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A theory is a set of sentences in some language. Usually, theories relate to models. For example, Euclidian geometry has a model consisting of an infinite set of dots. A subset of the Euclidian theory may have different models satisfying it (e.g. a finite set of points, etc). Most of mathematical theories we learn about in school have models like that. Sentences in a theory could be seen as "compressing" information on how parts of these models behave, e.g. Euler's formula V-E+F=2 dictates that if something is a convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

In practice, on the very basic level, first one works with the model and observes certain laws. Then, notation is introduced to describe general statements on the model and theorems are deduced. Finally one may want to find what axioms fully derive the model that we initially started from.

Does it ever happen that theories are useful on their own - without caring about what they model? One could imagine an edge case, where operating on certain abstract rules is used to derive a sentence in some theory, without caring what their model is and at the end this somehow turns out to be useful. If the answer is affirmative, any examples would also be helpful. (This question is not about whether a model exist and can be constructed, I'm assuming that if the theory is consistent, that implies some kind of model can be constructed)

Note: Fine comments below corrected some incorrect assumptions I made when asking the this question. If a theory is consistent, a model can be constructed - true but highly non-trivial. If a theory is complete, it does not mean there's just one model, it can have multiple non-isomorphic models.

anon2328
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  • "I'm assuming that if the theory is consistent that implies some kind of model can be constructed" This is true but highly nontrivial - see e.g. here. – Noah Schweber Mar 09 '21 at 16:40
  • Yes! Thank you for that. I had a second thought before writing that sentence (that it may not be trivial at all) – anon2328 Mar 09 '21 at 16:42
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    It's definitely nontrivial, and in my opinion it is the first really surprising result in logic. Neither incompleteness nor downward Lowenheim-Skolem were nearly as surprising to me. (That said, I think most people find downward Lowenheim-Skolem more counterintuitive than completeness.) – Noah Schweber Mar 09 '21 at 16:46
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    Modal logic was developed syntactically before a framework of mathematical models was understood. Of course there was still an informal motivation behind what ideas the symbols were intended to express, so not really sure if this counts. Also, in set theory the “standard model” was at first (and arguably still) too vague to reason about informally in a systematic and coherent way, so formal axiomatization led intuition rather than trailing it, to a greater extent than usual. – spaceisdarkgreen Mar 10 '21 at 05:33
  • I don't know that either of those count though, because it certainly wasn't the case that people 'didn't care what the axioms modeled'... they just didn't necessarily have a sharp conception of what they were intended to model. – spaceisdarkgreen Mar 10 '21 at 05:42
  • @spaceisdarkgreen so it's a scale, not a 0 or 1 thing. On the one end, for a complete theory, one has full realization on what the model is. Closer to the other end is the modal logic example you mentioned. Also, if a certain theorem depends only on a subset of the axioms (e.g. a theorem in geometry that does not depend on the 5th postulate in Euclidian geometry), then this could qualify as well, as the model is ambiguous? – anon2328 Mar 10 '21 at 13:35
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    A complete theory such as the theory of algebraically closed fields or real closed fields can have many non-isomorphic models. So you observation that "for a complete theory, full reallzation on what the model is" is wrong. – Rob Arthan Mar 10 '21 at 23:25
  • Thanks for clearing that up, I edited the question to include the comments. – anon2328 Mar 11 '21 at 13:28

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