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## Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of . . .

Venue: | INSTITUTE FOR MATHEMATICAL SCIENCES, NATIONAL UNIVERSITY OF SINGAPORE, WORLD SCIENTIFIC |

Citations: | 3 - 1 self |

### Citations

161 |
Soare, Π01 classes and degrees of theories,
- I
- 1972
(Show Context)
Citation Context ...roved this result to show that every instance X of WKL has a solution that is strictly weaker than X ′ (in the sense of Turing reducibility). In their celebrated Low Basis Theorem, Jockusch and Soare =-=[105]-=- showed that, in fact, every instance X of WKL has a solution Y such that Y ′ (and even (X ⊕ Y )′) is computable in X ′. These and other results on the complexity of WKL, which we will further discuss... |

117 |
On notation for ordinal numbers.
- Kleene
- 1938
(Show Context)
Citation Context ... that function. It thus forms the theoretical underpinning of the common programming practice of having a routine make recursive calls to itself. See e.g. Soare [196] for a proof. Theorem 2.1 (Kleene =-=[110]-=-). Let f be a total computable function. Then there is an e such that Φe = Φf(e). The halting problem relative to X is X ′ = {e : ΦXe (e)↓}. Thus the unrelativized halting problem is denoted by ∅′. We... |

112 |
Finite sums from sequences within cells of a partition of
- Hindman
- 1974
(Show Context)
Citation Context ...rem, another of the many Ramsey theoretic principles that have proved to be of reverse mathematical interest. For X ⊆ N, let S(X) be the set of sums of finite nonempty subsets of X. Hindman’s Theorem =-=[83]-=- states that if we partition N into sets C0, . . . , Cn, then there is an infinite X ⊆ N such that S(X) ⊆ Ci for some i 6 n. Blass, Hirst, and Simpson [5] showed that Hindman’s Theorem implies ACA0 ov... |

81 |
Foundations of Constructive Mathematics,
- Bishop
- 1967
(Show Context)
Citation Context ...As is almost always done in mathematical practice, we will use classical logic, although computable mathematics is of course related to constructive mathematics. See Section 4 of Bridges and Palmgren =-=[12]-=- for a discussion of the field of constructive reverse mathematics. As to the choice of base theory T , there are a few natural desiderata. We wish to use T to prove theorems of the forms T ` Q → P an... |

76 | On the strength of Ramsey’s theorem for pairs, The
- Cholak, Jockusch, et al.
(Show Context)
Citation Context ...lems in several areas of reverse mathematics. Reading the original papers discussed in expository articles such as this one is of course also important. Jockusch [97] and Cholak, Jockusch, and Slaman =-=[20]-=-, for example, are classics that will amply reward the reader. Other major papers in this line of research, including more recent ones answering open questions in [20], will be mentioned below. 2 Gath... |

74 | The word problem.
- Boone
- 1959
(Show Context)
Citation Context ...e famous DPRM Theorem, giving a negative solution to Hilbert’s 10th Problem; see Poonen [162]) and the set encoding the word problem for finitely presented groups (as shown by Novikov [156] and Boone =-=[8]-=-). We can take a given c.e. set A and encode it into a finitely presented group G so that the word problem in G has the same degree as A, but neither G nor its particular word problem would likely be ... |

68 |
Model theory. Third edition.
- Chang, Keisler
- 1990
(Show Context)
Citation Context ...al core that connects them with principles of the kind we have been discussing. We will see examples of this process in this subsection. We begin by reviewing a few definitions. See Chang and Kiesler =-=[17]-=- or Marker [132] for an introduction to model theory, and Ash and Knight [3] and Harizanov [77] for more on computable model theory. In this subsection, by a theory we mean a consistent set of sentenc... |

63 |
ENDERTON, A mathematical introduction to logic,
- B
- 1972
(Show Context)
Citation Context ...such that S ` ϕ ↔ ψ. Conclude that S has a deductive closure T that is a complete theory, and that A and B are structures (i.e., the elementary diagrams of these structures exist). [See e.g. Enderton =-=[53]-=- for more on quantifier elimination, in particular the fact that one can restrict attention to those formulas ψ of the form ∃x θ, where θ is a conjunction of atomic formulas and negations of atomic fo... |

59 |
Ramsey’s theorem and Recursion Theory.
- Jockusch
- 1972
(Show Context)
Citation Context ...talbán [146] also discusses open problems in several areas of reverse mathematics. Reading the original papers discussed in expository articles such as this one is of course also important. Jockusch =-=[97]-=- and Cholak, Jockusch, and Slaman [20], for example, are classics that will amply reward the reader. Other major papers in this line of research, including more recent ones answering open questions in... |

56 | What's so special about Kruskal's theorem and the ordinal ra? A survey of some results in proof theory.
- GALLIER
- 1991
(Show Context)
Citation Context ...cially felt due to the fact that this theorem figures prominently in computer science, because it is the main tool for showing that sets of rewrite rules are terminating [. . . ]” (They quote Gallier =-=[68]-=- as a place “where this challenge is offered”.) Friedman (see [186]) showed that Kruskal’s Theorem is not provable in ATR0. Rathjen and Weiermann [168] pinpointed the reverse mathematical strength of ... |

55 |
Two recursively enumerable sets of incomparable degrees of unsolvability
- Friedberg
- 1957
(Show Context)
Citation Context ...two c.e. degrees known, the degree of the computable sets, and the degree of the halting problem. Post [164] asked whether there are any others. Post’s Problem was answered independently by Friedberg =-=[57]-=- and Muchnik [150]. The method they introduced in their proofs, known as the priority method, became one of the hallmarks of computability theory, much like forcing in set theory following the work of... |

54 | Partition relations for cardinal numbers
- Erdős, Hajnal, et al.
- 1965
(Show Context)
Citation Context ...nteresting example, due to Carlson [13], is mentioned in Montalbán [146]. Miller (see [146]) suggested the reverse mathematical study of a family of principles introduced by Erdős, Hajnal, and Rado =-=[54]-=-. These principles are of the form RTnk,j, stating that for each k-coloring c of [N]n, there is an infinite set S such that the restriction of c to [S]n uses only j many colors. Montalbán [146] notes... |

49 | Combinatorial principles weaker than Ramsey’s theorem for pairs - Hirschfeldt, Shore |

