### BibTeX

@MISC{Bovykin_briefintroduction,

author = {Andrey Bovykin},

title = {Brief introduction to unprovability},

year = {}

}

### OpenURL

### Abstract

### Citations

149 | Hilbert’s Tenth Problem - Matiyasevich - 1993 |

115 | Metamathematics of First-Order Arithmetics - Hájek, Pudlák - 1993 |

45 | Accessible independence results for Peano Arithmetic
- Kirby, Paris
- 1982
(Show Context)
Citation Context ...les[14] and [15] by the author and A. Weiermann. PH is an arithmetical version of large cardinals The historical prototypes of the Paris-Harrigton Principle and the earlier PA-unprovablestatements of =-=[70]-=- are large cardinal axioms. In the case of arithmetic, closedness properties postulated by large cardinal axioms correspond to closedness properties of initialsegments of models of arithmetic under th... |

41 |
Combinatorial theorems on classifications of subsets of a given set
- Erdős, Rado
- 1952
(Show Context)
Citation Context ...= x 1 F-1m (x) . In particular, KM(2)log is provable but PH(2)log is unprovable. The reason for I\Sigma 1-provability of PH(k+1)log(n) and KM(k+1)log(n) for large n comes from the Erd"os-Rado theorem =-=[28]-=-, which implies that an upper bound for the Ramsey number R(m, k + 1, m) is 2n(m) for some large enough n depending only on k. Given m and c,let ` = max{m, c}. Consider any colouring f : [0, 2n(`)]k+1... |

38 |
Nonprovability of certain combinatorial properties of finite trees
- Simpson
- 1985
(Show Context)
Citation Context ...skal's Theorem and well-quasi-orders Very often an unprovable statement can be viewed as a `miniaturisation' of an infinitarytheorem. A spectacular example of miniaturisation is H. Friedman's Theorem =-=[79]-=- on unprovability of a finite version of Kruskal's Theorem. Define a (nonplane, rooted) tree asa partially ordered set with the least element and such that the set of all predecessors of every point i... |

36 |
The metamathematics of the graph minor theorem
- Friedman, Robertson, et al.
(Show Context)
Citation Context ...ults, to the author's knowledge, haven't yet been written down by anyone.Another important example is a theorem by H. Friedman, N. Robertson and P. Seymour on unprovability of the Graph Minor Theorem =-=[32]-=-. For multigraphs G and H, we say that H is a minor of G if H can be obtained from G by a succession of three elementaryoperations: edge removal, edge contraction and removal of an isolated vertex. Th... |

34 |
A finite combinatorial principle which is equivalent to the l-consistency of predicative analysis
- FRIEDMAN, MCALOON, et al.
- 1982
(Show Context)
Citation Context ...ts [25], Cichon's treatmentof Goodstein sequences [24], Bigorajska-Kotlarski series of results on partitioning ff-largesets (see e.g. [8], [9]), the Friedman-McAloon-Simpson early fundamental article =-=[30]-=- and the subsequent paper by S. Shelah [84], J.-P. Ressayre's statements unprovable in I\Sigma 1, PAand ZF that state existence of finite approximations of models of these theories [77], the treatment... |

34 |
Rapidly growing Ramsey functions
- Ketonen, Solovay
- 1981
(Show Context)
Citation Context ... of large cardinals. J. Ketonen's manuscripts [49], [51] provide an alternative way toview such analogy, using ordinals, a theme that is closely connected with the fundamental Ketonen-Solovay article =-=[50]-=-. Fragments of Peano Arithmetic If we restrict the induction scheme to \Sigma n-formulas, that is, arithmetical formulas of theform 9 x18x29x3 . . . '(x1, x2, . . . , xn, y), where ' is preceded by no... |

28 |
Hierarchies of provably recursive functions. Handbook of proof theory
- Fairtlough, Wainer
- 1998
(Show Context)
Citation Context ...y different proofs of G"odel's Theorem. The second category is well explained in an earlyarticle by W. Buchholz and S. Wainer [18], in the Friedman-Sheard article [33], in the Fairtlough-Wainer paper =-=[29]-=- and in the recent article by A. Weiermann [91].Apparently, there is also a third category of unprovability proofs, proofs that interpret directly independent statements as reduction strategies for pr... |

