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A COMPLETE BOUNDED THEORY WITH UNBOUNDED TYPES
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 23 April 2026, pp. 1-14
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- Article
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One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say that a theory is bounded if there is an axiomatization involving only
$\forall _n$-formulas for some finite n, and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated automorphism orbit. Our main result, however, shows that this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact 
$\forall _1$-axiomatizable, which has unbounded types.
