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With this second understanding of theory, the answer is a little more subtle. In the empty language, for example, every model is just a naked set, with no structure. There are exactly countably many countable models in this language(the : one of each finite ones, plus the size and one countably infinite one), and thus model. If φn is the assertion that there are exactly countably many n objects, then for any set A of natural numbers, we may form the theory TA, which asserts that ¬φn for each n in A. These theories are all inequivalent, and all true in this languageany infinite model. If the language has a binary relation symbolM is any model, then there will be are continuum many countable modelstheories TA that are true in M.

This shows that in fact every model M, since one can interpret this relation symbol as an equivalence relationin any language, and consider the possible countable models with an equivalence relationsatisfies at least continuum many deductively closed theories. By counting the sizes of

If the equivalence classeslanguage is larger, one can see that with uncountable size κ, then either there are continuum uncountably many countable modelsrelation symbols, uncountably many function symbols or uncountably many constant symbols. In generaleach case, suppose that M it is a model fun exercise to form 2κ many inequivalent theories T in a the languageof size κ. Let . Given any model M, let σ be any sentence false in M. For any theory T containing σ, we may form the theory T' = { σ implies φ | φ in T }. This theory is true in M, since σ is false in M. And Thus, by counting theories in this manner, one can show that there are 2κ many inequivalent theories true in M, except in trivial cases.
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You are using the terms "model" and "theory" in an idiosyncractic way.

In model theory, a model is a first-order structure, that is, a set with some functions, relations and perhaps distinguished elements, called constants. A theory, in contrast, is a collection of assertions, a set of sentences in this language. A given theory, which can be thought of as a set of axioms in the sense that you mentioned, can give rise to many models. And indeed, the Lowenheim-Skolem theorem says that if a theory has an infinite model, then it has infinite models of arbitrarily large cardinality. (Thus, except in trivial cases one cannot uniquely specify a model by giving "some (countable) number of axioms" as you said, since the same axioms will have models of many different sizes.)

Suppose that M is a model in a language of size κ, meaning that the language has κ many possible assertions. In this case, since any given assertion is either true in M or its negation is true in M, the complete theory of M, that is, the set Th(M) consisting of all sentences true in M, will also have size κ. Any subset S of Th(M) will also be true in M, of course. Thus, there are 2κ many theories true in M. For example, if the language has countably many symbols in it, then any given model in this language will satisfy continuum 2ω many theories.

For

But this answer counts theories as different, I had said that a theory is just a set when they are different merely as sets of sentences, which is a common definition. However, many even when these theories in this sense will have exactly the same consequences, and hence exactly the same models. But for the purposes of counting theories, so you it may not want be more sensible to count them as different. Another use another common definition is that a of theory, which is a set of sentences closed under consequence. This definition amounts to considering only equivalence classes of theories, where two identifying theories are equivalent if they that have the same models. And when you are counting theories, perhaps this definition is more sensible.

With this second understanding of theory, the answer is more subtle. In the empty language, for example, every model is just a naked set, with no structure. There are exactly countably many countable models in this language (the finite ones, plus the countably infinite one), and thus there are exactly countably many theories in this language. If the language has a relation symbol, even a unary binary relation symbol, then there will be continuum many countable models, since one can interpret this relation symbol as an equivalence relation, and consider the possible countable models with an equivalence relation. By counting the sizes of the equivalence classes, one can see that there are continuum many countable models.

In general, suppose that M is a model in a language of size κ. Let σ be any sentence false in M. For any theory T containing σ, we may form the theory T' = { σ implies φ | φ in T }. This theory is true in M, since σ is false in M. And by counting theories in this manner, one can show that there are 2κ many inequivalent theories true in M, except in trivial cases.

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You are using the terms "model" and "theory" in an idiosyncractic way.

In model theory, a model is a first-order structure, that is, a set with some functions, relations and perhaps distinguished elements, called constants. A theory, in contrast, is a collection of assertions, a set of sentences in this language. A given theory, which can be thought of as a set of axioms in the sense that you mentioned, can give rise to many models. And indeed, the Lowenheim-Skolem theorem says that if a theory has an infinite model, then it has infinite models of arbitrarily large cardinality. (Thus, except in trivial cases one cannot uniquely specify a model by giving "some (countable) number of axioms" as you said, since the same axioms will have models of many different sizes.)

Suppose that M is a model in a language of size κ, meaning that the language has κ many possible assertions. In this case, since any given assertion is either true in M or its negation is true in M, the complete theory of M, that is, the set Th(M) consisting of all sentences true in M, will also have size κ. Any subset S of Th(M) will also be true in M, of course. Thus, there are 2κ many theories true in M. For example, if the language has countably many symbols in it, then any given model in this language will satisfy continuum 2ω many theories.

For this answer, I had said that a theory is just a set of sentences, which is a common definition. However, many theories in this sense will have exactly the same consequences, and hence exactly the same models, so you may not want to count them as different. Another common definition is that a theory is a set of sentences closed under consequence. This definition amounts to considering only equivalence classes of theories, where two theories are equivalent if they have the same models. And when you are counting theories, perhaps this definition is more sensible.

With this second understanding of theory, the answer is more subtle. In the empty language, for example, every model is just a naked set, with no structure. There are exactly countably many countable models in this language (the finite ones, plus the countably infinite one), and thus there are exactly countably many theories in this language. If the language has a relation symbol, even a unary relation symbol, then there will be continuum many countable models, since one can interpret this relation symbol as an equivalence relation, and consider the possible countable models with an equivalence relation. By counting the sizes of the equivalence classes, one can see that there are continuum many countable models.

In general, suppose that M is a model in a language of size κ. Let σ be any sentence false in M. For any theory T containing σ, we may form the theory T' = { σ implies φ | φ in T }. This theory is true in M, since σ is false in M. And by counting theories in this manner, one can show that there are 2κ many inequivalent theories true in M, except in trivial cases.