48 | Some systems of second order arithmetic and their use
- Friedman
- 1975
(Show Context)
Citation Context ...ee more such examples once we have introduced the complementary viewpoint of reverse mathematics. 4 Gauging our strength: Reverse mathematics The reverse mathematics program was initiated by Friedman =-=[59, 60]-=- and further developed by Simpson, his students and collaborators, and, more recently, an increasing number of researchers from computability theory, proof theory, and other areas of mathematical logi... |

40 |
A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis
- Friedman, McAloon, et al.
- 1982
(Show Context)
Citation Context ...X0 = X and Xi+1 = {(m0, . . . ,mk) : ϕ(m0, . . . ,mk, Xi)}. McAloon [133] noted that a proof of the following theorem can be extracted from that of a related theorem in Friedman, McAloon, and Simpson =-=[64]-=-. Other proofs can be found in Mileti [138], as a consequence of his computability theoretic analysis of another version of Ramsey’s Theorem called the Canonical Ramsey Theorem; and in Dzhafarov and H... |

38 |
The metamathematics of the graph minor theorem,
- Friedman, Robertson, et al.
- 1987
(Show Context)
Citation Context ... this fascinating part of the reverse mathematical universe. The papers cited above are a good starting point for further study; other related papers include those by Friedman, Robertson, and Seymour =-=[65]-=- on the celebrated Robertson-Seymour graph minor theorem, and by Neeman [154, 155] on Jullien’s indecomposability theorem. A particularly interesting aspect of [155] is that it gives an example of a r... |

36 |
Recursively presentable prime models,
- Harrington
- 1974
(Show Context)
Citation Context ... us discuss an important technique that it illustrates. The proof of Theorem 9.55 is based on that of its earlier computability theoretic analog. Theorem 9.56 (Goncharov and Nurtazin [72]; Harrington =-=[79]-=-). Let T be a complete decidable atomic theory such that there is a computable listing of the principal types of T . Then T has a decidable atomic model. The proof of Theorem 9.56 uses a Henkin constr... |

35 |
Systems of second order arithmetic with restricted induction,
- Friedman
- 1976
(Show Context)
Citation Context ...(x), both with parameters from M and no free variables other than x. We begin with ACA0. Recall that P − 0 is the collection of basic first order axioms from Definition 4.1. 117 Theorem 7.4 (Friedman =-=[60]-=-). If M satisfies P−0 and arithmetic induction, then it is an ω-submodel of some model of ACA0. Proof. Let SN be the set of X ⊆ M that are arithmetically definable over M. Let N be the ω-extension of ... |

34 |
A criterion for completeness of degrees of unsolvability.
- Friedberg
- 1957
(Show Context)
Citation Context ...le. A proof of Theorem 6.76 can be found in Odifreddi [158, Theorem V.4.3]. The last statement in the theorem is not explicitly stated, but follows from the proof given there. Theorem 6.77 (Friedberg =-=[58]-=-). If d > 0′ then there is an a such that d = a′. Theorem 6.77 was the first of several jump inversion theorems. The following is another; others can be found in [40, 158, 196], for instance. Theorem ... |

32 |
Countable algebra and set existence axioms,
- Friedman, Simpson, et al.
- 1983
(Show Context)
Citation Context ...0 and ACA0, and it turns out be our old friend Weak König’s Lemma. Definition 4.13. The system WKL0 consists of RCA0 together with Weak König’s Lemma. 21 Exercise 4.14 (Friedman, Simpson, and Smith =-=[67]-=-). Adapt the solution to Exercise 3.25 to show that the following statement is equivalent to WKL0 over RCA0: If T is an infinite tree and there is a function f such that for every σ ∈ T , we have σ(n)... |

32 | Combinatorics in subsystems of second order arithmetic - Hirst - 1987 |

31 |
Some unifying principles in Ramsey theory
- Carlson
- 1988
(Show Context)
Citation Context ...in than previously known ones. Of course, there are other Ramsey theoretic principles still awaiting computability theoretic and reverse mathematical analysis. One interesting example, due to Carlson =-=[13]-=-, is mentioned in Montalbán [146]. Miller (see [146]) suggested the reverse mathematical study of a family of principles introduced by Erdős, Hajnal, and Rado [54]. These principles are of the form ... |

30 |
Degrees of generic sets
- Jockusch
(Show Context)
Citation Context ...ng σ. We say that G meets or avoids every Σ0n set of strings. (Note that for any set A of strings, the strings σ such that either σ ∈ A or τ /∈ A for all τ extending σ form a dense set.) See Jockusch =-=[100]-=- for more on this topic. The theory of algorithmic randomness may also be seen as arising from effectivizing a notion of forcing (see Downey and Hirschfeldt [40, Section 7.2.5]). We will not be discus... |

28 | 2011a) Weihrauch degrees, omniscience principles and weak computability
- Brattka, Gherardi
(Show Context)
Citation Context ...209] in the context of computable analysis, and widely studied since, including for purposes of computability theoretic comparison of mathematical principles; see 22 for instance Brattka and Gherardi =-=[10, 11]-=-. Therefore, it is sometimes denoted by 6W instead of 6u. For further discussion of Weihrauch reducibility in our context, and a proof of equivalence between uniform reducibility and (a special case o... |

28 |
Soare, Degrees of members of Π01 classes
- Jockusch, I
- 1972
(Show Context)
Citation Context ... corresponding to the completions of PA is then a universal Π01 class, in the sense that any set that can compute an element of P can compute an element of any nonempty Π01 class. (Jockusch and Soare =-=[105, 106]-=- showed that for every Π 0 1 class P there is a computably axiomatizable first order theory T such that the class of degrees of members of P coincides with the class of degrees of complete extensions ... |

25 | Computability Theory and Linear Orderings - Downey - 1998 |

24 | 2011b) Effective choice and boundedness principles in computable analysis
- Brattka, Gherardi
(Show Context)
Citation Context ...209] in the context of computable analysis, and widely studied since, including for purposes of computability theoretic comparison of mathematical principles; see 22 for instance Brattka and Gherardi =-=[10, 11]-=-. Therefore, it is sometimes denoted by 6W instead of 6u. For further discussion of Weihrauch reducibility in our context, and a proof of equivalence between uniform reducibility and (a special case o... |

