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Computability Learning SeminarExport this event to calendar

Tuesday, February 25, 2020 — 2:00 PM EST

Dino Rossegger, Department of Pure Mathematics, University of Waterloo

"Iterated Reflection principlesĀ 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.

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