Church's Undecidability Theorem is one of the meta-theoretical results of the mid-third decade of the last century, which along with other limiting theorems such as those of Gödel and Tarski have generated endless reflections and analyzes, both within the framework of the formal sciences, that is, mathematics, logic and theoretical computation, as well as outside them, especially the philosophy of mathematics, philosophy of logic and philosophy of mind. We propose, as a general purpose of this article, to formulate Church's Undecidability (...) Theorem and present the main ideas of its demonstration. In order to carry out the first objective, we need to introduce and explain the notions of recursive function and numbering used by Gödel, which will allow to formally and rigorously enunciate Church's Theorem. After we enunciate Church's Theorem of Unspeakability in a formal and rigorous manner, we will present the main ideas of the proof of Church's Undecidability Theorem for First Order Logic, which uses Robinson's axiomatic system for arithmetic and four facts about himself: In Robinson's system for arithmetic recursive functions are representable Robinson's system is undecidable, The number of axioms proper to the Robinson system is finite and The logical calculation of the Robinson system is equal to the calculation of the first-order logic. (shrink)

The present paper has three objectives: Presenting an actualization of a proof of the decidability of monadic predicates logic in the contemporary model theory context; Show examples of decidable and undecidable fragments inside First order logic, offering an original proof of the following theorem: Any formula of First of order logic is decidable if its prenex normal form is in the following form: ∀x1,…,∀xn∃y1,…,∃ymφ; Presenting a theorem that characterizes the validity of First order logic by the tautologicity of Propositional logic, (...) said result is interesting since immediately arises the doubt of how to conciliate said characterization with Alonzo Church’s Undecidability Theorem for First Order Logic. (shrink)