24 |
Models of Peano arithmetic, Oxford Logic Guides, 15
- Kaye
- 1991
(Show Context)
Citation Context ...n be proved in Σ01-PA, and hence in RCA0 (see Hájek and Pudlák [75]). Although PH seems quite similar to FRT, Paris and Harrington [160] showed that it cannot be proved in PA (see also [75] or Kaye =-=[109]-=-). As mentioned in Section 4.6, PA is the first order part of ACA0, so ACA0 does not prove PH. Combining this fact with Exercise 6.25 shows that RT is strictly stronger than ACA0 over RCA0. Indeed, we... |

22 | Degrees in which the recursive sets are uniformly recursive - Jockusch - 1972 |

22 |
Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion
- Jr, Lerman, et al.
- 1989
(Show Context)
Citation Context ...) if it is c.e. relative to X and computes X. A set A is n-CEA(X) if there is a set B that is (n − 1)-CEA(X) and such that A is c.e. relative to B and computes B. Jockusch, Lerman, Soare, and Solovay =-=[102]-=- extended Theorem 3.18 by showing that if A is n-CEA(X) for some n and has PA degree relative to X, then A >T X ′. 48 Exercise 6.47 (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman [86]). Assum... |

21 |
Logical analysis of some theorems of combinatorics and topological dynamics
- Blass, Hirst, et al.
- 1987
(Show Context)
Citation Context ...ite nonempty subsets of X. Hindman’s Theorem [83] states that if we partition N into sets C0, . . . , Cn, then there is an infinite X ⊆ N such that S(X) ⊆ Ci for some i 6 n. Blass, Hirst, and Simpson =-=[5]-=- showed that Hindman’s Theorem implies ACA0 over RCA0, and can be proved in the system ACA + 0 introduced at end of Section 6.3, which is strictly stronger than ACA0 but weaker than ATR0. I Open Quest... |

20 |
Well-partial orderings and hierarchies
- Jongh, Parikh
- 1977
(Show Context)
Citation Context ...stion 10.19. Does ATR0 ` EXT(η)? Does RCA0 + EXT(η) ` ATR0? Maximal linear extensions. Another example of a principle equivalent to ATR0 is the maximal linear extension theorem of de Jongh and Parikh =-=[35]-=-. A partial order (P,4) is a well-partial-order (wpo) if for every f : N→ P there are m < n such that f(m) 4 f(n) (in other words, there are no infinite descending sequences and no infinite antichains... |

19 |
Linear extensions of ordered sets
- Bonnet, Pouzet
- 1981
(Show Context)
Citation Context ... fails to hold. There are also countable order types that are weakly extendible but not strongly extendible, such as ω + 1. For these and other classical facts on extendibility, see Bonnet and Pouzet =-=[7]-=-. Note also that EXT(τ) is clearly equivalent to EXT(τ ∗). There are characterizations of the strongly extendible and weakly extendible order types, due to Bonnet [6] and Jullien [108], respectively. ... |

19 | Pure computable model theory,
- Harizanov
- 1998
(Show Context)
Citation Context ...de range of topics in computable mathematics. We will have occasion to cite in particular the articles by Cenzer and Remmel [15] on Π01 classes, Downey [39] on computable linear orders, and Harizanov =-=[77]-=- on computable model theory. Other articles in those volumes related to our topics include the ones by Gasarch [70] on computable combinatorics and Simpson and Rao [193] on the reverse mathematics of ... |

18 |
Weak comparability of well orderings and reverse mathematics.
- Friedman, Hirst
- 1990
(Show Context)
Citation Context ...thematics. Choice principles in reverse mathematics are discussed in Section VII.6 of Simpson [191], and versions of the axiom of choice related to well-orders have been studied by Friedman and Hirst =-=[63]-=- and Hirst [92]. Dzhafarov and Mummert [51] studied combinatorial equivalents of the axiom of choice, finding several examples that are either equivalent to ACA0 or strictly stronger. In [52], they st... |

18 | The atomic model theorem and type omitting
- Hirschfeldt, Shore, et al.
(Show Context)
Citation Context ...ll our heuristic. Hyperimmunity is a computability theoretic natural concept, so we should not be surprised to find that it can be characterized by other means. Indeed, Hirschfeldt, Shore, and Slaman =-=[89]-=- showed that a version of the Omitting Types Theorem of basic model theory, called OPT, is equivalent over RCA0 to the statement that for every set Y , there is a set that has hyperimmune degree relat... |

16 |
Bounding prime models
- Csima, Hirschfeldt, et al.
(Show Context)
Citation Context ...dense predicates D0, D1, . . . on 2<N, there is a G ∈ 2N that meets each Di. In this form, this principle was considered by Shinoda and Slaman [178] and later by Csima, Hirschfeldt, Knight, and Soare =-=[33]-=- in connection with the study of atomic models, which we will discuss in the next subsection. 74 Exercise 9.45 (Dzhafarov and Mummert [52]). Show that RCA0 + Π 0 1G ` FIP. [Hint: Given a nontrivial fa... |

16 |
A ∆02 set with no infinite low subset in either it or its complement,
- Downey, Hirschfeldt, et al.
- 2001
(Show Context)
Citation Context ..., if relativizable, would allow us to construct an ω-model of RCA0 + SRT 2 2 consisting entirely of low sets, which would then not be a model of RT22. However, Downey, Hirschfeldt, Lempp, and Solomon =-=[41]-=- showed that this idea cannot work. Theorem 6.37 (Downey, Hirschfeldt, Lempp, and Solomon [41]). There is a ∆02 set such that neither it nor its complement has an infinite low subset. To be more preci... |

16 | Located Sets and Reverse Mathematics.
- Giusto
- 2000
(Show Context)
Citation Context ...R is the statement that for each set A there is a function that is diagonally noncomputable relative to A (i.e., a function f such that f(e) 6= ΦAe (e) for all e). 62 Exercise 9.7 (Giusto and Simpson =-=[71]-=-). Show that WKL0 ` DNR. Giusto and Simpson [71] showed that in fact WWKL0 ` DNR; Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman [2] showed that RCA0 + DNR 0 WWKL0. As mentioned in Section 3.1, every 0,... |

15 |
Contribution à l’étude des types d’ordre dispersés
- Jullien
- 1969
(Show Context)
Citation Context ...ng uncountable ones, so this notion is not amenable to treatment in the framework of classical computability theory and reverse mathematics. Thus we work with the following notion, studied by Jullien =-=[108]-=-, among others. Definition 10.12. Let τ be an order type. We say that τ is weakly extendible if every countable partial order with no linear suborder of type τ has a linear extension with no suborder ... |

