02/23/2008, 05:46 PM
You do realize that that's just the Ackermann function applied to omega, right? w[w]w = A(w,w), which corresponds with phi_2(0). That's definitely computable, although not primitive recursive. (Primitive recursive functions correspond with the operations of finite order.)
But, as I've said above, the Ackermann function is but a mere scratch at the top 0.01% of the tip of the iceberg. As it stands, I've only barely made it 1% of the way down from the top of the tip of the iceberg---I still have no idea how to computationally get to phi_phi_1(0). And that's but one puny step out of infinitely-many needed to reach the Feferman-Schütte ordinal. (And this is disregarding the fact that every subsequent step is immensely larger than all previous steps combined.)
But, as I've said above, the Ackermann function is but a mere scratch at the top 0.01% of the tip of the iceberg. As it stands, I've only barely made it 1% of the way down from the top of the tip of the iceberg---I still have no idea how to computationally get to phi_phi_1(0). And that's but one puny step out of infinitely-many needed to reach the Feferman-Schütte ordinal. (And this is disregarding the fact that every subsequent step is immensely larger than all previous steps combined.)
