If that's true, then Yesenin-Volpin was just playin the role of a Turing machine trying to determine if a certain program halts (the program that counts from 0 to the input). If 3^^^3 (say) for a finitist doesn't exists, then s/he really has a different concept of number than you have (for example, they are not axiomatized by the Peano Arithmetic). It's fun to observe that ultrafinitism is axiomatic: if it's a coherent point of view, it cannot prove that a certain number doesn't exists, only assume it. I also suspect (but finite model theory is not my field at all) that they have an 'inner' model that mimics standard natural numbers...