## Computability-Theoretic and Proof-Theoretic Aspects of Partial and Linear Orderings (0)

Venue: | Israel Journal of mathematics |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Downey_computability-theoreticand,

author = {Rodney G. Downey and Denis R. Hirschfeldt and Steffen Lempp and D. Reed Solomon},

title = {Computability-Theoretic and Proof-Theoretic Aspects of Partial and Linear Orderings},

journal = {Israel Journal of mathematics},

year = {},

volume = {138},

pages = {2003}

}

### OpenURL

### Abstract

Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a well-partial ordering always have a well-ordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a well-partial ordering always has a well-ordered extension.

### Citations

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(Show Context)
Citation Context ...valent to ATR 0 , and the existence of a decomposition of a countable abelian group into a maximal divisible subgroup and a reduced group is equivalent to 1 1 -CA 0 . We refer the reader to Simpson [=-=Si98] for -=-more details. Again we can ask \What is the point of all this?" At one level, we can mention the greater insight one obtains from calibrating the precise resources needed to prove a theorem. We c... |

126 |
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(Show Context)
Citation Context ...for any i, any X i -computable linear extensionsofscontains an innites- 8 COMPUTABILITY- AND PROOF-THEORETIC ASPECTS OF ORDERINGS descending chain Turing computable ins. Since, by Jockusch and Soare [=-=JS7-=-2] and Simpson [Si98, Theorem VIII.2], there is a model of WKL 0 whose second-order part consists of all sets in the Turing ideal generated by a sequence X 0 T X 1 T : : : of uniformly low, uniforml... |

96 |
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(Show Context)
Citation Context ...ded partial ordering has a computably well-ordered linear extension which is computable in 0 0 , the Turing degree of the halting problem. (See Rosenstein [Ro84] for all these results, and Rosenstein =-=[Ro82] for more ba-=-ckground.) We sharpen these results as follows: Theorem 1. (1) \! is extendible" is provable in ACA 0 . (2) \! is extendible" proves WKL 0 over RCA 0 . (3) \! is extendible" is not p... |

83 |
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(Show Context)
Citation Context ...e mathematics, referring to Simpson [Si98] where necessary, and assume that the reader is familiar with the rudiments of computability theory, as found in an initial segment of Soare [So86] or Rogers =-=[Ro67]-=-. 2. The proof of Theorem 1 To prove part (1) of Theorem 1, simply observe that the proof of Kierstead and Rosenstein [Ro84] (see also [Do98, p. 909]) can be used: Fix a partial ordering P = hN;si. (N... |

76 |
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(Show Context)
Citation Context ...rems are \harder" then others. A beautiful example of this is the work on \fast growing Ramsey functions". One result in this area is the celebrated ParisHarrington version of thesnite Ramse=-=y Theorem [PH77]-=-, which is not provable in Peano Arithmetic. Here one shows that the theorem is equivalent to ACA 0 , and hence although the theorem is concerned only withsnite sets, any proof must nevertheless use i... |

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Computable algebra, general theory and theory of computable fields
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Citation Context ...antee that one can eectively generate a related structure B of a particular kind?" A pretty example can be obtained from the work of Rabin, Frohlich and Shepherdson, and Metakides and Nerode. Rab=-=in [Ra60-=-] demonstrated that if one is computably given aseld hF; ; +; 1 ; 0; 1i (so that F is a computable set coded by the natural numbers, upon which the normalseld operations are computable) then one can e... |

42 |
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(Show Context)
Citation Context ...s.) Now suppose fa s g s2N is as-descending sequence (coded in the model N ). By Ramsey's Theorem, we may assume without loss of generality that fa s j s 2 N g is as-antichain. (Note that by Jockusch =-=[Jo72]-=-, Ramsey's Theorem can be used inside the model N .) We may also assume that fa s g s2N issusual ordering of N ). We can now construct (inside the model N ) a subsequence fa j t g t2N and as-descendin... |

37 |
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(Show Context)
Citation Context ...rdered semiring, as above, and 0 1 -induction, as we dene below.) In his address 4 COMPUTABILITY- AND PROOF-THEORETIC ASPECTS OF ORDERINGS to the International Congress of Mathematicians, Friedman [F=-=r-=-74] identiedsve systems of second-order arithmetic specied by starting with the basic axioms of a discretely ordered semiring and the 0 1 -induction scheme ('(0) ^ 8x('(x) ! '(x + 1))) ! 8n('(n)) whe... |

29 |
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(Show Context)
Citation Context ...gave rise to Godel's powerful incompleteness theorems and, indirectly, to computer science through the work of Turing, von Neumann, and others. Another classical example of such questions was Dehn's [=-=De12]-=- word, conjugacy, and isomorphism problems insnitely presented group theory, which led to the formation of the subject of combinatorial group theory. In such studies one asks questions like \If one is... |

25 |
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(Show Context)
Citation Context ...markable fact that almost all of the classical theorems of mathematics are equivalent to one of thesve comprehension schemes above. Pursuing ourseld example, we note that Friedman, Simpson, and Smith =-=[FSS83] re--=-interpreted and extended the computability results of Rabin-Frohlich-Shepherdson-Metakides-Nerode mentioned above to show that the statement \Every countableseld has an algebraic closure" is prov... |

