Abstract
It is well known how to generalize the meagre ideal replacing \(\aleph _0\) by a (regular) cardinal \(\lambda > \aleph _0\) and requiring the ideal to be \(({<}\lambda )\)-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing \(\aleph _0\) by \(\lambda \). So naturally, to call it a generalization we require it to be \(({<}\lambda )\)-complete and \(\lambda ^+\)-c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of \(\lambda \)-Borel sets modulo the ideal. Common wisdom have said that there is no such thing because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive \(=\) existence answer for a generalization of the null ideal for a “mild” large cardinal \(\lambda \)—a weakly compact one. Second, we try to show that this together with the meagre ideal (for \(\lambda \)) behaves as in the countable case. In particular, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no \(\lambda \)-Cohen; Random \(\lambda \)-reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. ar**v:1603.08362 [math.LO]) and a joint work with Baumhauer and Goldstern.
Similar content being viewed by others
References
Bartoszyński, T., Judah, H.: Set Theory: On the Structure of the Real Line. A K Peters, Wellesley (1995)
Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, pp. 395–490. Springer, New York (2010)
Brendle, J., Shelah, S.: Ultrafilters on \(\omega \)-their ideals and their cardinal characteristics. Trans. Am. Math. Soc. 351, 2643–2674 (1999). ar**v:math.LO/9710217
Cohen, S., Shelah, S.: On a parallel of random real forcing for inaccessible cardinals (2016) ar**v:1603.08362 [math.LO]
Cummings, J., Shelah, S.: Cardinal invariants above the continuum. Ann. Pure Appl. Log. 75, 251–268 (1995). ar**v:math.LO/9509228
Fremlin, D.H.: Measure Theory, vol. 1–5. Torres Fremlin, Colchester (2004). https://www.essex.ac.uk/maths/people/fremlin/mt.htm
Grossberg, R., Shelah, S.: On cardinalities in quotients of inverse limits of groups. Math. Jpn. 47, 189–197 (1998). ar**v:math/9911225 [math.LO]
Halko, A., Shelah, S.: On strong measure zero subsets of \({}^\kappa 2\). Fundam. Math. 170, 219–229 (2001). ar**v:math.LO/9710218
Magidor, M., Shelah, S., Stavi, J.: On the standard part of nonstandard models of set theory. J. Symb. Log. 48, 33–38 (1983)
Magidor, M., Shelah, S., Stavi, J.: Countably decomposable admissible sets. Ann. Pure Appl. Log. 26, 287–361 (1984). Proceedings of the 1980/1 Jerusalem Model Theory year
Malliaris, M., Shelah, S.: Constructing regular ultrafilters from a model-theoretic point of view. Trans. Am. Math. Soc. 367, 8139–8173 (2015)
Malliaris, M., Shelah, S.: Cofinality spectrum theorems in model theory, set theory and general topology. J. Am. Math. Soc. 29, 237–297 (2016). ar**v:1208.5424
Matet, P., Rosłanowski, A., Shelah, S.: Cofinality of the nonstationary ideal. Trans. Am. Math. Soc. 357, 4813–4837 (2005). ar**v:math.LO/0210087
Oxtoby, J.C.: Measure and Category. A Survey of the Analogies Between Topological and Measure Spaces. Graduate Texts in Mathematics, vol. 2. Springer, New York (1980)
Rosłanowski, A., Shelah, S.: Norms on possibilities II: More ccc ideals on \(2^{\textstyle \omega }\). J. Appl. Anal. 3, 103–127 (1997). arxiv:math.LO/9703222
Rosłanowski, A., Shelah, S.: Sweet & sour and other flavours of ccc forcing notions. Arch. Math. Log. 43, 583–663 (2004). ar**v:math.LO/9909115
Rosłanowski, A., Shelah, S.: How much sweetness is there in the universe? Math. Log. Q. 52, 71–86 (2006). ar**v:math.LO/0406612
Rosłanowski, A., Shelah, S.: Reasonably complete forcing notions. Quad. Mat. 17, 195–239 (2006). ar**v:math.LO/0508272
Rosłanowski, A., Shelah, S.: Sheva–Sheva–Sheva: large creatures. Isr. J. Math. 159, 109–174 (2007). arxiv:math.LO/0210205
Rosłanowski, A., Shelah, S.: Generating ultrafilters in a reasonable way. Math. Log. Q. 54, 202–220 (2008). ar**v:math.LO/0607218
