Some Interesting Properties of Field theories with an Infinite Number of   Fields
  We give an indication that gravity coupled to an infinite number of fields might be a renormalizable theory. A toy model with an infinite number of interacting fermions in four-dimentional space-time is analyzed. The model is finite at any order in perturbation theory. However, perturbation theory is valid only for external momenta smaller than $\lambda ^{-\frac{1}{2}}$ , where $\lambda$ is the coupling constant. 
N.Itzhaki