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PUNCTUAL CATEGORICITY AND UNIVERSALITY
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 22 October 2020, pp. 1427-1466
- Print publication:
- December 2020
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- Article
- Export citation
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We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is
${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is 
$0''$. We also prove that, with a bit of work, the latter result can be pushed beyond 
$\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.
