The Consistency of Peano Arithmetic

Specialeforsvar: Bjørn Jackson Jakobsen

Titel: The Consistency of Peano Arithmetic
An Introduction to Gentzen’s First Order Logic, Peano Arithmetic and Proof Theory

Abstract:  Gödel dealt a great blow to mathematics when he proved that Peano arithmetic - arithmetic on the natural numbers - cannot prove its own consistency, at the time destroying any hope of proving Peano Arithmetic finitistic which informally means
proving arithmetic without the use of infinite arguments. That is until Gentzen came with a proof, he claimed to do just this. It does so by assigning ordinals to proofs, where these ordinals have size less then "0, which informally is the limit of all ordinals
we can write. He then showed that a proof of contradiction could not exist since it cannot be assigned one such ordinal. It has since been hotly debated, if such a proof is finitistic, since it is using induction up to an ordinal that is not finitistic (but all
proofs would have assigned a finitistic ordinal). This thesis will study Gentzen’s approach to first order logic as well as show some
general results of first order logic. It will then dive into Peano Arithmetic and show Gentzen’s consistency proof. Since some disagree with the finitistic nature of the proof, this thesis will then end with a proof of consistency of Peano Arithmetic, where
we limit proofs by the number of quantifiers used. In doing so, we will only need induction up to a finitistic ordinal. By picking the maximum number of quantifiers to be the number of atoms in the universe, we can at least assure that humanity will never find a proof of inconsistency of Peano Arithmetic.

Vejleder: Asger Dag Törnquist
Censor:     Jesper Bengtson, IT Universitetet