Computability Learning Seminar

Dino Rossegger, Department of Pure Mathematics, University of Waterloo

"Iterated Reflection principles and Feferman's Completeness Theorem"

G\"odel's first incompleteness theorem is proved by constructing a sentence $\phi$ which is true but not provable in the given theory $T$. But what if one adds $\phi$ to $T$? Then one obtains a stronger theory. One question that arises is whether $T+\phi$ can prove that $T$ is consistent. A natural choice of $\phi$ is the statement that $T$ is consistent, i.e. $Con(T)$. The process of adding consistency statements to $T$ is usually referred to as a reflection principle. Turing explored the idea of iterating this process through the transfinite, i.e. let $T_0=T$ and $T_{\alpha}=T+\bigcup_{\beta<\alpha}Con(T_\beta)$. He showed that for every true $\Pi^0_1$ formula $\phi$ of $PA$, there is $a\in \mathcal O$ with $|a|=\omega+1$ such that $PA_a$ proves $\phi$ and conjectured that every true statement of $PA$ is provable in a theory obtained by iterating reflection on $PA$.

This conjecture was answered negatively by Feferman. However, he showed that if one modified the reflection principle slightly that for every true statement of $PA$ there is an ordinal notation $a\in\mathcal O$ with $|a|=\omega^{\omega^{\omega+1}}$ such that the statement is provable in the theory obtained by iterating this reflection principle along $|a|$. This result is known as Feferman's completeness theorem.

In this talk we will discuss reflection principles with a focus on Turing's and Feferman's results. If time permits I will sketch the proof of a new result by Pakhomov and Rossegger that obtains tight bounds on the order type of the ordinal in Feferman's completeness theorem. No prior knowledge of proof theory is required.

MC 5413