Is there any mathematical theory without any axioms?

Theories such as set theory, number theory etc., all has axioms in it.

I have confusion between mathematical theory and the word theory in usage such as Automata theory.

  • $\begingroup$ You have not to conflate "formalized theory" with "theory" meaning "the study of ...". Automata theory is the study of the mathematical properties of abstract machines or automata, like linear algebra is the study of the math properties of matrices. $\endgroup$ – Mauro ALLEGRANZA Feb 17 at 14:10
  • $\begingroup$ If you consider instead a formal system for some branch of mathematics, based on e.g. first order logic, obviously we need some (at least one) mateìhematical axioms. Without them, what we have is simply the underlying logic, e.g. predicate calculus with equality. $\endgroup$ – Mauro ALLEGRANZA Feb 17 at 14:11
  • $\begingroup$ @MauroALLEGRANZA If both are different, is there any reason for making such nomenclature? $\endgroup$ – hanugm Feb 17 at 14:11
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    $\begingroup$ Theory is a term with a very broad meaning, used in many contexts. $\endgroup$ – Mauro ALLEGRANZA Feb 17 at 14:14
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    $\begingroup$ Among the many usage of "theory", we can find Axiomatic system : "any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. " This is quite different from e.g. the Theory of art. $\endgroup$ – Mauro ALLEGRANZA Feb 17 at 14:20

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