theorem. [ DeLong page,
162]
Wikipedia - The first incompleteness theorem states that no consistent
system of axioms whose theorems can be listed by an "effective
procedure" (e.g., a computer program, but it could be any sort of
algorithm) is capable of proving all truths about the relations of the
natural numbers (arithmetic). For any such system, there will always be
statements about the natural numbers that are true, but that are
unprovable within the system. The second incompleteness theorem, an
extension of the first, shows that such a system cannot demonstrate its
own consistency.
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Gödel Numbering
1
3
5
7
9
11
13
17
19
23
‘0’
‘’’
‘-’
‘=>’
‘V’
‘(‘
‘)
‘x’
‘y’
‘z’
29
31
37
41
43
47
53
…
‘=‘
‘+’
‘.’
‘x1’
‘y1’
‘z1’
‘z2’
…
1 = (0)’ = 211 x 31 x 513 x 73
The following proof would be a sequence of sequences of symbols
which would correspond to a single Gödel number (2g1 x 3g2 x 5g3)
which is the proof of 2 + 1 = 3. See DeLong page 167 for another
example
Dr. Philip Cannata
g1
(0’’ 0 0’’)
g2
(0’’ 0’ (0’’ 0 x)’)
g3 (0’’ 0’ (0’’)’) => (0’’ 0’ 0’’’)
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Pages 166 – 170 in DeLong
Proof in primitive recursive arithmetic
xPy
(2g1 x 3g2 x 5g3) P (2 + 1 = 3)
Formula provable in A (see page 127 for the definition of A)
Prov(x) = (∃y)(yPx)
Primitive Recursive Substitution Function
Sb(x yz)
(let ((y z)) x)
Universal Generalization
xGeny
Example:
x+0=x
(∀x) x + 0 = x is 17Genc1 – the first equation on page 167 is c1
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Correspondence Lemma:
Page 171 in DeLong
For every primitive recursive relation P(x1,…,xn) (e.g., xPy) there exists (in A) a
formula (with Gödel number r) which contains free variables with Godel numbers
g1, …,gn such that for all x1,…,xn:
P(x1,…,xn) -> Prov(let ((17 Nml(x1)) (19 Nml(x2))…(gn Nml(xn))) r))
not P(x1,…,xn) -> Prov(Neg(let ((17 Nml(x1)) (19 Nml(x2))…(gn Nml(xn))) r)))
where 17 is ‘x’ and 19 is ‘y’
Example:
Consider 1 + 2 = 3
x + y = z is a primitive recursive relation
((x) + (y)) = (z) is the corresponding formula in A such that
0’ + 0’’ = 0’’’ is provable
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not xP(let ((19 Nml(y))) y) -> Prov(let ((17 Nml(x))(19 Nml(y))) q))
xP(let ((19 Nml(y))) y) -> Prov(Neg(let ((17 Nml(x))(19 Nml(y))) q)))
not 0P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(0))(19 Nml(p))) q))
not 1P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(1))(19 Nml(p))) q))
not 2P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(2))(19 Nml(p))) q))
.
.
.
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GA is (∀x) not xP(let ((19 Nml(y))) y)
G is formula in A that has Gödel number (let ((19 Nml(y))) y)
GM is “G is not provable in A”.
Case 1: Suppose G is provable in A, then GA is false and
there is an n such that
nP(let ((19 Nml(y))) y),
but from 2) on the previous page this would mean
Prov(Neg(let ((17 Nml(n))(19 Nml(y))) q)))
However, if G is provable, it also means that
Prov((let ((17 Nml(n))(19 Nml(y))) q)))
(see 5) on the next page)
Therefore, A is inconsistent or G is not provable in A.
I.e., GA is TRUE, and G is not provable if A is consistent.
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Case 2: Suppose not G is provable in A,
Then GA is true which implies
(∀x) Prov(let ((17 Nml(x))(19 Nml(p))) q)
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GA is TRUE if A is consistent, but
under the interpretation of A as
primitive recursive arithmetic,
GA = G. Therefore G is TRUE and
not provable.
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Gödel's Incompleteness Theorem
If “proof” is a proof of “statement”
then P is True.
If you have a statement g with variable x and if, when
you substitute g for x, you produce “statement” then Q is
True.
not P(proof, statement) && Q(x, statement) = g
Let g be the
Gödel number for
this statement,
A recursive notion.
But now science, spurred on by its powerful delusion, hurtles inexorably towards its limits where the
optimism hidden in the essence of logic founders. For the periphery of the circle of science has an
infinite number of points and while there is no telling yet how the circle could ever be fully surveyed,
the noble and gifted man, before he has reached the middle of his life, still inevitably encounters such
peripheral limit points and finds himself staring into an impenetrable darkness. If he at that moment
sees to his horror how in these limits logic coils around itself and finally bites its own tail - then the
new form of knowledge breaks through, tragic knowledge, which in order to be tolerated, needs art as a
protection and remedy.
Friedrich Nietzsche (1844 - 1900) The Birth of Tragedy
not P(proof, statement) && Q(g, statement) = s
Let s be the Gödel number
for this statement but by
the definition of Q that
means “statement” is “s”.
not P(proof, s) && Q(g, s) - I am a statement that is not provable.
There are Predicate Logic Statements that are True that can’t be proved True
(Incompleteness) and/or there are Predicate Logic Statements that can be proved True that
are actually False ( Inconsistent Axioms or Unsound inference rules).
i.e., If Gödel's statement is true, then it is a example of something that is true for which there is no proof.
If Gödel's statement is false, then it has a proof and that proof proves the false Gödel statement true.
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Gödel's Incompleteness Theorem
I am a statement that is not provable.
There are Predicate Logic Statements that are True that can’t be proved True
(Incompleteness) and/or there are Predicate Logic Statements that can be proved True that
are actually False ( Inconsistent Axioms or Unsound inference rules).
i.e., If Gödel's statement is true, then it is a example of something that is true for which there is no proof.
If Gödel's statement is false, then it has a proof and that proof proves the false Gödel statement true.
Logic/Math/CS
Physics
Theology
Philosophy
Plotinus
Unsound
Superposition
S
L
F
T
P
Consubstantial
G
W
F
S
The ONE
Is nothing else but The ONE,
it can’t even be finite.
H
Plato
The Forms (e.g. Justice)
Opposite is Excluded Middle
~p or p
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Self
Other
Trace of
The One
Finite
…
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Good Books to Have for a Happy Life 
From Frege to Gödel:
Dr. Philip Cannata
My Favorite
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