True, but the generation will never terminate (as the list is countably infinite). Your next argument might be "oh I just need to generate T until I hit f_i" --

True, but the generation will never terminate (as the list is countably infinite). Your next argument might be "oh I just need to generate T until I hit f_i" -- which is true, but then I can introduce another function f_i_i which will bungle up your system again :) But this gets into the Second Incompleteness Theorem.