A proof for completeness of Łukasiewicz logic

In this paper we give a new proof for the completeness of infinite valued propositional Łukasiewicz logic introduced by Łukasiewicz and Tarski in 1930. Our approach employs a Hilbert-style proof that relies on the concept of maximal consistent extensions, and unlike classical logic, in this context, the maximal extensions are not required to include all formulas or their negations. To illustrate this point, we provide examples of such formulas.