# On what I do not understand (and have something to say): Part I

Fundamenta Mathematicae (2000)

- Volume: 166, Issue: 1-2, page 1-82
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topShelah, Saharon. "On what I do not understand (and have something to say): Part I." Fundamenta Mathematicae 166.1-2 (2000): 1-82. <http://eudml.org/doc/212475>.

@article{Shelah2000,

abstract = {This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdotes and opinions. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept to a minimum ("see ..." means: see the references there and possibly the paper itself). The base were lectures in Rutgers, Fall '97, and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Rosłanowski for many helpful comments.},

author = {Shelah, Saharon},

journal = {Fundamenta Mathematicae},

keywords = {set theory; cardinal arithmetic; pcf theory; forcing; iterated forcing; large continuum; nep; nicely definable forcing; combinatorial set theory; Boolean algebras; set-theoretic algebra; partition calculus; Ramsey theory},

language = {eng},

number = {1-2},

pages = {1-82},

title = {On what I do not understand (and have something to say): Part I},

url = {http://eudml.org/doc/212475},

volume = {166},

year = {2000},

}

TY - JOUR

AU - Shelah, Saharon

TI - On what I do not understand (and have something to say): Part I

JO - Fundamenta Mathematicae

PY - 2000

VL - 166

IS - 1-2

SP - 1

EP - 82

AB - This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdotes and opinions. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept to a minimum ("see ..." means: see the references there and possibly the paper itself). The base were lectures in Rutgers, Fall '97, and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Rosłanowski for many helpful comments.

LA - eng

KW - set theory; cardinal arithmetic; pcf theory; forcing; iterated forcing; large continuum; nep; nicely definable forcing; combinatorial set theory; Boolean algebras; set-theoretic algebra; partition calculus; Ramsey theory