13 | Reverse mathematics and the equivalence of definitions for well and better quasi-orders, The
- Cholak, Marcone, et al.
(Show Context)
Citation Context ...artial-order (wpo) if for every f : N→ P there are m < n such that f(m) 4 f(n) (in other words, there are no infinite descending sequences and no infinite antichains; see Cholak, Marcone, and Solomon =-=[21]-=- for an analysis of the reverse mathematical aspects of equivalent definitions of wpo). A partial order is a wpo iff all its linearizations are well-orders. (See [21] for a proof and reverse mathemati... |

13 |
Nurtazin, Constructive models of complete decidable theories (Russian), Algebra i Logika
- Goncharov, T
- 1973
(Show Context)
Citation Context ...explained below. For more on priority arguments, including infinite injury arguments, see for instance Soare [196]. Theorem 9.11 (Tennenbaum, see Rosenstein [172]; Denisov, see Goncharov and Nurtazin =-=[72]-=-). There is a computable infinite linear order with no computable infinite ascending or descending sequence. Proof. We build a computable linear order (N,4). We will ensure that no infinite c.e. set c... |

12 | The strength of the rainbow Ramsey theorem
- Csima, Mileti
(Show Context)
Citation Context ...nsidered. We have already mentioned the Canonical Ramsey Theorem [137, 138] and the Polarized Ramsey Theorem [22, 48]; other examples include the Dual Ramsey Theorem [143], the Rainbow Ramsey Theorem =-=[29, 34, 38, 142, 205, 206, 207]-=-, and “infinite exponent” versions discussed in Sections V.9, VI.6, and VI.7 of Simpson [191]. Gasarch [personal communication] has pointed out that the work of Mileti [137, 138] on the Canonical Rams... |

12 | Computability-theoretic and proof-theoretic aspects of partial and linear orderings
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...s not true in M. Thus EXT(ω∗) does not hold in RCA0. In fact, we have the following result, which can be proved simultaneously with Theorem 10.8. Theorem 10.9 (Downey, Hirschfeldt, Lempp, and Solomon =-=[42]-=-). EXT(ω∗) implies WKL0 over RCA0. 164 Proof of Theorems 10.8 and 10.9. Let T be an infinite binary tree. For σ, τ ∈ T , let σ 4P τ iff σ extends τ . We show that any linear extension 4L of 4P has an ... |

12 | Ramsey’s theorem and cone avoidance
- Dzhafarov, Jockusch
(Show Context)
Citation Context ... class of sets that do not compute ∅′ to give another proof of Corollary 6.54. 101 We give a newer proof of Theorem 6.61, considerably simpler than the original one, taken from Dzhafarov and Jockusch =-=[50]-=- (who also showed that the proof of Theorem 6.61 can be combined with that of Theorem 6.53 to construct infinite homogeneous sets for computable colorings of pairs that are both cone avoiding and low2... |

11 | The metamathematics of stable Ramsey’s theorem for pairs
- Chong, Slaman, et al.
- 2012
(Show Context)
Citation Context ...ther SRT 2 2 implies RT 2 2 (or, equivalently, whether SRT22 implies COH) over RCA0, however, remained open and actively pursued for many years. It has recently been solved by Chong, Slaman, and Yang =-=[24]-=-. Theorem 6.36 (Chong, Slaman, and Yang [24]). SRT22 does not imply RT 2 2 over RCA0, or even over WKL0. The proof of this theorem is a striking application of computability theory on nonstandard (fir... |

11 |
1993]: `Interpretability and fragments of arithmetic
- H¶ajek
(Show Context)
Citation Context ...rvativity results over stronger systems can also be quite interesting, as we saw in Theorems 6.83 and 6.91. Both WKL and COH have been shown to be Π11-conservative over both BΣ 0 2 and IΣ 0 2 (Hájek =-=[74]-=- and Chong, Slaman, and Yang [23]), in addition to being conservative over RCA0. We will see another example of a principle with this property in Section 9.3, as well as another principle that is Π11-... |

11 | Turing degrees of certain isomorphic images of computable relations
- Harizanov
(Show Context)
Citation Context ...etween “pure computability theory” and computable mathematics is a two-way street. It is not difficult to see that the ω part of any computable linear order of type ω+ω∗ is ∆02. Conversely, Harizanov =-=[78]-=- showed that every ∆02 degree contains the ω part of some computable linear order of type ω + ω∗. Jockusch (see [88]) noted that we can therefore argue as follows. Fix a ∆02 degree a, and let 4 be a l... |

10 | The strength of infinitary Ramseyan principles can be accessed by their densities, accepted for publication in Ann. Pure Appl. Logic, http: //logic.pdmi.ras.ru/~andrey/research.html - Bovykin, Weiermann - 2005 |

10 | Degree spectra of prime models
- Csima
(Show Context)
Citation Context ...as an ω-model consisting entirely of low sets, so the following is one consequence of Theorem 9.52, originally proved directly. (We will see another proof in Exercise 9.59 below.) Theorem 9.54 (Csima =-=[31]-=-). Every complete decidable atomic theory has an atomic model whose elementary diagram is low. The analysis of AMT provides a good example of the practice of stripping away the area-specific details o... |

10 | A survey of recursive combinatorics
- Gasarch
(Show Context)
Citation Context ...d Remmel [15] on Π01 classes, Downey [39] on computable linear orders, and Harizanov [77] on computable model theory. Other articles in those volumes related to our topics include the ones by Gasarch =-=[70]-=- on computable combinatorics and Simpson and Rao [193] on the reverse mathematics of algebra. The standard textbook in reverse mathematics is Simpson’s classic Subsystems of Second Order Arithmetic, n... |

10 |
Hindman’s theorem, ultrafilters, and reverse mathematics
- Hirst
(Show Context)
Citation Context ...ch as Hindman’s original one) were studied by Blass, Hirst, and Simpson [5] and Towsner [202]. Another proof uses ultrafilters, and was examined from the point of view of reverse mathematics by Hirst =-=[91]-=- and Towsner [201]. Yet another proof involves topological dynamics, using a theorem that, like Hindman’s Theorem itself, follows from ACA+0 and may be equivalent to ACA0; see [5, 146] for details. Fu... |

