Re: Growth Rate of Level-k Goodstein Function


r.e.s. wrote:

The version using F_a seems to be another way of writing what I
posted 2006-09-21 (my x_i are your c_i, and my p is your m) ...

g_2(n) = f_p^x_p f_(p-1)^x_(p-1) ... f_0^x_0 a - a

that is,
B_2(n) = f_p^x_p f_(p-1)^x_(p-1) ... f_0^x_0 a

where many parentheses are omitted for easier reading, and where

f_0 (t) = t + 1
f_(k+1) (t) = f_k^(t+1) (t).


Hmm... I think for this to work, there has to be an increment on the
limit ordinals as well. I've seen the GW hierarchy with t+1 as the
exponent, but not in the fundamental sequence.



A question about terminology ...

I see the F_a hierarchy variously called the Wainer hierarchy,
the extended Grzegorczyk hierarchy, and the Grzegorczyk-Wainer
hierarchy; but, as I recall, the f_a (appropriately defined at
limit ordinals) were used by Wainer in his 1989 JSL paper
"Slow Growing versus Fast Growing". Are the f_a sometimes also
called "the Wainer hierarchy"?

I'd say they would also be called by any of the above names, just like
there are numerous variants of the Ackermann function.

.

Relevant Pages

  • Re: Growth Rate of Level-k Goodstein Function
    ... where H is the Hardy hierarchy, ... H_a = H_) for a limit, and athe fundamental sequence ... I see the F_a hierarchy variously called the Wainer hierarchy, ... the extended Grzegorczyk hierarchy, and the Grzegorczyk-Wainer ...
    (comp.theory)
  • Re: Growth Rate of Level-k Goodstein Function
    ... No limit ordinals are used in the above recursion for g_2, ... nonnegative integers -- the extended hierarchy isn't needed here. ... I see the F_a hierarchy variously called the Wainer hierarchy, ...
    (comp.theory)