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_posts/2022-05-03-mopa-tba.html

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+---
+layout: talk
+title: MOPA
+talk_title: The structural complexity of models of PA
+categories: MOPA
+date: 2022-05-03
+semester: spring-2020
+speaker_first: Dino  
+speaker_last: Rossegger
+speaker_website: "https://drossegger.github.io/"
+affiliation: UC Berkeley and TU Wien
+abstract: "
+<p>
+The Scott rank of a countable structure is the least ordinal $&#92;alpha$ such that all
+automorphism orbits of the structure are definable by infinitary
+$&#92;Sigma_{&#92;alpha}$ formulas. Montalbán showed that the Scott rank of
+a structure is a robust measure of the structural and computational complexity
+of a structure by showing that
+various different measures are equivalent. For example, a structure has
+Scott rank $&#92;alpha$ if and only if it has a $&#92;Pi_{&#92;alpha+1}$ Scott sentence if
+and only if it is uniformly $&#92;pmb &#92;Delta_&#92;alpha^0$ categorical if and only if
+all its automorphism orbits are $&#92;Sigma_&#92;alpha$ infinitary definable.
+</p><p>
+In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that 
+non-standard models of PA have Scott rank at least $&#92;omega$, but, other
+than that, there are no limits to their complexity. Given a completion $T$ of
+$PA$ we give a reduction via bi-interpretability of the class of linear orders
+to the models of $T$. This allows us to exhibit models of $T$ of Scott rank
+$&#92;alpha$ for every $&#92;omega&#92;leq &#92;alpha&#92;leq &#92;omega_1$. In particular, every
+completion of $T$ has models of high Scott rank.
+</p><p>
+This is joint work with Antonio Montalbán.
+</p>
+<p><strong><a href=''>Video</a></strong></p>
+ ​"
+excerpt_separator: <!--more-->
+talk: yes
+note: "<strong>2:00pm</strong> NY time<br>
+<font color='red' size='3'><strong>Virtual</strong> </font>(email <a href='mailto:vgitman@nylogic.org'>Victoria Gitman</a> for meeting id)"
+---