28 | Finite functions and the necessary use of large cardinals
- Friedman
- 1998
(Show Context)
Citation Context ... [19], L. Carlucci's recent uniform treatment of hydras, wormsand sequences with gap condition [23] and H. Friedman's many results on combinatorial statements unprovable in ZFC+ large cardinals [31], =-=[34]-=-. (Many new unprovable state-ments are announced regularly on H. Friedman's webpage, so we shall not even attempt to give an up-to-date account of these rapid developments here.)The subject of first-o... |

24 |
Ordinal numbers and the Hilbert basis theorem
- Simpson
- 1988
(Show Context)
Citation Context ...fication" of why the corresponding ordinal descent principle is true. This ordinal nowadays turns up in various natural mathematical con-texts, see for example Higman's lemma and S. Simpson's article =-=[82]-=- on the Robson Basis Theorem.One more family of independence results comes from braid-theoretic analogues of the hydra battle. Termination of the following game on positive braids is unprovable in I\S... |

23 |
Some problems in logic and number theory, and their connections
- Woods
- 1981
(Show Context)
Citation Context ...rm n2 + 1) and its formulation alreadyprovides some necessary ingredients for the unprovability proof. ( I\Sigma 1-unprovability of hy-pothesis H was already conjectured in the PhD thesis of A. Woods =-=[93]-=-). Braids Braids are very popular and interesting objects in today's mainstream mathematics. The n-strand braid group Bn is a group with the following presentation: Bn = hoe1, . . . , oen-1; oeioej = ... |

21 |
Models of Peano Arithmetic. Oxford Logic Guides, 15. Oxford Science Publications
- Kaye
- 1991
(Show Context)
Citation Context ...gma k) implies ConI\Sigma k .) A good exposition of 1-consistency and reflection principles is in the old Smorynski's paper [85], a good exposition of the satisfaction predicate is in Kaye's textbook =-=[48]-=-. Threshold results for PH and KM Let log(n)(x) = log2(log2 . . . log2-- -z "" n times (x)) and the tower-function 2n(x) be defined as 20(x) = x, 2n+1(x) = 22n(x). Also, define log*(m) as the minimal ... |

19 |
Elementary descent recursion and proof theory. Annals of Pure and Applied Logic 71
- Friedman, Sheard
- 1995
(Show Context)
Citation Context ... of both approaches as well as of many different proofs of G"odel's Theorem. The second category is well explained in an earlyarticle by W. Buchholz and S. Wainer [18], in the Friedman-Sheard article =-=[33]-=-, in the Fairtlough-Wainer paper [29] and in the recent article by A. Weiermann [91].Apparently, there is also a third category of unprovability proofs, proofs that interpret directly independent stat... |

19 |
On Gödel incompleteness and finite combinatorics
- Kanamori, McAloon
- 1987
(Show Context)
Citation Context ...[56], the arboreal statement by G. Mills [68], P. Pudlák’s Principle [74, 45], the kiralic and regal principles by P. Clote and K. McAloon [27]. An important PA-unprovable statement was introduced in =-=[50]-=- by A. Kanamori and K. McAloon. A function f in m arguments is called regressive if f(x0,x1,...,xm−1) ≤ x0 for all x0 <x1 < ··· <xm−1. For regressive functions of m arguments, we cannot guarantee exis... |

18 |
Hilbert’s Tenth Problem, Foundations of Computing Series
- Matiyasevich
- 1993
(Show Context)
Citation Context ...ition, I would like to mention unprovable statements obtained by translating consistency statements for various theories into non-solvability assertions about corresponding Diophantine equations (see =-=[65]-=-), unprovable statements about Diophantine games [65], the Schütte-Simpson treatment of finite sequences of natural numbers with gapcondition [85], K. McAloon’s [66], [67] and Z. Ratajczyk’s [76], [77... |

17 |
Provably computable functions and the fastgrowing hierarchy
- Buchholz, Wainer
- 1985
(Show Context)
Citation Context ...[58] by H. Kotlarski is an exposition of both approaches as well as of many different proofs of G"odel's Theorem. The second category is well explained in an earlyarticle by W. Buchholz and S. Wainer =-=[18]-=-, in the Friedman-Sheard article [33], in the Fairtlough-Wainer paper [29] and in the recent article by A. Weiermann [91].Apparently, there is also a third category of unprovability proofs, proofs tha... |

