Generally, a mathematical theory is characterized by its axioms (like group theory). Computability theory is characterized by definitions (such as those of Turing machines or general recursive functions) and doesn't introduce any new unique axioms of its own. So my question is: are there any other mathematical theories that share this feature or is it just computability theory?
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1Introduce new axioms to what? Which axioms do you assume to be given? ZFC? – Philipp Aug 07 '21 at 06:48
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I cannot think of any axioms of combinatorics, so I guess you have another example there. – Marc van Leeuwen Aug 07 '21 at 07:45
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The list of group "axioms" is really just a definition of what a group is. – Noah Schweber Aug 10 '21 at 05:33
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A group is usually defined as a pair $(G, \cdot)$ where $G$ is a set, $\cdot$ an operation on $G$, which together satisfy certain conditions. In this way it does not introduce any axioms, it only says things about objects which meet the definition.
In other words, the distinction between definition and axiom is often kind of arbitrary.
MennoK
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