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 Kurt Gödel 1906-1978 define some foundational concepts: Constant Variable signs Numerical variables: signs Represented as primes • A Formal System is a strict language 1= ~ greater than 13 x, y, z,… 2= V Undefined numbers used to describe something with 3= x=1, y=2, z=3 formal rules. It has four components: 4= ⱻ - Syntax: the base signs and symbols of the 5= ⊃= Sentential variables language – for example: +, x 6= 0 (represented as the - Rules of formation: the “grammar” used to 7= s squared of primes combine symbols. For example: x+y not ++xy= 8= ( greater than 13 - Initial axioms: required as a starting point to 9= ) p, q, r,… derive all other mathematical sentences. For • Principia Mathematica was a formal 10= , example: x+y = y+x system devised by Bertrand Russell and 11= + - Rules of transformation: methods which Alfred North Whitehead. They 12= x Propositional expressions allow you to combine axioms or rules. For x+y=z, 1+2=3 believed their three volumes example: x+y = y+x & x = 2 2+y = y+2 Predicate Variables encompassed all of arithmetic. *assignment of Gödel (classes of classes): numbers always • Consistency within a system: all proceed in magnitude represented as the cube statements derived from the initial axioms • Godel disproved this belief in his 1931 2, 3, 5, 7, etc. with of primes greater than 13 P, Q, R,… are TRUE within that system. This means, paper. He employed a common paradox the constant signs’ known by many names: Russel’s, numbers being raised to the power of the prime for example, that we should not be able to prove that 1=0 Epimenides’, Liar’s. The basic premise An Example of Gödel’s Numbering: is this: a statement which contradicts A 243,000,000 D 6 5 6 B 64x243x15,625 E 0=0 • Completeness within a system: all truths itself (such as “this sentence is false”). C 2^6x3^5x5^6 in the system are derivable from the initial As you can see each number and each formula has a • Gödel accomplished this by using specific string of finite symbols which allowed Gödel to axioms and rules of the system. Although express statements about the system of PM WITHIN the this makes logical sense, it is not Gödel Numbering: a method to very system of PM and this time of mapping brought correspond all syntax and variables great strength to his proof. guaranteed. (predicates, and sentences, and numeric Citations Delong, Howard. (1970). A profile of mathematical logic. Massachusetts. variables) with unique numbers which Dover Publications Incompleteness: when all truths are Hofstadter, Douglas. (1979). Gödel, Escher, Bach. Basic Books Inc. not derivable from the axioms and rules preserved the truth value of statements. Nagel, Ernest; Newman, James R. ( 2001). Gödel's proof. New York city. New York University Press. within that system. Angelica Hope Advisor: Edwin Herman .