18 |
Computability theory and linear orderings
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(Show Context)
Citation Context ...stability of thesrst-order theory. From a computability-theoretic point of view, however, partial and linear orderings are very interesting as they allow a wide variety of codings (see, e. g., Downey =-=[Do98]-=-). In this paper, we prove some computability-theoretic results, as well as some corollaries for reverse mathematics, on partial and linear orderings. The starting point of our investigations is Szpil... |

18 |
Sur l’extension de l’ordre partiel”, Fund
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(Show Context)
Citation Context ... some computability-theoretic results, as well as some corollaries for reverse mathematics, on partial and linear orderings. The starting point of our investigations is Szpilrajn's Theorem (Szpilrajn =-=[Sz30]-=-). Any partial order P = hS;si has a linear extension L = hS;si. Given a property P of partial orderings, it is natural to ask whether P satisfying property P implies that L can be chosen to satisfy p... |

12 |
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(Show Context)
Citation Context ... well as independently by Galvin, Kostinsky, and McKenzie in the United States. A complete characterization of the countable extendible linear order types was obtained by Bonnet [Bo69]. In his thesis =-=[Ju-=-69], Jullien obtained a characterization of all countable weakly extendible linear order types (i. e., those such that any countable partial ordering P = hS;si not containing a chain of order type ... |

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5 |
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(Show Context)
Citation Context ...enstein, any computable, computably well-founded partial ordering has a computably well-ordered linear extension which is computable in 0 0 , the Turing degree of the halting problem. (See Rosenstein =-=[Ro84] for all-=- these results, and Rosenstein [Ro82] for more background.) We sharpen these results as follows: Theorem 1. (1) \! is extendible" is provable in ACA 0 . (2) \! is extendible" proves WKL 0 ... |

4 |
The Metamathematics of the Graph Minor Theorem, Logic and Combinatorics
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- 1987
(Show Context)
Citation Context ...e although the theorem is concerned only withsnite sets, any proof must nevertheless use innite sets. An even more striking example of this phenomenon is the work of Friedman, Robertson, and Seymour [=-=FRS87-=-], who proved that the Graph Minor Theorem (even for graphs of bounded tree-width) is not provable in 1 1 -CA 0 and hence the very complicated iterated minimal bad sequence arguments are, in some sen... |

4 | Extending partial orders to dense linear orders
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(Show Context)
Citation Context ... in which we will be analyzing extensions of partial orderings to linear ones. Already, we know that this area should be full of metamathematical complexities because of the work of Slaman and Woodin =-=[SW-=-98]. They answered a question ofsLos, who had asked for a classication of those partial orderings with a dense linear extension. They showed that the collection is not Borel, that is, not 1 1 , and h... |

2 |
Eective procedures in theory
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(Show Context)
Citation Context ... (so that F is a computable set coded by the natural numbers, upon which the normalseld operations are computable) then one can eectivelysnd a computable algebraic closure. Frohlich and Shepherdson [F=-=S56]-=- showed that one can be given two computable algebraic closures of the same computableseld which are not computably the same. This is interesting because the usual method of generating algebraic closu... |

1 |
Strati et extension des genres de cha^nes denombrables
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- 1969
(Show Context)
Citation Context ...d Pouzet in France, as well as independently by Galvin, Kostinsky, and McKenzie in the United States. A complete characterization of the countable extendible linear order types was obtained by Bonnet =-=[Bo6-=-9]. In his thesis [Ju69], Jullien obtained a characterization of all countable weakly extendible linear order types (i. e., those such that any countable partial ordering P = hS;si not containing a ... |

1 |
Extension et strati d'ensembles disperses
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(Show Context)
Citation Context ... an additional oracle for X i . This concludes the proof of part (3) of Theorem 1. 3. The proof of Theorems 2 and 2A To prove part (1) of Theorem 2, simply observe that the proof of Bonnet and Pouzet =-=[BP6-=-9] (see also [BP82, p. 140]) can be adapted here: Given a partial ordering P = hP;si, we say an element a of P iss-good if hP (P a);si has a scattered linear extension; and >P -good if hP (P a);si has... |

1 |
Linear extensions of ordered sets, Ordered Sets (Proceedings of a
- Bonnet, Pouzet
- 1981
(Show Context)
Citation Context ...chain scattered, then \ is extendible" simply means that any scattered partial ordering can be extended to a scattered linear ordering. (We refer the reader to the survey papers by Bonnet and Pou=-=zet [BP82]-=- and by Downey [Do98] for more background on linear extensions of partial orderings, and on computability-theoretic aspects of linear orderings, respectively.) 1 In the literature, this is sometimes r... |

1 |
Metakides and Anil Nerode, Eective content of theory
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- 1979
(Show Context)
Citation Context ...s is to adjoin roots, and this necessarily species a unique computable closure. So, in particular,sRabin's theorem must use a dierent method of constructing algebraic closures. Metakides and Nerode [M=-=N79]-=- explained the phenomenon by proving that a computable eld has a computably unique computable algebraic closure iff it has a (separable) splitting algorithm, which means, roughly speaking, that a comp... |