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Mathematical theories are always based on some axioms. What else makes up an axiomatic theory? I mean , except from the axioms, we need some logical rules to draw conclusions and some definitions. What exactly are these definitions? (define definition!) I mean, can we use these axioms and define whatever we want? Are these "objects" we have defined part of the axiomatic theory? When is a definition considered correct or not ( or just correct for a specific axiomatic theory)? Can we always add new definitions to a theory? And finally, can a theory without definitions be useful ?

Thanks

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# Axiomatic theory

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