Rosłanowski, A., Shelah, S.: Lords of the iteration. In: Set Theory and Its Applications, volume 533 of Contemporary Mathematics (CONM), pp. 287–330. American Mathematical Society (2011). arxiv:math.LO/0611131
Rosłanowski, A., Shelah, S.: Reasonable ultrafilters, again. Notre Dame J. Form. Log. 52, 113–147 (2011). arxiv:math.LO/0605067
Rosłanowski, A., Shelah, S.: More about \(\lambda \)-support iterations of \(({<}\lambda )\)-complete forcing notions. Arch. Math. Log. 52, 603–629 (2013). ar**v:1105.6049
Shelah, S.: Bounding forcing with chain conditions for uncountable cardinals. http://shelah.logic.at/E82_abs.html. Accessed 15 Feb 2017
Shelah, S.: Creature iteration for inaccesibles. http://shelah.logic.at/1100_abs.html. Accessed 15 Feb 2017
Shelah, S.: Iterations adding no \(\lambda \)-Cohen (in preparation)
Shelah, S.: On CON(\(\mathfrak{d}_\lambda >\text{cov}_\lambda \)(meagre)). Trans. Am. Math. Soc. arxiv:0904.0817
Shelah, S.: Random \(\lambda \)-reals for inaccessible continued (in preparation)
Shelah, S.: The null ideal for uncountable cardinals (in preparation)
Shelah, S.: Can the fundamental (homotopy) group of a space be the rationals? Proc. Am. Math. Soc. 103, 627–632 (1988)
Shelah, S.: Classification theory and the number of nonisomorphic models. In: Barwise, J., Keisler, H.J., Suppes, P., Troelstra, A.S. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 92. North-Holland, Amsterdam (1990)
Shelah, S.: Vive la différence I: nonisomorphism of ultrapowers of countable models. In: Set Theory of the Continuum, volume 26 of Mathematical Sciences Research Institute Publications, pp. 357–405. Springer, Berlin (1992). arxiv:math.LO/9201245
Shelah, S.: How special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null. Isr. J. Math. 88, 159–174 (1994). ar**v:math.LO/9303208
Shelah, S.: Proper and improper forcing. In: Feferman, S., Hodges, W.A., Lerman, M., Macintyre, A.J., Magidor, M., Moschovakis, Y.M. (eds.) Perspectives in Mathematical Logic. Springer, Berlin (1998)
Shelah, S.: Consistently there is no non trivial ccc forcing notion with the Sacks or Laver property. Combinatorica 21, 309–319 (2001). ar**v:math.LO/0003139
Shelah, S.: Strong dichotomy of cardinality. Results Math. 39, 131–154 (2001). ar**v:math.LO/9807183
Shelah, S.: On nice equivalence relations on \({}^\lambda 2\). Arch. Math. Log. 43, 31–64 (2004). ar**v:math.LO/0009064
Shelah, S.: Properness without elementaricity. J. Appl. Anal. 10, 168–289 (2004). ar**v:math.LO/9712283
Shelah, S.: Quite complete real closed fields. Isr. J. Math. 142, 261–272 (2004). ar**v:math.LO/0112212
Shelah, S.: On nicely definable forcing notions. J. Appl. Anal. 111(1), 1–17 (2005). ar**v:math.LO/0303293
Shelah, S.: The combinatorics of reasonable ultrafilters. Fundam. Math. 192, 1–23 (2006). ar**v:math.LO/0407498
Shelah, S.: The spectrum of characters of ultrafilters on \(\omega \). Colloq. Math. 111(2), 213–220 (2008). ar**v:math.LO/0612240
Shelah, S.: Polish algebras, shy from freedom. Isr. J. Math. 181, 477–507 (2011). ar**v:math.LO/0212250
Shelah, S.: The character spectrum of \(\beta (N)\). Topol. Appl. 158, 2535–2555 (2011). ar**v:1004.2083
Shelah, S., Väisänen, P.: On equivalence relations second order definable over \(H(\kappa )\). Fundam. Math. 174, 1–21 (2002). ar**v:math.LO/9911231
Shelah, S., Väisänen, P.: The number of \(L_{\infty \kappa }\)-equivalent nonisomorphic models for \(\kappa \) weakly compact. Fundam. Math. 174, 97–126 (2002). ar**v:math.LO/9911232
Solovay, R.M.: A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92, 1–56 (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of James E. Baumgartner (1943–2011)
Research supported by the United-States-Israel Binational Science Foundation (Grant Nos. 2006108, 2010405). Publication 1004.
Rights and permissions
About this article
Cite this article
Shelah, S. A parallel to the null ideal for inaccessible \(\lambda \): Part I. Arch. Math. Logic 56, 319–383 (2017). https://doi.org/10.1007/s00153-017-0524-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-017-0524-0