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SCOTT SENTENCE COMPLEXITIES OF LINEAR ORDERINGS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 13 December 2024, pp. 1-30
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- Article
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We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman–Stanley embedding on Scott sentence complexity and show that it only preserves
$\Pi ^{\mathrm {in}}_{\alpha }$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except 
$\Sigma ^{\mathrm {in}}_{3}$ and 
$\Sigma ^{\mathrm {in}}_{\lambda +1}$ for 
$\lambda $ a limit ordinal. We show that the former cannot be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.