17 | A classification of rapidly growing Ramsey functions
- Weiermann
(Show Context)
Citation Context ... Also, define PHF as 8k PH(k)F and KMF as 8k KM(k)F . Itis easy to see that for every strictly increasing F , PHF implies PH and KMF implies KMthus making these statements PA-unprovable. A. Weiermann =-=[89]-=- proved that for every n,PH log(n) is PA-unprovable but PHlog* is PA-provable. In the case of fixed dimension, thestory is more complex. The following interesting result was first proved by Gyesik Lee... |

16 |
A mathematical incompleteness
- Paris, Harrington
- 1977
(Show Context)
Citation Context ...athematical' character (notreferring to arithmetisation of syntax and provability) appeared in 1976 in the work of J. Paris (building upon joint work with L. Kirby [52]) and led to the formulation in =-=[69]-=- ofthe Paris-Harrington Principle (denoted PH): "for any numbers m, n and c, there exists anumber N such that for every colouring f of m-subsets of {0, 1, . . . , N - 1} into c colours,there is an f -... |

14 |
On the Necessary Use of Abstract Set Theory
- Friedman
- 1981
(Show Context)
Citation Context ... Hydra [19], L. Carlucci's recent uniform treatment of hydras, wormsand sequences with gap condition [23] and H. Friedman's many results on combinatorial statements unprovable in ZFC+ large cardinals =-=[31]-=-, [34]. (Many new unprovable state-ments are announced regularly on H. Friedman's webpage, so we shall not even attempt to give an up-to-date account of these rapid developments here.)The subject of f... |

14 |
Ein in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen, Archiv für Mathematische Logik und Grundlagenforschung
- Schütte, Simpson
- 1985
(Show Context)
Citation Context ...tions about corresponding Diophantine equations (see[63]), unprovable statements about Diophantine games [63], the Sch"utte-Simpson treatment of finite sequences of natural numbers with gap-condition =-=[78]-=-, K. McAloon's [64],[65] and Z. Ratajczyk's [75],[76] theories of iterations of the Paris-Harrington Principle and R. Sommer's related result on transfinite induction [86], independence results on poi... |

13 | An application of graphical enumeration to PA
- Weiermann
(Show Context)
Citation Context ... 12 log2 i then the statement becomes I\Sigma 1-provable but for thecondition | Ti| <= k + 4 log2 i, the statement is PA-unprovable. What happens between 12and 4 was recently resolved by A. Weiermann =-=[88]-=-. Let ff be Otter's constant (ff = 1ae , where ae is the radius of convergence of P1i=0 tizi, where ti is the number of finite trees ofsize i), ff ss 2.955765 . . .. Then for any primitive recursive r... |

12 | A model-theoretic approach to ordinal analysis
- Avigad, Sommer
- 1997
(Show Context)
Citation Context ... u ^bi . . .. If a < bi then w[k] is defined as . . . bmi-2i (bi - 1)3 wkbi-1,min u (bi - 1) u ^bi . . . , otherwise w[k] is . . . bmi-1i (bi + 1)3 wkbi+1,max u (bi + 1) u ^bi . . .. The sequence w[1]=-=[2]-=- . . . [k]consists of positive even braid words and decreases with respect to the braid ordering OE. Eventually it terminates but its termination is unprovable in I\Sigma 2. There are versionsof this ... |

12 |
Why are braids orderable?, Panoramas et Synthèses 14, Soc. math. française
- Dehornoy, Dynnikov, et al.
- 2002
(Show Context)
Citation Context ...There are several independence results about braids, relying on P. Dehornoy's leftinvariant ordering OE of positive braids as !! ! and S. Burckel's ordering of n-strand positive braids as !! n-2, see =-=[27]-=-. Some unprovability results immediately follow from I\Sigmas2-unprovability of transfinite induction up to !! ! . The statement "for every K there is N such that for any sequence B1, B2, . . . , BN o... |

11 | finite sequences
- Friedman, Long
- 1983
(Show Context)
Citation Context ...r every K-sequence x1, x2, . . . , xK of elements of {1, 2, . . . , m}, there are i < j < K2 suchthat xi, xi+1, . . . , x2i is a subsequence of xj , xj+1, . . . , x2j" is unprovable in I\Sigma 2 (see =-=[36]-=-).Another series of H. Friedman's results under the general name of "boolean relation theory" can be found in [38]: if we list all (second-order) statements of a certain simpleshape (all of them a pri... |

