We show the existence of a phase transition at the level of measures for the generalized dimension of the maximal entropy measure in a model that was considered by F. Hofbauer and which is related to a model of M. Fisher. The model presented here is related to the one-dimensional Ising model in which a wall effect is assumed. In this situation, the problem has to be considered in the one-dimensional lattice N. In general there is no first-order transition for the Ising model in the lattice 77, but under our assumptions such transitions can occur. The Ising model has the purpose of explaining the magnetization of ferromagnetic systems at low temperatures. The main difference of our result from a previous result of F. Hofbauer is that the transition is analyzed in the setting of the generalized dimension. This setting is more closely related to the observables. The main purpose of this paper is to explain another mathematical model for phase transition using the mathematical results obtained by F. Hofbauer. We also use results of the thermodynamic formalism in an essential way.