## Journal of Symbolic Logic

### Saccharinity

#### Abstract

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. As an application, we introduce a new method to force (weak) measurability of all definable sets with respect to a certain (non-ccc) ideal.

#### Article information

Source
J. Symbolic Logic, Volume 76, Issue 4 (2011), 1153-1183.

Dates
First available in Project Euclid: 11 October 2011

https://projecteuclid.org/euclid.jsl/1318338844

Digital Object Identifier
doi:10.2178/jsl/1318338844

Mathematical Reviews number (MathSciNet)
MR2895391

Zentralblatt MATH identifier
1247.03109

#### Citation

Kellner, Jakob; Shelah, Saharon. Saccharinity. J. Symbolic Logic 76 (2011), no. 4, 1153--1183. doi:10.2178/jsl/1318338844. https://projecteuclid.org/euclid.jsl/1318338844

#### References

• Jörg Brendle, Mad families and iteration theory, Logic and algebra, Contemporary Mathematics, vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 1–31.
• Jörg Brendle, Lorenz Halbeisen, and Benedikt Löwe, Silver measurability and its relation to other regularity properties, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 138 (2005), no. 1, pp. 135–149.
• Martin Goldstern, A taste of proper forcing, Set theory (Curaçao, 1995; Barcelona, 1996), Kluwer Academic Publishers, Dordrecht, 1998, pp. 71–82.
• M. Groszek and T. Jech, Generalized iteration of forcing, Transactions of the American Mathematical Society, vol. 324 (1991), no. 1, pp. 1–26.
• Marcia Groszek, $\omega^*_1$ as an initial segment of the c-degrees, Journal of Symbolic Logic, vol. 59 (1994), no. 3, pp. 956–976.
• Thomas Jech and Saharon Shelah, A complete Boolean algebra that has no proper atomless complete subalgebra, Journal of Algebra, vol. 182 (1996), no. 3, pp. 748–755.
• Vladimir Kanovei, On non-wellfounded iterations of the perfect set forcing, Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 551–574.
• J. Kellner, Non elementary proper forcing, preprint, family http://arxiv.org/abs /0910.2132.
• Andrzej Rosłanowski and Saharon Shelah, Norms on possibilities. I. Forcing with trees and creatures, Memoirs of the American Mathematical Society, vol. 141 (1999), no. 671, pp. xii+167.
• ––––, Sweet & sour and other flavours of ccc forcing notions, Archive for Mathematical Logic, vol. 43(2004), no. 5, pp. 583–663.
• Saharon Shelah, Can you take Solovay's inaccessible away?, Israel Journal of Mathematics, vol. 48 (1984), no. 1, pp. 1–47.
• ––––, Properness without elementaricity, Journal of Applied Analysis, vol. 10(2004), no. 2, pp. 169–289.
• ––––, Two cardinal invariants of the continuum $(\mathfrakd<\mathfraka)$ and FS linearly ordered iterated forcing, Acta Mathematica, vol. 192(2004), no. 2, pp. 187–223.
• Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics. Second Series, vol. 92 (1970), pp. 1–56.
• Jindřich Zapletal, Forcing idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008.