11 |
Initial segments of models of Peano’s axioms. In: Set theory and hierarchy theory
- Kirby, Paris
- 1977
(Show Context)
Citation Context ...irst PA-unprovable statements of ‘mathematical’ character (not referring to arithmetisation of syntax and provability) appeared in 1976 in the work of J. Paris (building upon joint work with L. Kirby =-=[57]-=-) and led to the formulation in [72] of the Paris-Harrington Principle (denoted PH): “for any numbers m, n and c, there exists a number N such that for every colouring f of m-subsets of {0, 1,...,N − ... |

10 |
A short proof of two recently discovered independence results using recursion theoretic methods
- Cichon
- 1983
(Show Context)
Citation Context ...iseinduction by T. Arai [1], J. Avigad's statement with update procedures [4], P. Clote's PhD thesis and his anti-basis approach to independence results [25], Cichon's treatmentof Goodstein sequences =-=[24]-=-, Bigorajska-Kotlarski series of results on partitioning ff-largesets (see e.g. [8], [9]), the Friedman-McAloon-Simpson early fundamental article [30] and the subsequent paper by S. Shelah [84], J.-P.... |

9 |
Transfinite induction within Peano arithmetic
- Sommer
- 1995
(Show Context)
Citation Context ...l numbers with gap-condition [78], K. McAloon's [64],[65] and Z. Ratajczyk's [75],[76] theories of iterations of the Paris-Harrington Principle and R. Sommer's related result on transfinite induction =-=[86]-=-, independence results on pointwiseinduction by T. Arai [1], J. Avigad's statement with update procedures [4], P. Clote's PhD thesis and his anti-basis approach to independence results [25], Cichon's ... |

9 |
Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results
- Weiermann
(Show Context)
Citation Context ...Sigma 1-provable; 2. if r > 1log2 ff then the statement is PA-unprovable. 5sSimilar results have been proved by A. Weiermann for plane trees, binary plane trees,Higman's lemma [92], ordinal notations =-=[90]-=-, [88]. Several general theorems have recently been proved by the author [12], giving an answer to the following question: if X is awell-quasi-ordered combinatorial class (i.e., there is a fixed notio... |

8 |
Generalizations of the Kruskal-Friedman Theorems
- Gordeev
- 1990
(Show Context)
Citation Context ...heorem for binary trees and two labels unprovable in ATR0. (There are also important results by L. Gordeev about the strengthof Kruskal's theorem with gap-condition and other unprovability results in =-=[39, 40, 41]-=-.) It was later shown by M. Loebl and J. Matou^sek [60] that if the condition |Ti| <= k + iis replaced by | Ti| <= k + 12 log2 i then the statement becomes I\Sigma 1-provable but for thecondition | Ti... |

8 |
Sharp thresholds for the Phase Transition between Primitive Recursive and Ackermannian Ramsey Numbers, Preprint, 2005 (submitted to JCTA). Available at: http://www.cs.bgu.ac.il/ kojman/paperslist.html
- Kojman, Lee, et al.
(Show Context)
Citation Context ... (A.Weiermann [89]); 2. if n > k then I\Sigma 1 proves PH(k+1)log(n) (G.Lee [59]). 4sThere are even slightly sharper threshold results for this case in [22]. Here is the picturein the case k = 1 (see =-=[56]-=-): if A-1 is the inverse of the Ackermann function and {Fm}m2!is the Grzegorczyk hierarchy of primitive recursive functions then: 1. I\Sigma 1 6` PH(2)log A-1 ; 2. for every m 2 !, I\Sigma 1 ` PH(2)lo... |

7 |
A slow growing analogue to Buchholz’ proof
- Arai
- 1991
(Show Context)
Citation Context ...ssible sequencesof rational numbers. E.g. in the case of the sine-function, what we need is that every function is approximated by sine on some subset: for any " > 0, K, n and any function g : [K]n ! =-=[-1, 1]-=-, there is a sequence of rational numbers ha1, a2, . . . , aKi such that forany i1 < . . . < in <= K, |g(i1, . . . , in) - sin(ai1 * . . . * ain)| < ". This situation where allpossible patterns (e.g. ... |

