Axiomatic Foundations of Mathematics Ryan Melton Dr. Clint Richardson, Faculty Advisor Stephen F. Austin State University As Bertrand Russell once said, Gödel's Method Pure mathematics is the subject in which we Consider the expression First, Gödel assigned a unique natural number to
do not know what we are talking about, or 2 + 3 = 5 each of the logical symbols and numbers. whether what we are saying is true. Russell’s statement begs from us one major This expression is mathematical; it belongs to the field For example: if the symbol '0' corresponds to we call arithmetic and is composed of basic arithmetic question: the natural number 1, '+' to 2, and '=' to 3, then symbols. '0 = 0' '0 + 0 = 0' What is Mathematics founded on? On the other hand, the sentence and
'2 + 3 = 5' is an arithmetical formula. 1 3 1 1 2 1 3 1 so each expression corresponds to a sequence. Axioms and Axiom Systems is metamathematical; it is constructed outside of mathematics and labels the expression above as a Then, for this new sequence x x x …x of formula in arithmetic. 1 2 3 n An axiom is a belief taken without proof, and positive integers, we associate a Gödel number thus an axiom system is a set of beliefs as follows: x1 x2 x3 xn taken without proof. enc( x1x2x3...xn ) = 2 3 5 ... pn Since Principia Mathematica was such a bold where the encoding is the product of n factors, Consistent? Complete? leap in the right direction--although proving each of which is found by raising the j-th prime nothing about consistency--several attempts at to the xj power. An axiom system is: proving consistency were made by other ---consistent if no valid statement in the mathematicians of the time, the most notable of Thus, '0 = 0' corresponds to 1, 3, 1 theory is both provably true and provably these was… which is encoded as 1 3 1 false. Kurt Gödel! 2 3 5 = 270 ---complete if each valid statement in the and theory is provably true or provably false. Kurt Gödel (1906 - 1978) '0 + 0 = 0' corresponds to 1, 2, 1, 3, 1 was a German logician, which is encoded as The Issue and a personal friend of 21 32 51 73 111 = 339,570. Einstein. As a refugee Can an axiomatic system be both from the Nazi Party, By the Fundamental Theorem of Arithmetic, Gödel fled to the United consistent and complete? this representation as a product of powers of States, and came to the primes is unique. Institute of Advanced Study in New Jersey.
The answer (first attempt): There, he proceeded to develop a fascinating Thus, any formula can be uniquely represented Principia Mathematica! proof that would amaze all who were struggling as a Gödel number, and any given Gödel with a solution to consistency. number can be used to produce the original The NEW Issue formula. At its heart, Gödel's argument hinges on the This allowed Gödel to show a correspondence following question: However, according to Principia between statements about natural numbers and
Mathematica, the consistency of a given statements about the provability of theorems Can Metamathematics be discussed system relies on the consistency of formal in the context of mathematics? about natural numbers; this the key observation logic itself. of the proof. That is, Ultimately, metamathematics could be Thus, we (still) know NOTHING about addressed in the context of arithmetic! consistency! Can you speak ABOUT mathematics WITH mathematics? What is Metamathematics? CONCLUSION References Gödel's final result was the fact that no According to Hilbert, metamathematics is axiom system that could support arithmetic the language ABOUT mathematics. Goldstein, Rebecca. Incompleteness: The Proof and Paradox of Kurt Gödel. New York: W.W. Norton & Company, 2005. Print. could be both consistent and complete. Nagel, Ernest and James Newman. Gödel's Proof. New York: New York University Metamathematical statements are Press, 2001. Print. Weisstein, Eric W. "Complete Axiomatic Theory." From MathWorld--A Wolfram Web Thus, consistency and completeness cannot statements about the signs occuring within a Resource 3 Mar 2010. Web. 20 Mar 2010. formalized mathematical system. Weisstein, Eric W. "Consistency." From MathWorld--A Wolfram Web Resource. 3 Mar 2010. coexist within the foundations of axiomatic Web. 20 Mar 2010. mathematics. However, the goal is to understand that foundation.