UR - http://eudml.org/doc/212475

ER -

## References

top- [1] U. Abraham, Aronszajn trees on ${\aleph}_{2}$ and ${\aleph}_{3}$, Ann. Pure Appl. Logic 24 (1983), 213-230.
- [2] U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ${\aleph}_{1}$-dense real order types, Ann. Pure Appl. Logic 29 (1985), 123-206. Zbl0585.03019
- [3] U. Abraham and S. Shelah, Isomorphism types of Aronszajn trees, Israel J. Math. 50 (1985), 75-113. Zbl0566.03032
- [4] U. Abraham and S. Todorčević, Partition properties of ${\omega}_{1}$ compatible with CH, Fund. Math. 152 (1997), 165-181. Zbl0879.03015
- [5] M. Ajtai, The complexity of the pigeonhole principle, Combinatorica 14 (1994), 417-433. Zbl0811.03042
- [6] U. Avraham [U. Abraham], K. J. Devlin and S. Shelah, The consistency with CH of some consequences of Martin’s axiom plus ${2}_{0}^{\aleph}>{\aleph}_{1}$, Israel J. Math. 31 (1978), 19-33. Zbl0382.03040
- [7] T. Bartoszyński and H. Judah, Set Theory: On the Structure of the Real Line, A K Peters, Wellesley, MA, 1995. Zbl0834.04001
- [8] T. Bartoszyński, A. Rosłanowski and S. Shelah, After all, there are some inequalities which are provable in ZFC, math.LO/9711222; J. Symbolic Logic 65 (2000), 803-816. Zbl0960.03042
- [9] T. Bartoszyński, A. Rosłanowski and S. Shelah, Adding one random real, ibid. 61 (1996), 80-90;math.LO/9406229. Zbl0859.03023
- [10] J. E. Baumgartner, Decomposition of embedding of trees, Notices Amer. Math. Soc. 17 (1970), 967.
- [11] J. E. Baumgartner, All ${\aleph}_{1}$-dense sets of reals can be isomorphic, Fund. Math. 79 (1973), 101-106. Zbl0274.02037
- [12] J. E. Baumgartner, Ultrafilters on ω, J. Symbolic Logic 60 (1995), 624-639. Zbl0834.04005
- [13] J. Baumgartner, A. Hajnal and S. Todorčević, Extensions of the Erdős-Rado Theorems, in: Finite and Infinite Combinatorics in Set Theory and Logic, Kluwer, 1993, 1-18. Zbl0846.03021
- [14] S. Ben David, On Shelah's compactness of cardinals, Israel J. Math. 31 (1978), 34-56 and 394. Zbl0384.03036
- [16] A. Blass and S. Shelah, Ultrafilters with small generating sets, Israel J. Math. 65 (1989), 259-271. Zbl0681.03033
- [17] R. Bonnet and D. Monk, Handbook of Boolean Algebras, Vols. 1-3, North-Holland, 1989.
- [18] J. Brendle and S. Shelah, Ultrafilters on ω--their ideals and their cardinal characteristics, Trans. Amer. Math. Soc. 351 (1999), 2643-2674; math.LO/9710217. Zbl0927.03073
- [19] T. J. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118 (1993), 577-586. Zbl0787.03037
- [20] K. Ciesielski, Set theoretic real analysis, J. Appl. Anal. 3 (1997), 143-190. Zbl0901.26001
- [21] K. Ciesielski and S. Shelah, Category analog of sub-measurability problem, ibid., to appear; math.LO/9905147.
- [22] J. Cummings, M. Džamonja and S. Shelah, A consistency result on weak reflection, Fund. Math. 148 (1995), 91-100; math.LO/9504221. Zbl0830.03024
- [23] J. Cummings and M. Foreman, The tree property, Adv. Math. 133 (1998), 1-32. Zbl0949.03039
- [25] J. Cummings and S. Shelah, Cardinal invariants above the continuum, Ann. Pure Appl. Logic 75 (1995), 251-268; math.LO/9509228. Zbl0835.03013
- [26] K. J. Devlin and S. Shelah, A weak version of ◊ which follows from ${2}_{0}^{\aleph}<{2}_{1}^{\aleph}$, Israel J. Math. 29 (1978), 239-247. Zbl0403.03040
- [27] T. Dodd and R. B. Jensen, The covering lemma for K, Ann. Math. Logic 22 (1982), 1-30. Zbl0492.03014
- [30] M. Džamonja and S. Shelah, On squares, outside guessing of clubs and ${I}_{<}f\left[\lambda \right]$, Fund. Math. 148 (1995), 165-198; math.LO/9510216. Zbl0839.03031
- [31] M. Džamonja and S. Shelah, Saturated filters at successors of singulars, weak reflection and yet another weak club principle, Ann. Pure Appl. Logic 79 (1996), 289-316; math.LO/9601219. Zbl0874.03059
- [32] P. C. Eklof and A. Mekler, Almost Free Modules; Set Theoretic Methods, North-Holland Library, 1990. Zbl0718.20027
- [33] P. Erdős and A. Hajnal, Unsolved problems in set theory, in: Axiomatic Set Theory, Proc. Sympos. Pure Math., 13, part I, Providence, RI, 1971, Amer. Math. Soc., 17-18. Zbl0228.04001
- [34] P. Erdős, A. Hajnal, A. Maté and R. Rado, Combinatorial Set Theory: Partition Relations for Cardinals, Stud. Logic Found. Math. 106, North-Holland, Amsterdam, 1984. Zbl0573.03019
- [35] W. G. Fleissner and S. Shelah, Collectionwise Hausdorff: incompactness at singulars, Topology Appl. 31 (1989), 101-107. Zbl0659.54016
- [36] M. Foreman and H. Woodin, The generalized continuum hypothesis can fail everywhere, Ann. Math. 133 (1991), 1-36. Zbl0718.03040