7 |
Generalizations of the one-dimensional version of the Kruskal-Friedman theorems
- Gordeev
- 1989
(Show Context)
Citation Context ...heorem for binary trees and two labels unprovable in ATR0. (There are also important results by L. Gordeev about the strengthof Kruskal's theorem with gap-condition and other unprovability results in =-=[39, 40, 41]-=-.) It was later shown by M. Loebl and J. Matou^sek [60] that if the condition |Ti| <= k + iis replaced by | Ti| <= k + 12 log2 i then the statement becomes I\Sigma 1-provable but for thecondition | Ti... |

7 |
A Ramsey-type theorem
- Neˇsetˇril, Valtr
- 1994
(Show Context)
Citation Context ... functions occur in this hierarchy, the result follows)) or from thestudy of well-quasi-ordered sets [79]. Most of the proofs in the volume [80] are of this category, as well as the articles [61] and =-=[62]-=- by M. Loebl and J. Ne^set^ril. Proofs in this categorywork very well (often better than model-theoretic proofs) at the lower end of consistency strength and for well-quasi-orders, like trees, whose o... |

7 |
Paris–Harrington incompleteness and progressions of theories, in
- McAloon
- 1985
(Show Context)
Citation Context ...g Diophantine equations (see[63]), unprovable statements about Diophantine games [63], the Sch"utte-Simpson treatment of finite sequences of natural numbers with gap-condition [78], K. McAloon's [64],=-=[65]-=- and Z. Ratajczyk's [75],[76] theories of iterations of the Paris-Harrington Principle and R. Sommer's related result on transfinite induction [86], independence results on pointwiseinduction by T. Ar... |

7 |
Hierarchies of cuts in models of arithmetic. Model Theory of Algebra and
- Paris
- 1980
(Show Context)
Citation Context ... H ae [a, b] of sizeat least n". The statement KM is unprovable in PA and is equivalent to PH. Indicators First independence results grew out of indicator theory (best references are [53], [52] and 2s=-=[71]-=-). In a model M ffl I\Sigma 1, an initial segment I is called semi-regular if for every a 2 I andevery function f : [0, a] ! M (i.e., an M -coded set of pairs), the image of f is bounded in I.Every se... |

7 | On undecidability of the weakened Kruskal’s Theorem - Loebl, Matouˇsek - 1987 |

6 |
Internal finite tree embeddings. In: Reflections on the foundations of mathematics
- Friedman
- 1998
(Show Context)
Citation Context ...n the 6ssequence becoming order (or embeddability) relation between chunks of this single object.For a tree T , let T [i] be the tree of nodes of T of height <= i. It has been proved byH. Friedman in =-=[35]-=- that the statement "for all n and k there is K such that whenever T is a rooted nonplane tree labeled by {1, 2, . . . , n} and every non-leaf has degree k thenthere are i < j <= K such that T [i] is ... |

6 |
Subsystems of true arithmetic and hierarchies of functions
- RATAJCZYK
- 1993
(Show Context)
Citation Context ...63]), unprovable statements about Diophantine games [63], the Sch"utte-Simpson treatment of finite sequences of natural numbers with gap-condition [78], K. McAloon's [64],[65] and Z. Ratajczyk's [75],=-=[76]-=- theories of iterations of the Paris-Harrington Principle and R. Sommer's related result on transfinite induction [86], independence results on pointwiseinduction by T. Arai [1], J. Avigad's statement... |

6 |
2007 Phase transition thresholds for some Friedman-style independence results
- Weiermann
- 2007
(Show Context)
Citation Context ...Ti| <= k + r log i is I\Sigma 1-provable; 2. if r > 1log2 ff then the statement is PA-unprovable. 5sSimilar results have been proved by A. Weiermann for plane trees, binary plane trees,Higman's lemma =-=[92]-=-, ordinal notations [90], [88]. Several general theorems have recently been proved by the author [12], giving an answer to the following question: if X is awell-quasi-ordered combinatorial class (i.e.... |

5 | Unprovability of sharp versions of Friedman’s sine-principle
- Bovykin
- 2007
(Show Context)
Citation Context ...nion of A and D together with thestatement that they are disjoint. Sine, zeta-function, diophantine approximation and universality The results in this section spring from H. Friedman's sine-principle =-=[11]-=-. For every n >= 1and every function F of one argument, let us introduce the statement SPnF : "for all m, thereis N such that for any sequence a1, a2, . . . , aN of rational numbers, there is H ` A of... |

