On Formally Undecidable Propositions of Principia Mathematica and Related Systems
Gödel’s First Incompleteness Theorem: any consistent formal system, S, within which a certain amount of arithmetic can be carried out,
is incomplete with regard to the elementary arithmetic; there are statements which can be neither nor disproved within the system S.
To understand this theorem, we first
define some foundational concepts:
Kurt Gödel 1906-1978
Constant
signs
• A Formal System is a strict language
used to describe something with
formal rules. It has four components:
- Syntax: the base signs and symbols of the
language – for example: +, x
- Rules of formation: the “grammar” used to
combine symbols. For example: x+y not ++xy=
- Initial axioms: required as a starting point to
derive all other mathematical sentences. For
example: x+y = y+x
- Rules of transformation: methods which
allow you to combine axioms or rules. For
example: x+y = y+x & x = 2
2+y = y+2
• Consistency within a system: all
statements derived from the initial axioms
are TRUE within that system. This means,
for example, that we should not be able to
prove that 1=0
• Completeness within a system: all truths
in the system are derivable from the initial
axioms and rules of the system. Although
this makes logical sense, it is not
guaranteed.
Incompleteness: when all truths are
not derivable from the axioms and rules
within that system.
• Principia Mathematica was a formal
system devised by Bertrand Russell and
Alfred North Whitehead. They
believed their three volumes
encompassed all of arithmetic.
• Godel disproved this belief in his 1931
paper. He employed a common paradox
known by many names: Russel’s,
Epimenides’, Liar’s. The basic premise
is this: a statement which contradicts
itself (such as “this sentence is false”).
• Gödel accomplished this by using
Gödel Numbering: a method to
correspond all syntax and variables
(predicates, and sentences, and numeric
variables) with unique numbers which
preserved the truth value of statements.
1= ~
2= V
3= ⊃
4= ⱻ
5= =
6= 0
7= s
8= (
9= )
10= ,
11= +
12= x
Variable signs
Numerical variables:
Represented as primes
greater than 13 x, y, z,…
Undefined numbers
x=1, y=2, z=3
Sentential variables
(represented as the
squared of primes
greater than 13
p, q, r,…
Propositional expressions
x+y=z, 1+2=3
Predicate Variables
*assignment of Gödel
(classes
of
classes):
numbers always
represented
as
the
cube
proceed in magnitude
2, 3, 5, 7, etc. with of primes greater than 13
P, Q, R,…
the constant signs’
numbers being raised to the power of the prime
An Example of Gödel’s Numbering:
A 243,000,000
D 656
B 64x243x15,625
E 0=0
C 2^6x3^5x5^6
As you can see each number and each formula has a
specific string of finite symbols which allowed Gödel to
express statements about the system of PM WITHIN the
very system of PM and this time of mapping brought
great strength to his proof.
Citations
Delong, Howard. (1970). A profile of mathematical logic.
Dover Publications
Hofstadter, Douglas. (1979). Gödel, Escher, Bach. Basic
Nagel, Ernest; Newman, James R. ( 2001). Gödel's proof.
New York University Press.
Angelica Hope
Massachusetts.
Books Inc.
New York city.
Advisor: Edwin Herman