2008-01-28, 20:08

Recently I've been studying the correspondence between large (countable) ordinals and what I call "large number functions" over the natural numbers. It seems that when Cantor studied the ordinals, he wasn't aware of higher-order operations like tetration (or perhaps he chose not to use them for some reason?), so starting with w, w+1, w+2, ... w+w, w*3, w*4, ... w^2, w^3, w^4, ... w^w, w^w^w, w^w^w^w, ... he eventually stopped at . But really, this is simply w tetrated to itself, which is w pentated to 2. One can easily envision continuing the sequence: w pentated to 3, w pentated to 4, ... w pentated to w, and then w hexated to 2, w hexated to 3, w hexated to 4, ....

Now, my question, Cantor did define subsequent epsilon ordinals, but where exactly do they lie in the hierarchy of higher ordinal operations? For example, is defined to be the second ordinal satisfying . Does that mean it is equal to w pentated to 3? Or perhaps w pentated to w?

Also, what about the Ackermann function as applied to the ordinals? Does A(w,w) correspond with some ordinal in the Veblen hierarchy?

I wonder what operation the Feferman-Schütte ordinal corresponds with---if we map it back to functions over the natural numbers, it would seem to me that this would correspond with an extremely fast-growing function. I suspect this function would still be computable, since it is still below the Church-Kleene ordinal. I also wonder if the Church-Kleene ordinal corresponds with the Busy Beaver function. (These considerations are part of my attempt to see how close I can get to the Busy Beaver function while still staying within the realm of computability.)

Now, my question, Cantor did define subsequent epsilon ordinals, but where exactly do they lie in the hierarchy of higher ordinal operations? For example, is defined to be the second ordinal satisfying . Does that mean it is equal to w pentated to 3? Or perhaps w pentated to w?

Also, what about the Ackermann function as applied to the ordinals? Does A(w,w) correspond with some ordinal in the Veblen hierarchy?

I wonder what operation the Feferman-Schütte ordinal corresponds with---if we map it back to functions over the natural numbers, it would seem to me that this would correspond with an extremely fast-growing function. I suspect this function would still be computable, since it is still below the Church-Kleene ordinal. I also wonder if the Church-Kleene ordinal corresponds with the Busy Beaver function. (These considerations are part of my attempt to see how close I can get to the Busy Beaver function while still staying within the realm of computability.)