5 |
Consistency and games - in search of new combinatorial principles
- Pudlák
(Show Context)
Citation Context ...ing strategy. Original sources are [52] and [71]. For another example connecting games and independence results, see the general idea and the Peano Arithmeticsection in a recent article by P. Pudl'ak =-=[74]-=-. There is much more left to say about games and independence results, e.g., about unprovability of existence of a winning strategy incertain games. Indicator theory ideas are also useful in the model... |

5 |
gaps and hydras
- Worms
- 2005
(Show Context)
Citation Context ...d [16] by the author and A. Weiermann, the Kochen-Kripke proof with ultrapowers [58], the Buchholz Hydra [18], L. Carlucci’s recent uniform treatment of hydras, worms and sequences with gap condition =-=[22]-=- and H. Friedman’s many results on combinatorial statements unprovable in ZFC+ large cardinals [32], [33]. (Many new unprovable statements are announced regularly on H. Friedman’s webpage, so we shall... |

5 |
Finite Sequences
- Long
- 2001
(Show Context)
Citation Context ...such that for every K-sequence x1,x2,...,xK of elements of {1, 2,...,m}, there are i<j< K 246 ANDREY BOVYKIN such that xi,xi+1,...,x2i is a subsequence of xj,xj+1,...,x2j” is unprovable in I Σ2 (see =-=[34]-=-). Another series of H. Friedman’s results under the general name of “boolean relation theory” can be found in [37]: if we list all (second-order) statements of a certain simple shape (all of them a p... |

5 |
The incompleteness theorems, Handbook of mathematical logic
- Smorynski
- 1977
(Show Context)
Citation Context ...ödel’s Second Incompleteness Theorem: put ϕ to be ∃x x̸= x to observe that RFNΣ1 (I Σk) implies ConI Σk .) A good exposition of 1-consistency and reflection principles is in the old Smorynski’s paper =-=[86]-=-, a good exposition of the satisfaction predicate is in Kaye’s textbook [51]. Threshold results for PH and KM. Let log (n) (x) = log2 (log2 ...log2(x)) } {{ } n times and the tower-function 2n(x) be d... |

5 | Regressive Ramsey numbers are Ackermannian - Kojman, Shelah - 1999 |

4 | The model-theoretic ordinal analysis of theories of predicative strength
- Avigad, Sommer
- 1999
(Show Context)
Citation Context ...rrently absent. Apart from the original articles we mentioned above, other good sources reportingon proofs of the first category would be the book [43] by P. H'ajek and P. Pudl'ak and the papers [2], =-=[3]-=- by J. Avigad and R. Sommer (proof-theoretic aspects). A recent bookmanuscript [58] by H. Kotlarski is an exposition of both approaches as well as of many different proofs of G"odel's Theorem. The sec... |

4 | Update procedures and the 1-consistency of arithmetic
- Avigad
- 2002
(Show Context)
Citation Context ...of the Paris-Harrington Principle and R. Sommer's related result on transfinite induction [86], independence results on pointwiseinduction by T. Arai [1], J. Avigad's statement with update procedures =-=[4]-=-, P. Clote's PhD thesis and his anti-basis approach to independence results [25], Cichon's treatmentof Goodstein sequences [24], Bigorajska-Kotlarski series of results on partitioning ff-largesets (se... |

4 |
A new proof-theoretic proof of the independence of Kirby-Paris’ Hydra Theorem
- Carlucci
- 2003
(Show Context)
Citation Context ...vability proofs, proofs that interpret directly independent statements as reduction strategies for proof systems. We have littleto say about it and refer the reader to a recent article by L. Carlucci =-=[21]-=-, an old paper by H. Jervell [46] and the articles [44] and [45] by M. Hamano and M. Okada.Once we have one unprovable statement, it is usually possible to find a series of mathematical statements in ... |

4 | Classifying the phase transition threshold for regressive Ramsey functions
- Carlucci, Lee, et al.
(Show Context)
Citation Context ... case of fixed dimension, thestory is more complex. The following interesting result was first proved by Gyesik Lee [59] for all n 6= k - 1 and later completed by L. Carlucci, G. Lee and A. Weiermann =-=[22]-=-: if k >= 2 then 1. if n <= k - 1 then KM(k+1)log(n) is I\Sigma k-unprovable; 2. if n > k - 1 then I\Sigma 1 proves KM(k+1)log(n) . Similar theorems hold for the family PH(k)log(n) : 1. if n <= k then... |