## Ordinal Notations and Well-Orderings in Bounded Arithmetic (2002)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Beckmann02ordinalnotations,

author = {Arnold Beckmann and Samuel R. Buss and Chris Pollett},

title = {Ordinal Notations and Well-Orderings in Bounded Arithmetic},

booktitle = {Annals of Pure and Applied Logic},

year = {2002},

pages = {2003}

}

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### Abstract

this paper. 2 General orderings This section states a couple results about general orderings. By a "general ordering" we mean any order defined by a # 1 -formula; by comparison the results of sections 3 and 4 concern specific natural well-orderings based on ordinal notations

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Citation Context ...lly-ordered words over a finite alphabet. Finally, in section 6, we discuss a corollary of our theorems for the class Polynomial Local Search, PLS, introduced by Johnson, Papadimitriou and Yannakakis =-=[9]-=-. Buss-Krajcek [5] showed that the # b 1 -definable functions of T 1 2 are precisely the (multivalued) functions which are definable as the composition of a projection function and a PLS function. We ... |

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Citation Context ...over a finite alphabet. Finally, in section 6, we discuss a corollary of our theorems for the class Polynomial Local Search, PLS, introduced by Johnson, Papadimitriou and Yannakakis [9]. Buss-Krajcek =-=[5]-=- showed that the # b 1 -definable functions of T 1 2 are precisely the (multivalued) functions which are definable as the composition of a projection function and a PLS function. We define the classes... |

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Citation Context ...al theories of arithmetic is far too extensive for us to survey here. We shall include the basic definitions for ordinal notations of ordinals below # 0 and # 0 , and the reader can refer to Feferman =-=[6, 7]-=- or the textbooks of Schutte [12] or Pohlers [11] for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow inductio... |

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Citation Context ...us to survey here. We shall include the basic definitions for ordinal notations of ordinals below # 0 and # 0 , and the reader can refer to Feferman [6, 7] or the textbooks of Schutte [12] or Pohlers =-=[11]-=- for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow induction only on certain 1 Dept. of Mathematics, Univ. o... |

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Citation Context ...uch as S 1 2 , T 1 2 , S 2 2 and T 2 2 as introduced by the second author in [3]. We presume the reader is familiar with these theories of bounded arithmetic: the necessary background can be found in =-=[3, 4, 8, 10]-=-. Ordinal notations have been extensively used in the study of strong theories, ranging from primitive recursive arithmetic, to Peano arithmetic and to fragments of second-order arithmetic; however, t... |

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Citation Context ...such as S 1 2, T 1 2 , S 2 2 and T 2 2 as introduced by the second author in [3]. We presume the reader is familiar with these theories of bounded arithmetic: the necessary background can be found in =-=[3, 4, 8, 10]-=-. Ordinal notations have been extensively used in the study of strong theories, ranging from primitive recursive arithmetic, to Peano arithmetic and to fragments of second-order arithmetic; however, t... |

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Citation Context ...arithmetic theories [1]. His dynamic ordinals are based on exponential notations for integers, which are similar to the Cantor normal form representation of ordinals less than # 0 . It is known (c.f. =-=[13]-=-) that if one formalizes transfinite induction on ordinals in the usual way, then induction on ordinals below # # (or even induction on just # 2 ) is su#cient to give the full strength of primitive re... |

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Citation Context ...uch as S 1 2 , T 1 2 , S 2 2 and T 2 2 as introduced by the second author in [3]. We presume the reader is familiar with these theories of bounded arithmetic: the necessary background can be found in =-=[3, 4, 8, 10]-=-. Ordinal notations have been extensively used in the study of strong theories, ranging from primitive recursive arithmetic, to Peano arithmetic and to fragments of second-order arithmetic; however, t... |

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Citation Context ... too extensive for us to survey here. We shall include the basic definitions for ordinal notations of ordinals below ɛ0 and Γ0, and the reader can refer to Feferman [6, 7] or the textbooks of Schütte =-=[12]-=- or Pohlers [11] for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow induction only on certain 1 Dept. of Math... |

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Citation Context ...uch as S 1 2 , T 1 2 , S 2 2 and T 2 2 as introduced by the second author in [3]. We presume the reader is familiar with these theories of bounded arithmetic: the necessary background can be found in =-=[3, 4, 8, 10]-=-. Ordinal notations have been extensively used in the study of strong theories, ranging from primitive recursive arithmetic, to Peano arithmetic and to fragments of second-order arithmetic; however, t... |

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Citation Context ...al theories of arithmetic is far too extensive for us to survey here. We shall include the basic definitions for ordinal notations of ordinals below # 0 and # 0 , and the reader can refer to Feferman =-=[6, 7]-=- or the textbooks of Schutte [12] or Pohlers [11] for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow inductio... |

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Citation Context ...o extensive for us to survey here. We shall include the basic definitions for ordinal notations of ordinals below # 0 and # 0 , and the reader can refer to Feferman [6, 7] or the textbooks of Schutte =-=[12]-=- or Pohlers [11] for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow induction only on certain 1 Dept. of Math... |