- [38] D. Fremlin, Real-valued-measurable cardinals, in: Set Theory of the Reals, Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 151-304.
- [39] L. Fuchs, Infinite Abelian Groups, Vols. I, II, Academic Press, New York, 1970, 1973.
- [40] S. Garcia-Ferreira and W. Just, Two examples of relatively pseudocompact spaces, Questions Answers Gen. Topology 17 (1999), 35-45. Zbl0943.54025
- [41] M. Gitik, All uncountable cardinals can be singular, Israel J. Math. 35 (1980), 61-88. Zbl0439.03036
- [47] M. Gitik and S. Shelah, Less saturated ideals, Proc. Amer. Math. Soc. 125 (1997), 1523-1530; math.LO/9503203. Zbl0864.03031
- [48] M. Goldstern and S. Shelah. Many simple cardinal invariants, Arch. Math. Logic 32 (1993), 203-221; math.LO/9205208. Zbl0786.03030
- [49] R. Graham, B. L. Rothschild and J. Spencer, Ramsey Theory, Wiley, New York, 1980.
- [50] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41(3) (1976), 663-671. Zbl0347.02044
- [51] R. Grossberg and S. Shelah, On the structure of $Ex{t}_{p}(G,\mathbf{Z})$, J. Algebra 121 (1989), 117-128. See also [52] below. Zbl0668.20044
- [52] R. Grossberg and S. Shelah, On cardinalities in quotients of inverse limits of groups, Math. Japonica 47 (1998), 189-197. Zbl0911.20040
- [53] A. Hajnal, True embedding partition relations, in: Finite and Infinite Combinatorics in Sets and Logic, Kluwer, 1993, 135-152.
- [54] A. Hajnal and P. Hamburger, Halmazelmélet [Set Theory], Tankönyvkiaó Vállalat, Budapest, 1983 (in Hungarian).
- [55] A. Hajnal, I. Juhász and Z. Szentmiklóssy, On the structure of CCC partial orders, Algebra Universalis, to appear. Zbl0938.06001
- [56] J. D. Hamkins, Every group has a terminating transfinite automorphism tower, Proc. Amer. Math. Soc. 126 (1998), 3223-3226. Zbl0904.20027
- [57] L. A. Harrington, M. D. Morley, A. Ščedrov and S. G. Simpson (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Stud. Logic Found. Math. 117, North-Holland, 1985. Zbl0588.03001
- [58] W. Hodges, For singular λ, λ-free implies free, Algebra Universalis 12 (1981), 205-220.
- [59] J. Ihoda [H. Judah] and S. Shelah, Souslin forcing, J. Symbolic Logic 53 (1988), 1188-1207.
- [60] T. Jech and S. Shelah, Possible pcf algebras, ibid. 61 (1996), 313-317; math.LO/9412208. Zbl0878.03036
- [61] H. Judah and S. Shelah, MA(σ-centered): Cohen reals, strong measure zero sets and strongly meager sets, Israel J. Math. 68 (1989), 1-17. Zbl0693.03032
- [62] H. Judah and S. Shelah, Baire Property and Axiom of Choice, Israel J. Math. 84 (1993), 435-450; math.LO/9211213. Zbl0797.03049
- [64] I. Juhász, Cardinal functions, in: Recent Progress in General Topology (Prague, 1991), North-Holland, Amsterdam, 1992, 417-441.
- [65] I. Juhász and S. Shelah, How large can a hereditarily separable or hereditarily Lindelöf space be?, Israel J. Math. 53 (1986), 355-364. Zbl0607.54002
- [66] W. Just, A. R. D. Mathias, K. Prikry and P. Simon, On the existence of large p-ideals, J. Symbolic Logic 55 (1990), 457-465. Zbl0715.04002
- [67] W. Just, S. Shelah and S. Thomas, The automorphism tower problem III: Closed groups of uncountable degree, Adv. Math., accepted.
- [68] A. S. Kechris and S. Solecki, Approximation of analytic by Borel sets and definable countable chain conditions, Israel J. Math. 89 (1995), 343-356. Zbl0827.54023
- [69] J. Ketonen, On the existence of P-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91-94. Zbl0339.54035
- [71] M. Kojman and S. Shelah, The universality spectrum of stable unsuperstable theories, Ann. Pure Appl. Logic 58 (1992), 57-72; math.LO/9201253. Zbl0769.03019
- [72] P. Komjáth, On second-category sets, Proc. Amer. Math. Soc. 107 (1989), 653-654. Zbl0672.28002
- [73] D. W. Kueker, Countable approximations and Löwenheim-Skolem theorems, Ann. Math. Logic 11 (1977), 57-103. Zbl0364.02009
- [75] K. Kunen, Large homogeneous compact spaces, in: Open Problems in Topology, Elsevier, 1990, 261-270.
- [76] R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1976), 151-169. Zbl0357.28003
- [77] M. Magidor and S. Shelah, Length of Boolean algebras and ultraproducts, Math. Japonica 48 (1998), 301-307; math.LO/9805145. Zbl0931.03061
- [78] A. R. D. Mathias, $0$ and the p-point problem, in: Higher Set Theory (Oberwolfach, 1977), Lecture Notes in Math. 669, Springer, Berlin, 1978, 375-383.
- [79] A. H. Mekler, A. Rosłanowski and S. Shelah, On the p-rank of Ext, Israel J. Math. 112 (1999), 327-356; math.LO/9806165 Zbl0942.20037
- [81] A. H. Mekler and S. Shelah, Almost free algebras, Israel J. Math. 89 (1995), 237-259; math.LO/9408213. Zbl0821.08008
