  In this paper we give a new proof for the completeness of infinite valued
propositional \L ukasiewicz logic introduced by \L ukasiewicz 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.
