On the bounding, splitting, and distributivity numbers

Alan S. Dow; Saharon Shelah

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 3, page 331-351
  • ISSN: 0010-2628

Abstract

top
The cardinal invariants 𝔥 , 𝔟 , 𝔰 of 𝒫 ( ω ) are known to satisfy that ω 1 ≤ 𝔥 ≤ min { 𝔟 , 𝔰 } . We prove that all inequalities can be strict. We also introduce a new upper bound for 𝔥 and show that it can be less than 𝔰 . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

How to cite

top

Dow, Alan S., and Shelah, Saharon. "On the bounding, splitting, and distributivity numbers." Commentationes Mathematicae Universitatis Carolinae 64.3 (2023): 331-351. <http://eudml.org/doc/299239>.

@article{Dow2023,
abstract = {The cardinal invariants $ \mathfrak \{h\}, \mathfrak \{b\},\mathfrak \{s\}$ of $ \mathcal \{P\} (\omega )$ are known to satisfy that $\omega _1 \le \mathfrak \{h\} \le \min \lbrace \mathfrak \{b\}, \mathfrak \{s\}\rbrace $. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak \{h\}$ and show that it can be less than $\mathfrak \{s\}$. The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).},
author = {Dow, Alan S., Shelah, Saharon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cardinal invariants of the continuum; matrix forcing},
language = {eng},
number = {3},
pages = {331-351},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the bounding, splitting, and distributivity numbers},
url = {http://eudml.org/doc/299239},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Dow, Alan S.
AU - Shelah, Saharon
TI - On the bounding, splitting, and distributivity numbers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 3
SP - 331
EP - 351
AB - The cardinal invariants $ \mathfrak {h}, \mathfrak {b},\mathfrak {s}$ of $ \mathcal {P} (\omega )$ are known to satisfy that $\omega _1 \le \mathfrak {h} \le \min \lbrace \mathfrak {b}, \mathfrak {s}\rbrace $. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak {h}$ and show that it can be less than $\mathfrak {s}$. The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).
LA - eng
KW - cardinal invariants of the continuum; matrix forcing
UR - http://eudml.org/doc/299239
ER -

References

top
  1. Balcar B., Pelant J., Simon P., 10.4064/fm-110-1-11-24, Fund. Math. 110 (1980), no. 1, pages 11–24. MR0600576DOI10.4064/fm-110-1-11-24
  2. Baumgartner J. E., Dordal P., 10.2307/2273792, J. Symbolic Logic 50 (1985), no. 1, 94–101. MR0780528DOI10.2307/2273792
  3. Blass A., 10.1007/BFb0097329, Conf. Set Theory and Its Applications, Toronto, 1987, Lecture Notes in Math., 1401, Springer, Berlin, 1989, pages 18–40. MR1031763DOI10.1007/BFb0097329
  4. Blass A., Shelah S., 10.1007/BF02764864, Israel J. Math. 65 (1989), no. 3, 259–271. MR1005010DOI10.1007/BF02764864
  5. Brendle J., Fischer V., 10.2178/jsl/1294170995, J. Symbolic Logic 76 (2011), no. 1, 198–208. MR2791343DOI10.2178/jsl/1294170995
  6. Brendle J., Raghavan D., 10.1016/j.apal.2013.09.002, Ann. Pure Appl. Logic 165 (2014), no. 2, 631–651. MR3129732DOI10.1016/j.apal.2013.09.002
  7. Dow A., Shelah S., 10.1016/j.indag.2017.01.010, Indag. Math. (N.S.) 29 (2018), no. 1, 382–395. MR3739621DOI10.1016/j.indag.2017.01.010
  8. Dow A., Shelah S., 10.1007/s00153-019-00674-x, Arch. Math. Logic 58 (2019), no. 7–8, 1005–1027. MR4003647DOI10.1007/s00153-019-00674-x
  9. Fischer V., Friedman S. D., Mejía D. A., Montoya D. C., 10.1017/jsl.2017.20, J. Symb. Log. 83 (2018), no. 1, 208–236. MR3796283DOI10.1017/jsl.2017.20
  10. Fischer V., Mejia D. A., 10.4153/CJM-2016-021-8, Canad. J. Math. 69 (2017), no. 3, 502–531. MR3679685DOI10.4153/CJM-2016-021-8
  11. Fischer V., Steprāns J., The consistency of 𝔟 = κ and 𝔰 = κ + , Fund. Math. 201 (2008), no. 3, 283–293. MR2457482
  12. Goldstern M., Kellner J., Mejía D. A., Shelah S., 10.1007/s11856-021-2237-7, Israel J. Math. 246 (2021), no. 1, 73–129. MR4358274DOI10.1007/s11856-021-2237-7
  13. Ihoda J. I., Shelah S., 10.2307/2274613, J. Symbolic Logic 53 (1988), no. 4, 1188–1207. MR0973109DOI10.2307/2274613
  14. Jech T., Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003. Zbl1007.03002MR1940513
  15. Kunen K., Vaughan J. E., Handbook of Set-theoretic Topology, North-Holland Publishing Co., Amsterdam, 1984. Zbl0674.54001MR0776619
  16. Mejía D. A., 10.1007/s00153-012-0315-6, Arch. Math. Logic 52 (2013), no. 3–4, 261–278. MR3047455DOI10.1007/s00153-012-0315-6
  17. Shelah S., 10.1090/conm/031/763901, Conf. Axiomatic Set Theory, Boulder, 1983, Contemp. Math., 31, Amer. Math. Soc., Providence, 1984, pages 183–207. Zbl0583.03035MR0763901DOI10.1090/conm/031/763901

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.