9 |
Infinite chains and antichains in computable partial orderings
- Herrmann
(Show Context)
Citation Context ...0 ADS (indeed, it shows that even WKL0 0 ADS), so by Theorem 7.18, RCA0 + COH 0 ADS. The next exercise uses the following theorem to give an alternate proof that WKL0 0 ADS. 131 Theorem 9.5 (Herrmann =-=[82]-=-). There is an infinite computable partial order with no ∆02 infinite chain or antichain. 61 Exercise 9.6 (Hirschfeldt and Shore [88]). Show that for any computable partial order 6P , there is a compu... |

8 | On the role of the collection principle for Σ02-formulas in second-order reverse mathematics
- Chong, Lempp, et al.
(Show Context)
Citation Context ...parts is a fruitful technique. The first order consequences of ADS have also been studied. Hirschfeldt and Shore [88] showed that RCA0 + ADS ` BΣ02. This result was improved by Chong, Lempp, and Yang =-=[22]-=-, who showed that RCA0 + SADS ` BΣ02. Their method involves an analysis of the first order part of models of RCA0 in which BΣ02 fails; in particular, they introduce the notion of a bi-tame cut, and sh... |

8 | On uniform relationships between combinatorial problems
- Dorais, Dzhafarov, et al.
- 2012
(Show Context)
Citation Context ...sion of Weihrauch reducibility in our context, and a proof of equivalence between uniform reducibility and (a special case of) Weihrauch reducibility, see Dorais, Dzhafarov, Hirst, Mileti, and Shafer =-=[38]-=-. Again, we can also consider a notion of strong uniform reducibility (also known as strong Weihrauch reducibility), where the above definition is changed by having Y = ΨŶ ; see for instance [38, 85]... |

8 |
Issues and problems in reverse mathematics
- Friedman, Simpson
- 2000
(Show Context)
Citation Context ...thematics of EM, including its relationship with one of the model theoretic principles we will discuss in Section 9.3, see [125]. Free sets and thin sets. Friedman [61] (see also Friedman and Simpson =-=[66]-=-) introduced the reverse mathematical study of the following Ramsey theoretic principles. Definition 9.32. The Free Set Theorem (FS) is the statement that for every n and every f : [N]n → N, there is ... |

8 | The strength of some combinatorial principles related to Ramsey’s Theorem for Pairs
- Hirschfeldt, Jockusch, et al.
- 2008
(Show Context)
Citation Context ...olovay [102] extended Theorem 3.18 by showing that if A is n-CEA(X) for some n and has PA degree relative to X, then A >T X ′. 48 Exercise 6.47 (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman =-=[86]-=-). Assume that every ∆02 set has a subset of either it or its complement that is both lown for some n and ∆ 0 2, and that this fact is relativizable. Show that there is an ω-model of RCA0 + SRT 2 2 co... |

8 |
Upward closure and cohesive degrees
- Jockusch
- 1973
(Show Context)
Citation Context ...We can now get a full answer to our question from the following remarkably general theorem, which applies to homogeneous sets, cohesive sets, and many other classes of objects. Theorem 6.66 (Jockusch =-=[99]-=-). Let C be a collection of infinite sets that contains an arithmetic set, such that if A ⊂ B is infinite and B ∈ C then A ∈ C. Then the class of degrees of elements of C is closed upwards. It is also... |

7 |
The polarized Ramsey’s theorem
- Dzhafarov, Hirst
(Show Context)
Citation Context ...er proofs can be found in Mileti [138], as a consequence of his computability theoretic analysis of another version of Ramsey’s Theorem called the Canonical Ramsey Theorem; and in Dzhafarov and Hirst =-=[48]-=-, in connection with yet another such version, called the Polarized Ramsey Theorem. We give a direct proof obtained by Dzhafarov [personal communication] as a simplification of the argument in [48]. T... |

7 | Computable trees, prime models, and relative decidability
- Hirschfeldt
(Show Context)
Citation Context ...ion that each type of a complete theory T is computable is still quite strong, and makes it possible to find atomic models of T “almost effectively”, in the following sense. Theorem 9.70 (Hirschfeldt =-=[84]-=-). Let T be a complete decidable theory each of whose types is computable, and let X be a noncomputable set. Then T has an X-decidable atomic model. Proof. By Exercise 9.57, it is enough to take a com... |

6 |
Computability theory in arithmetic: Provability, structure and techniques’, Computability theory and its applications: current trends and open problems
- Chong, Yang
- 2000
(Show Context)
Citation Context ...ments and other computability theoretic constructions has been a fruitful 150 line of research, known as reverse computability theory (or, more commonly, reverse recursion theory); see Chong and Yang =-=[25, 26]-=-. In some cases, however, we can modify a priority construction to eliminate the need for Σ02-induction. One technique for doing so was originally developed in the context of the theory of computabili... |

6 |
Recursion theory in weak fragments of Peano arithmetic: a study of cuts
- Chong, Yang
- 1998
(Show Context)
Citation Context ... A ∧ 〈k, i, j〉 ∈ W ], and altering the other definitions accordingly. However, for our purposes these issues will not arise, so we may safely ignore them. For more on this subject, see Chong and Yang =-=[25, 26]-=-. Although its focus is different, another useful source on defining 47 computability theoretic notions in settings with limited induction is Sacks [175]; see in particular Section VII.3. Most people ... |

6 |
Reverse mathematics, computability, and partitions of trees
- Chubb, Hirst, et al.
(Show Context)
Citation Context ...denote by TSnk (TS(n) being TSnω in their notation). Ramsey theoretic principles on structures other than the natural numbers are also of interest. An example is versions of Ramsey’s Theorem on trees =-=[27, 30, 49, 134]-=-. Hummel and Jockusch [95] studied generalizations of the notion of cohesiveness. Dzhafarov [45] studied “almost all” versions of SRT22 using tools from effective measure theory. Flood [56] studied a ... |

6 |
Classifying model-theoretic properties
- Conidis
- 2008
(Show Context)
Citation Context ...idable atomic model. Theorem 9.51 (Csima, Hirschfeldt, Knight, and Soare [33]). A ∆02 set is atomic bounding iff it is not low2. Outside of the ∆02 degrees, the situation is more complex; see Conidis =-=[28]-=-. Let Y be a set of PA degree and let M be an ω-model of WKL0 all of whose sets are Y -computable, which exists by Theorem 4.22. Then Theorem 9.51 implies that M 2 AMT, so WKL0 0 AMT. Nevertheless, AM... |

