ON THE MINIMALITY OF NON-$\sigma$-SCATTERED ORDERS

Abstract

In this dissertation we study the minimality of non-$\sigma$-scattered orders. While there are insightful theorems, due to Laver, about $\sigma$-scattered orders, we will show the class of non-$\sigma$-scattered orders tend to be more chaotic by a number of consistency results. For instance, we show if there is a supercompact cardinal, there is a forcing extension in which there is no minimal non-$\sigma$-scattered linear order. This shows that Laver's theorem regarding $\sigma$-scattered linear orders is sharp. Our work also includes results concerning trees. For instance, we show it is consistent that there is a Kurepa tree which is minimal with respect to club embeddings. Moreover, we show it is consistent that there is a minimal non-$\sigma$-scattered linear order which does not contain any real or Aronszajn type. Working on these problems resulted in a few byproduct theorems as well.