# What are sufficient grounds for establishing a theory?

This question delves into the definition of a theory, but somewhat into the grounds of Set Theory as well. I was wondering on what grounds is theory established and accepted. To what degree do the fundamental assumptions that derive the axioms need to be "true" in order to be accepted?

In my eyes, the set the theory applies to must be well-defined, and the assumptions must all be situated within that well-defined set. Such as to make the proposed observations consistent, complete, and provable. However, in that case, how do we know that the assumptions are truly within the well-defined set and what do we do when exceptions arise?

• I'm sorry, but this question doesn't make sense because you are using terminology incorrectly, and I can't figure out what you really mean. A theory never becomes a theorem; a theory contains theorems. A theory (in the formal sense) is a set of the axioms and all of the statements/equations that follow from those axioms. Those statements/equations that follow from the axioms are the theorems. Aug 23, 2021 at 19:08
• I'm sorry, it should be building a theory from theorems correct? And would it make sense to ask about how assumptions become axioms? Aug 23, 2021 at 19:23
• Theorems are claims that can be deduced from axioms of a theory, so a theory can never become a theorem, nor is it built from them. Axioms of a mathematical theory can, in principle, be arbitrary. Empirical theories (from physics, chemistry, biology), when they are axiomatized, get their axioms by generalizing what happens in many observations and experiments. The more their theorems correctly predict what happens the more certain we become of their postulated axioms. If there are "exceptions" then they are beyond the scope of theory's applicability, and we need a different theory for them. Aug 23, 2021 at 19:29
• Well thank you for explaining the process. I'll try to reword my question in that sense. Aug 23, 2021 at 19:57
• See e.g. P.Maddy, [Defending the Axioms: ](google.it/books/edition/Defending_the_Axioms/FtoUDAAAQBAJ) Aug 24, 2021 at 8:48