Author:
ARAI TOSHIYASU,WAINER STANLEY S.,WEIERMANN ANDREAS
Abstract
AbstractFollowing our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA,
$\Sigma ^1_1$
-DC
$_0$
, ATR
$_0$
, up to ID
$_1$
. The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal form representations of positive integers.
Publisher
Cambridge University Press (CUP)
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