6 |
A survey of the reverse mathematics of ordinal arithmetic
- Hirst
(Show Context)
Citation Context ...ce principles in reverse mathematics are discussed in Section VII.6 of Simpson [191], and versions of the axiom of choice related to well-orders have been studied by Friedman and Hirst [63] and Hirst =-=[92]-=-. Dzhafarov and Mummert [51] studied combinatorial equivalents of the axiom of choice, finding several examples that are either equivalent to ACA0 or strictly stronger. In [52], they studied a princip... |

6 |
Countable retracing functions and Π 0 2 predicates
- Jockusch, McLaughlin
- 1972
(Show Context)
Citation Context ...(n−2). The above theorem can also be proved using the following computability theoretic fact, where a function g majorizes a function f if f(n) 6 g(n) for all n. Theorem 6.18 (Jockusch and McLaughlin =-=[104]-=-). For each k, there is an increasing function f ≡T ∅(k) such that any function that majorizes f computes f . 37 Exercise 6.19 (Hirschfeldt and Jockusch [85]). Prove Theorem 6.17 as follows. (i) Fix k... |

5 |
Recursive equivalence types
- Dekker, Myhill
- 1960
(Show Context)
Citation Context ...or RT22. As noted in Rogers [170], early computability theoretic discussion of cohesiveness and related notions (including results on the existence of cohesive sets) can be found in Dekker and Myhill =-=[36]-=- and Rose and Ullian [171]. Definition 6.40. We define the notion of Mathias forcing (P,4). The set P consists of the pairs of the form (F, I), where F ⊂ N is finite, I ⊆ N is infinite, and maxF < min... |

5 | On the proof theoretical strength of the Dushnik– Miller theorem for countable linear orderings
- Downey, Lempp
- 1999
(Show Context)
Citation Context ...ivial self-embedding of 4, we can take 160 an x such that x ≺ Φe(x) ≺ Φe(Φe(x)) and ensure that there are more elements between x and Φe(x) than between Φe(x) and Φe(Φe(x)).] Indeed, Downey and Lempp =-=[44]-=- built a computable linear order L of type ω such that any nontrivial self-embedding of L computes ∅′. Their construction can be transformed into a proof that the Dushnik-Miller Theorem implies ACA0 o... |

5 | On the strength of the finite intersection principle
- Dzhafarov, Mummert
(Show Context)
Citation Context ...nd Hirst [63] and Hirst [92]. Dzhafarov and Mummert [51] studied combinatorial equivalents of the axiom of choice, finding several examples that are either equivalent to ACA0 or strictly stronger. In =-=[52]-=-, they studied a principle of this sort that lives in the world below RT22 (at least when working with a slight amount of additional induction), known as the finite intersection principle (FIP). (They... |

5 |
Metamathematics of First-order Arithmetic, second printing, Perspect
- Hájek, Pudlák
- 1998
(Show Context)
Citation Context ... contain a copy of the standard model as an initial segment. Elements of this initial segment are called standard, and other elements of the domain of N are called nonstandard. See Hájek and Pudlák =-=[75]-=- for more on nonstandard models of fragments of Peano Arithmetic. One way to obtain subsystems of second order arithmetic is to restrict the comprehension and induction schemes. Definition 4.2. For a ... |

5 |
Π01 Classes and Rado’s Selection Principle
- Jockusch, Lewis, et al.
- 1991
(Show Context)
Citation Context ...computable” (e.g., low, of hyperimmunefree degree, cone-avoiding). The situation for the full version of König’s Lemma is quite different. The following fact is proved in Jockusch, Lewis, and Remmel =-=[103]-=-, where it is referred to as a “known result”, similar to one in Yates [210]. 9 Exercise 3.24. Show that there is a computable finitely branching tree T such that [T ] is a singleton, and the unique e... |

5 |
A cohesive set which is not
- Stephan
- 1993
(Show Context)
Citation Context ...so |X| 6∑j<k 2j < 2k. We will analyze COH further in the next subsection, but for now we note the following facts, which are weaker forms of Theorem 6.45 below. 43 Exercise 6.33 (Jockusch and Stephan =-=[107]-=-). Show that there is a sequence of uniformly computable sets R0, R1, . . . with no computable cohesive set. Thus RCA0 0 COH. [This exercise should be done before reading the following proof.] Theorem... |

4 | Π11-conservation of combinatorial principles weaker than Ramsey’s theorem for pairs
- Chong, Slaman, et al.
(Show Context)
Citation Context ... reason such double jump control proofs are interesting is that they can be converted into proofs of the kinds of conservativity results discussed in Sections 6.8 and 7 below. Chong, Slaman, and Yang =-=[23]-=- remark that for a principle P of the form ∀X [θ(X) → ∃Y ψ(X, Y )], there is a “heuristic correspondence” between the computability theoretic conclusion that every instance X of P has a solution Y suc... |

4 | Soare, Bounding homogeneous models
- Csima, Hirschfeldt, et al.
(Show Context)
Citation Context ...ound is basic model theory. In this subsection, we will focus on atomic models and type omitting, but there is also considerable related material on the reverse mathematics of homogeneous models (see =-=[32, 87, 120, 121, 122]-=-). The common thread in much of this work is the combinatorics of omitting and realizing types. It is possible to abstract away the model theoretic framework of several theorems in this area to reveal... |

4 | On notions of computability theoretic reduction between Π1 principles
- Hirschfeldt, Jockusch
(Show Context)
Citation Context ...tween principles where we do not use the power of the instance X, which is known as strong computable reducibility, is still of interest in some cases; see Dzhafarov [47] and Hirschfeldt and Jockusch =-=[85]-=-. (Of course, in many cases, the two notions coincide, for example when we can code X into X̂ so that any solution to X̂ computes X.) Also of interest is the idea of reducing P uniformly to Q. We say ... |

3 |
Stratifications et extension des genres de châınes dénombrables
- Bonnet
- 1969
(Show Context)
Citation Context ...f a partial order that could be quite useful to have preserved in a linearization is well-foundedness. As it turns out, the above construction already preserves well-foundedness. Theorem 10.6 (Bonnet =-=[6]-=-). Every well-founded countable partial order can be extended to a well-order. 162 Proof. Let 4P be a partial order with domain N and let 4L be built as above. Suppose that there is an infinite descen... |

