**Authors:** Daniel Cordero Grau

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.

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