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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?

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  • 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 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?
    – Andrew Su
    Aug 23 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.
    – Conifold
    Aug 23 at 19:29
  • Well thank you for explaining the process. I'll try to reword my question in that sense.
    – Andrew Su
    Aug 23 at 19:57
  • See e.g. P.Maddy, [Defending the Axioms: ](google.it/books/edition/Defending_the_Axioms/FtoUDAAAQBAJ) Aug 24 at 8:48
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Let me try to summarize my understanding of your questions before I respond to it.

You seem to be asking about the creation of theories, and how they are motivated by fundamental assumptions. Once a theory has come to be, it is (usually) expressible in math, and that means ultimately in sets, the ground of math. Now, since the theory is motivated by fundamental assumptions, shouldn't we be able to identify those assumptions in the theory, and be able to express them with sets, just like we can express theorems with sets?

But before I answer those question, let me try to tackle a second question you raised. There is also the question of why we are using these particular assumptions, and not other ones? What justifies our assumptions?

Here I would answer that if you want to look back and evaluate your assumptions, it matters why you want to make a theory in the first place. Some people make theories because they think they are beautiful, in which case the assumptions are justified by the perceived beauty of the formal system. Other people want to be able to make machines, in which case the functionality of those machines would justify the assumptions.

What I'm getting at here, is that the assumptions that created a formal system will always be in a realm outside of the formal system itself. Now I haven't myself run through the technical proof of Gödel's incompleteness theorems, but nevertheless I smell Gödel here: Suppose you have a formal system that can 1) determine whether things are true of false 2) express the assumptions that motivated the creation of the theory. Then you would be able to use that formal system to evaluate the truth of those assumptions. I'm claiming that this is essentially the same as something the 2nd incompleteness theory explicitly prohibits: "Second incompleteness theorem For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself." Ref. 1.

Further, if you had a formal system that was able to express the assumptions that caused the creation of itself, that means you should also be able to mathematically express the concept of beauty, or functionality, or any other reason why someone looked at the world, thought about it, and decided to turn what they were thinking about into a formal system.

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