3 |
On Mathias generic sets
- Cholak, Dzhafarov, et al.
- 2012
(Show Context)
Citation Context ...versions of forcing. These notions have played an important role in computability theory, and have also proved of interest to reverse mathematics. (See Jockusch [100] and Cholak, Dzhafarov, and Hirst =-=[18]-=- for computability theoretic versions of Cohen forcing and Mathias forcing, respectively.) When we say that D0, D1, . . . are uniformly Π 0 1 predicates on 2 <N, we mean that we fix a Π01 formula ϕ(i,... |

3 | Random reals, the rainbow Ramsey theorem, and arithmetic conservation
- Conidis, Slaman
(Show Context)
Citation Context ...nsidered. We have already mentioned the Canonical Ramsey Theorem [137, 138] and the Polarized Ramsey Theorem [22, 48]; other examples include the Dual Ramsey Theorem [143], the Rainbow Ramsey Theorem =-=[29, 34, 38, 142, 205, 206, 207]-=-, and “infinite exponent” versions discussed in Sections V.9, VI.6, and VI.7 of Simpson [191]. Gasarch [personal communication] has pointed out that the work of Mileti [137, 138] on the Canonical Rams... |

3 |
Cohesive avoidance and strong reductions
- Dzhafarov
(Show Context)
Citation Context ...ess, the notion of reducibility between principles where we do not use the power of the instance X, which is known as strong computable reducibility, is still of interest in some cases; see Dzhafarov =-=[47]-=- and Hirschfeldt and Jockusch [85]. (Of course, in many cases, the two notions coincide, for example when we can code X into X̂ so that any solution to X̂ computes X.) Also of interest is the idea of ... |

3 |
Reverse mathematics and properties of finite character
- Dzhafarov, Mummert
- 2011
(Show Context)
Citation Context ...hematics are discussed in Section VII.6 of Simpson [191], and versions of the axiom of choice related to well-orders have been studied by Friedman and Hirst [63] and Hirst [92]. Dzhafarov and Mummert =-=[51]-=- studied combinatorial equivalents of the axiom of choice, finding several examples that are either equivalent to ACA0 or strictly stronger. In [52], they studied a principle of this sort that lives i... |

3 |
The Inevitably of Logical Strength: strict reverse mathematics
- Friedman
- 2009
(Show Context)
Citation Context ...untable dense subsets of separable metric spaces. See [191] for more on this subject. A proposal for a version of reverse mathematics that does away with the need for coding has been made by Friedman =-=[62]-=-. In this strict reverse mathematics, there should be a separate theory for each area of mathematics, with the basic notions of that area taken as primitives, and logical axioms replaced by natural ma... |

3 | The Boolean algebra of logic - Hanf - 1975 |

3 |
The degrees of hyperhyperimmune sets
- Jockusch
- 1969
(Show Context)
Citation Context ...d in that subsection. By using the technique of forcing the jump, as in Exercise 2.8, we can in fact get C ′ 6T ∅′′, though it is also not difficult to do better than that: 46 Exercise 6.42 (Jockusch =-=[96]-=-). a. Proceed as outlined above to show that if R0, R1, . . . are uniformly computable then they have a cohesive set C such that C ′ 6T ∅′′. [This exercise should be attempted before reading the proof... |

2 |
Some questions arising from Hindman’s Theorem
- Blass
- 2005
(Show Context)
Citation Context ...like Hindman’s Theorem itself, follows from ACA+0 and may be equivalent to ACA0; see [5, 146] for details. Further computability theoretic questions related to Hindman’s Theorem can be found in Blass =-=[4]-=-. Hirst [93] analyzed a statement that seems quite similar to (an equivalent form of) Hindman’s Theorem, but is much weaker, being equivalent to BΣ02 over RCA0. The analysis of different proofs and ve... |

2 |
Π01 classes: structure and applications
- Cenzer, Jockusch
- 2000
(Show Context)
Citation Context ...notion of a Π01 class, which is closely related to Weak König’s Lemma. In the following subsection, we focus mainly on facts that will be useful later on in the article. For more on Π01 classes, see =-=[14, 15, 16]-=-. 3.1 Π01 classes, basis theorems, and PA degrees It is not difficult to see that a subset of 2N (or of NN) is closed iff it is equal to [T ] for some tree T . The notion of a Π01 class is an effectiv... |

2 |
Reverse mathematics and Ramsey’s property for trees
- Corduan, Groszek, et al.
(Show Context)
Citation Context ...denote by TSnk (TS(n) being TSnω in their notation). Ramsey theoretic principles on structures other than the natural numbers are also of interest. An example is versions of Ramsey’s Theorem on trees =-=[27, 30, 49, 134]-=-. Hummel and Jockusch [95] studied generalizations of the notion of cohesiveness. Dzhafarov [45] studied “almost all” versions of SRT22 using tools from effective measure theory. Flood [56] studied a ... |

2 |
Ramsey’s theorem for trees: the polarized tree theorem and notions of stability
- Dzhafarov, Hirst, et al.
(Show Context)
Citation Context ...denote by TSnk (TS(n) being TSnω in their notation). Ramsey theoretic principles on structures other than the natural numbers are also of interest. An example is versions of Ramsey’s Theorem on trees =-=[27, 30, 49, 134]-=-. Hummel and Jockusch [95] studied generalizations of the notion of cohesiveness. Dzhafarov [45] studied “almost all” versions of SRT22 using tools from effective measure theory. Flood [56] studied a ... |

2 |
two postings to the Foundations of Mathematics mailing list, http://www.cs.nyu.edu/pipermail/fom/1999-July/003257.html and http://www.cs.nyu.edu/pipermail/fom/1999-July/003263.html
- Friedman
(Show Context)
Citation Context ... of SADS. For more on the reverse mathematics of EM, including its relationship with one of the model theoretic principles we will discuss in Section 9.3, see [125]. Free sets and thin sets. Friedman =-=[61]-=- (see also Friedman and Simpson [66]) introduced the reverse mathematical study of the following Ramsey theoretic principles. Definition 9.32. The Free Set Theorem (FS) is the statement that for every... |

