Re: Growth Rate of Level-k Goodstein Function

"Deedlit" <roycepeng@xxxxxxxxxxx> wrote ...
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)]

The two equations are correct but are irrelevant to the ordinal
-- in each case n^^(n+1) is the correct base-n representation,
whereas the notations on the RHS are not in the required form.
(The definition of the unique hereditary representation is at
the link given in the first posting of this thread.)


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.

Matching up ordinal expressions seems to be about finding a
correspondence between Goodstein *ordinals* and a standard
ordinal notation -- which is NOT the same thing as finding a
correspondence between Goodstein *functions* and a standard
ordinal-indexed hierarchy of functions.
.

Relevant Pages

  • Re: What is the Size of V in this theory?
    ... Do the set of all countable ordinals exist in ZF? ... are axioms. ... I'll refer to that hierarchy. ... ordinals are well-ordered by the membership relation, ...
    (sci.logic)
  • Re: Request help in understanding levels.
    ... The levels are defined by transfinite recursion on the ordinals. ... Transfinite recursion is enabled by the axiom schema of replacement. ... Let 'e' be the membership symbol. ... Or once the hierarchy is defined, ...
    (sci.logic)
  • Re: Goodsteins Theorem
    ... PA^- plus Sigma_1 induction) to arithmetical transfinite induction on ... all ordinals below epsilon_0. ... given Goodstein sequence is bounded in Peano Arithmetic, ... as standard PA, ought not to be accepted as definitive. ...
    (sci.logic)
  • Re: Goodsteins Theorem
    ... all ordinals below epsilon_0. ... given Goodstein sequence is bounded in Peano Arithmetic, ... interpretation of Goodstein's Theorem as a number-theoretic assertion ... as standard PA, ought not to be accepted as definitive. ...
    (sci.logic)
  • Re: fixed point set theory
    ... are explaining for you the mistake you've made is ... none of the OP is standard, no standard ordinals for example. ... you call me childish, but its childish to correct me on things i am not even talking about ... ...
    (sci.math)