- [82] A. H. Mekler, S. Shelah and O. Spinas, The essentially free spectrum of a variety, Israel J. Math. 93 (1996), 1-8; math.LO/9411234. Zbl0845.08005
- [83] A. W. Miller, Arnie Miller's problem list, in: Set Theory of the Reals, Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 645-654. Zbl0828.03017
- [84] D. Monk, Cardinal Invariants of Boolean Algebras, Progr. Math. 142, Birkhäuser, Basel, 1996. Zbl0849.03038
- [85] J. Roitman, Adding a random or Cohen real: topological consequences and the effect on Martin's axiom, Fund. Math. 103 (1979), 47-60. Zbl0442.03034
- [92] A. Rosłanowski and S. Shelah, Norms on possibilities I: forcing with trees and creatures, Mem. Amer. Math. Soc. 671 (1999); math.LO/9807172. Zbl0940.03059
- [93] M. Rubin and S. Shelah, Combinatorial problems on trees: partitions, Δ-systems and large free subtrees, Ann. Pure Appl. Logic 33 (1987), 43-81. Zbl0654.04002
- [94] G. Sageev and S. Shelah, Noetherian ring with free additive groups, Abstracts Amer. Math. Soc. 7 (1986), 369.
- [96] J. D. Sharp and S. Thomas, Unbounded families and the cofinality of the infinite symmetric group, Arch. Math. Logic 34 (1995), 33-45. Zbl0818.03027
- [107] S. Shelah, Few non-minimal types and non-structure, in: Proc. 11 Internat. Congress of Logic, Methodology and Philosophy of Science (Krokow, 1999), Kluwer, to appear; math.LO/9906023.
- [112] S.Shelah, More constructions for Boolean algebras, Arch. Math. Logic,submitted; math.LO/9605235.
- [122] S. Shelah, On what I do not understand (and have something to say), model theory, Math. Japonica, accepted; math.LO/9910158.
- [143] S. Shelah, Classification theory for nonelementary classes, I. The number of uncountable models of $\psi \in {L}_{{\omega}_{1},\omega}$. Part B, Israel J. Math. 46 (1983), 241-273.
- [148] S. Shelah, More on proper forcing, ibid. 49 (1984), 1034-1038. Zbl0577.03026
- [166] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
- [168] S. Shelah, How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null, Israel J. Math. 88 (1994), 159-174; math.LO/9303208.
- [179] S. Shelah, Note on ω-nw-nep forcing notions, preprint.
- [180] S. Shelah, J. Saxl and S. Thomas, Infinite products of finite simple groups, Trans. Amer. Math. Soc. 348 (1996), 4611-4641; math.IG/9605202. Zbl0867.20026
- [181] S. Shelah and O. Spinas, On incomparability and related cardinal functions on ultraproducts of Boolean algebras, in preparation; math.LO/9903116. Zbl0971.03048
- [183] S. Shelah and L. Stanley, A theorem and some consistency results in partition calculus, Ann. Pure Appl. Logic 36 (1987), 119-152. Zbl0625.03029
- [184] S. Shelah and J. Steprāns, Somewhere trivial automorphisms, J. London Math. Soc. 49 (1994), 569-580; math.LO/9308214. Zbl0817.54005
- [185] S. Shelah and J. Steprāns, Martin's axiom is consistent with the existence of nowhere trivial automorphisms, Proc. Amer. Math. Soc., submitted. Zbl0993.03064
- [186] S. Shelah and S. Thomas, The cofinality spectrum of the infinite symmetric group, J. Symbolic Logic 62 (1997), 902-916; math.LO/9412230. Zbl0889.03037
- [187] S. Shelah and J. Zapletal, Canonical models for ${\aleph}_{1}$ combinatorics, Ann. Pure Appl. Logic 98 (1999), 217-259; math.LO/9806166. Zbl0935.03055
- [189] S. Solecki, Analytic ideals, Bull. Symbolic Logic 2 (1996), 339-348. Zbl0862.04002
- [190] R. M. Solovay, Real valued measurable cardinals, in: Axiomatic Set Theory, Proc. Symposia Pure Math. 13, part 1, Providence, RI, Amer. Math. Soc. 1971, 397-428.
- [191] S. G. Simpson (moderator), E-mail list for discussing foundations of mathematics, http://www.math.psu.edu/simpson/fom/.
- [192] J. Steprāns and S. W. Watson, Homeomorphisms of manifolds with prescribed behaviour on large dense sets, Bull. London Math. Soc. 19 (1987), 305-310. Zbl0637.03051
- [193] S. Thomas, The cofinalities of the infinite-dimensional classical groups, J. Algebra 179 (1996), 704-719. Zbl0846.20005
- [194] S. Todorčević, Partition Problems in Topology, Contemp. Math. 84, Amer. Math. Soc., Providence, RI, 1989. Zbl0659.54001
- [195] J. van Mill, An introduction to βω, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 503-567.
- [196] B. Veličković, Applications of the open coloring axiom, in: Set Theory of the Continuum (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ. 26, Springer, 1992, 137-154. Zbl0785.03032
- [197] S. W. Williams, Box products, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 169-200.
- [198] H. Woodin, The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal, de Gruyter Ser. Logic Appl. 1, de Gruyter, in press.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.