2 | Induction, bounding, weak combinatorial principles, and the Homogeneous Model Theorem
- Hirschfeldt, Lange, et al.
(Show Context)
Citation Context ...ree T is atomic if each node of T can be extended to an atom, and strongly atomic if for every σ0, . . . , σn ∈ T , each σi can be extended to an atom. 30 Exercise 4.35 (Hirschfeldt, Lange, and Shore =-=[87]-=-). Show that BΣ02 is equivalent over RCA0 to the statement that every atomic tree with no dead ends is strongly atomic. 60 Note that, despite being equivalent to BΣ02, the statement in the above exerc... |

1 |
reverse mathematics and combinatorics: open problems
- Computability
(Show Context)
Citation Context ... The exercise may seem as pointless as it is difficult, but of course mathematicians do set and perform tasks of this kind on a regular basis. “Use the Bolzano-Weierstrass Theorem to show that if f : =-=[0, 1]-=- → R is continuous, then f is uniformly continuous.” is a typical homework problem in analysis, and the question “Can Chaitin’s information-theoretic version of Gödel’s First Incompleteness Theorem b... |

1 |
Effectively Closed Sets, to be published
- Cenzer, Remmel
(Show Context)
Citation Context ...notion of a Π01 class, which is closely related to Weak König’s Lemma. In the following subsection, we focus mainly on facts that will be useful later on in the article. For more on Π01 classes, see =-=[14, 15, 16]-=-. 3.1 Π01 classes, basis theorems, and PA degrees It is not difficult to see that a subset of 2N (or of NN) is closed iff it is equal to [T ] for some tree T . The notion of a Π01 class is an effectiv... |

1 |
Free sets and reverse mathematics, in [189
- Cholak, Giusto, et al.
(Show Context)
Citation Context ...S(n) the statement that FS holds for n, and by TS(n) the statement that TS holds for n. The reverse mathematics of these and related principles has been studied by Cholak, Giusto, Hirst, and Jockusch =-=[19]-=-, but there are still many interesting open questions in this line of research, whose solutions may well require major methodological advances like the one that was required to prove Liu’s Theorem 6.7... |

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The finite intersection principle and genericity, to appear, available at time of writing at http://homepages.ecs.vuw.ac.nz/~downey/publications/ FIP paper.pdf
- Diamondstone, Downey, et al.
(Show Context)
Citation Context ...ions are pairs (σ, n), where σ ∈ N<N and⋂i<|σ|Aσ(i) contains some number 6 s, and where (τ, t) extends (σ, s) if σ is an initial segment of τ and t > s.] Diamondstone, Downey, Greenberg, and Turetsky =-=[37]-=- clarified the connection between FIP and genericity by tying FIP to the important computability notion of 1-genericity. We will discuss some further details of the connections between FIP, Π01G, RT 2... |

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On self-embeddings of computable linear orderings, Ann. Pure Appl. Logic 138
- Downey, Jockusch, et al.
- 2006
(Show Context)
Citation Context ...ω such that any nontrivial self-embedding of L computes ∅′. Their construction can be transformed into a proof that the Dushnik-Miller Theorem implies ACA0 over RCA0 (see Downey, Jockusch, and Miller =-=[43]-=-), and hence is equivalent to ACA0, by Exercise 10.2 below. For order type ω, the result of Downey and Lempp is the best possible, since given a computable linear order 4 of type ω, we can use ∅′ to c... |

1 | Stable Ramsey’s theorem and measure
- Dzhafarov
(Show Context)
Citation Context ... natural numbers are also of interest. An example is versions of Ramsey’s Theorem on trees [27, 30, 49, 134]. Hummel and Jockusch [95] studied generalizations of the notion of cohesiveness. Dzhafarov =-=[45]-=- studied “almost all” versions of SRT22 using tools from effective measure theory. Flood [56] studied a principle that is a hybrid of Weak König’s Lemma and Ramsey’s Theorem for pairs. Combinatorial ... |

1 |
Infinite saturated orders, Order 28
- Dzhafarov
- 2011
(Show Context)
Citation Context ...] studied a principle that is a hybrid of Weak König’s Lemma and Ramsey’s Theorem for pairs. Combinatorial principles of reverse mathematical interest can come from many different sources. Dzhafarov =-=[46]-=- gave an example arising from mathematical psychology. 87 Exercise 10.25. Find a theorem or (even better) family of theorems not yet studied from the computability theoretic and reverse mathematical p... |

1 |
Reverse mathematics and a Ramsey-type König’s
- Flood
(Show Context)
Citation Context ...7, 30, 49, 134]. Hummel and Jockusch [95] studied generalizations of the notion of cohesiveness. Dzhafarov [45] studied “almost all” versions of SRT22 using tools from effective measure theory. Flood =-=[56]-=- studied a principle that is a hybrid of Weak König’s Lemma and Ramsey’s Theorem for pairs. Combinatorial principles of reverse mathematical interest can come from many different sources. Dzhafarov [... |

1 |
The Complexity of Classical Theorems on Saturated Models
- Harris
- 2007
(Show Context)
Citation Context ... between being saturated bounding and being able to subuniformly enumerate the computable sets, in the sense of Theorem 3.23, but we do not know how close this connection is. 80 Exercise 9.74 (Harris =-=[80, 81]-=-). Use Theorems 3.23 and 9.73 to show that if X is either high or has PA degree, then X is saturated bounding. I Open Question 9.75. Is it the case that X is saturated bounding iff X is either high or... |

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On bounding saturated models, preprint, available at time of writing at http://people.cs.uchicago.edu/~kaharris/papers/ computability/sat.pdf
- Harris
(Show Context)
Citation Context ... between being saturated bounding and being able to subuniformly enumerate the computable sets, in the sense of Theorem 3.23, but we do not know how close this connection is. 80 Exercise 9.74 (Harris =-=[80, 81]-=-). Use Theorems 3.23 and 9.73 to show that if X is either high or has PA degree, then X is saturated bounding. I Open Question 9.75. Is it the case that X is saturated bounding iff X is either high or... |

1 |
Symbolic Logic 74
- Jockusch, Kastermans, et al.
- 2009
(Show Context)
Citation Context ...valent over RCA0. The above notion of stability is not necessarily the most intuitively natural one. (See [88] for more on this choice of definition.) Jockusch, Kastermans, Lempp, Lerman, and Solomon =-=[101]-=- defined a partial order to be weakly stable if each element is small, large, or isolated. As the following theorems show, these two notions of stability are computability theoretically different but ... |