8 research outputs found
無限ゲームと様相μ計算の部分体系についてのオートマトン理論的研究
要約のみTohoku University田中一之課
(Extra)ordinary equivalences with the ascending/descending sequence principle
We analyze the axiomatic strength of the following theorem due to Rival and
Sands in the style of reverse mathematics. "Every infinite partial order of
finite width contains an infinite chain such that every element of is
either comparable with no element of or with infinitely many elements of
." Our main results are the following. The Rival-Sands theorem for infinite
partial orders of arbitrary finite width is equivalent to over . For each fixed , the
Rival-Sands theorem for infinite partial orders of width is
equivalent to over . The Rival-Sands theorem for
infinite partial orders that are decomposable into the union of two chains is
equivalent to over . Here
denotes the recursive comprehension axiomatic system,
denotes the induction scheme, denotes the
ascending/descending sequence principle, and denotes the stable
ascending/descending sequence principle. To our knowledge, these versions of
the Rival-Sands theorem for partial orders are the first examples of theorems
from the general mathematics literature whose strength is exactly characterized
by , by , and by
. Furthermore, we give a new purely combinatorial result by
extending the Rival-Sands theorem to infinite partial orders that do not have
infinite antichains, and we show that this extension is equivalent to
arithmetical comprehension over