Re: Growth Rate of Level-k Goodstein Function


r.e.s. wrote:


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.

No limit ordinals are used in the above recursion for g_2, just
nonnegative integers -- the extended hierarchy isn't needed here.
(Your recursion for g_2 in terms of the F_a also doesn't involve
limit ordinals, although the Hardy hierarchy version does.)

Yes, of course, silly of me.

Incidentally, I thought about whether we could extend this to all g_k.
I thought
that perhaps we could use the Hardy hierarchy and index them by
Goodstein's
"majorant ordinals", but after some examination I don't think the
higher limit ordinals
will reduce to the same smaller ordinals - for example, take w^^(w+1).

For n = 2, n^^(n+1) = (n^^n)^n
For n = 3, n^^(n+1) = (n^^n)^[(n^^2)^(n*2 + 2)]

We would need to find an ordinal expression that matches up for all n,
and it doesn't
look like we can do that. So we can't match up the Goodstein function
with an
ordinal hierarchy like we did up to level 3.

.

Relevant Pages

  • Re: Growth Rate of Level-k Goodstein Function
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