How does a theory being finitely axiomatizable differ from merely axiomatizable?
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Get StartedHow does a theory being finitely axiomatizable differ from merely axiomatizable?
Options:
- Finitely axiomatizable means a finite set of axioms generates the theory's consequences, whereas axiomatizable may require infinitely many axioms
- Finitely axiomatizable means the theory has finitely many models
- Axiomatizable means derivable in finite time, finitely axiomatizable means decidable
- There is no difference; the terms are interchangeable
Correct answer: Finitely axiomatizable means a finite set of axioms generates the theory's consequences, whereas axiomatizable may require infinitely many axioms
Explanation: Finitely axiomatizable means one finite axiom set entails exactly the theory's sentences, while axiomatizable may need infinitely many axioms; example: PA is axiomatizable but not finitely axiomatizable.
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