Yeah I ' m just being glib for the lols . <| sep |> What do you have against sporks ? <| sep |> one could just believe it ' s false instead of true . No , that ' s the same thing . When you have an indecidable proposition P , it ' s negation nonP is also indecidable , you can accept either as a new axiom but obviously not both at the same time . Each time you find an indecidable proposition , if you want to add it to your theory , you have to make a choice if you admit P or if you admit nonP . For example if you only admit Euclid ' s first four axioms , then the proposition PIn a plane , given a line and a point not on it , exactly one line parallel to the given line can be drawn through the point is indecidable . If you admit P then you are doing Euclidean geometry . If you don ' t admit it but rather admit something else instead , you ' re doing other kinds of geometry . So is P true ? Is it false ? Neither and both , it depends on what you choose and the context . I ' m familiar with recursionmy background is mathematics , not philosophySo is mine , I ' m a maths teacher , but I happen to be specialized in mathematical logic . I ' m not sure what you mean here about it being a schema of infinitely many axioms . For each formula n , y 1 , y 2 , . yk that you can write in Peano ' s arithmetic where y are fixed parameters and n is your variable if you can show that 1 0 , y 1 , y 2 , . yk is true 2 n n , y 1 , y 2 , . yk n 1 , y 1 , y 2 , . yk is truethen you have 3 n n , y 1 , y 2 , . yk . That ' s one axiom for each formula . P . S I ' m not extremely rigorous in the description of the recursion principle , I ' m trying to make a compromise between rigor and intelligibility .