  Staying in the technical vein, several authors have noted a tension in the way Putnam's argument deals with the notion of finitude. On the one hand, Putnam needs to use this notion in order to characterize his model as an ω-model and (even) to make sense of the formal definitions of a first-order language and of the first-order satisfaction  relation.[47]  On the other hand, Putnam can't allow opponents of his argument to use this notion to specify what they think makes a model intended. If his opponents could use this notion, then they could define the notion of a model's being “well-founded,” and that would be enough to rule out the models generated by Putnam's theorem. In this sense, then, Putnam's argument seems to turn on an unmotivated asymmetry between the kinds of technical machinery that he himself uses and the kinds of machinery that he makes available to his critics. See Bays 2001 and Bellotti 2005 for developments of this point; see § 3.4 of Hafner 2005 for some critical reflections.