Godel1

GodcI's Theorem

Kurt Godeł (1931): Uber formal unentscheidbare Satze der Principia mathematica und vervandter Systeme; based on presentation by Ernest Nagel and James R. Newman: Gódel's Theorem.

Godefs Theorem: In any formal system adeąuate for number theory there is an undecidable formula, ie. a formula that is not provable and whose negation is not provable. Yet the formula must either be true or false.

The Magie of Prime Numbers:

Every number can be presented as a product of prime numbers, eg.:

113176140	2	Number 113176140	is equivalent to
56588070	2
28294035	3	2^2 * 3^1 * 5^1 * 7^ł * 11^2 x	17'x 19°x...xi3i^, = 113176140
9431345	5	4 x 3 x 5 x 7 x 121 x	17 x l x ... xi3i = 113176140
1886269	7	t
269467	11
24497	11
2227	17
131	131

Constants	Godeł numbers	Meaning
	1	it is not true that
V	2	or
—>	3	if...then
3	4	there is
=	5	equals
0	6	zero
s	7	successor
(	8	left paranthesis
)	9	right paranthesis
•	10	dot
Variables	Godeł numbers	Examples
X	11	0 name
y	13	sO name
z	17	y name
p	ll^2	0=0 formula
	13^2	(3x)(x=s0) formula
r	17^2	p—>q formula
P	11^J	is a prime number predicate
Q	13^J	is a compound number ditto
R	17^3	is greater than predicate