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Are there any formal theories that are not axiomatic? Could you give me some examples?

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    $\begingroup$ Not in first-order predicate calculus, thogh the axioms may be specified in a way that does not permit verifying whether a sentence is an axiom. For example one may choose to look at the theory determined by all sentences true in the natural numbers. $\endgroup$ – André Nicolas Jun 17 '14 at 19:53

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