## Final Report Summary - CLASSFORCING (Class forcing, internal consistency and the outer model program.)

The European Reintegration Grant (ERG) FP7-MC-ERG 224747 (Researcher: Jakob Kellner) was awarded for a reintegration period of three years (2008/09-2011/08) at the Kurt Goedel Research Center for Mathematical Logic (KGRC) at the University of Vienna, Austria, after Kellner's Marie Curie Intra-European Fellowships (EIF) postdoc at the Hebrew University of Jerusalem, Israel (2006/09-2008/08).

(A) Mathematical results:

The following publications were created with support by the ERG grant:

[1] J. Kellner. Non elementary proper forcing. preprint, arXiv: math/0910.2132.

[2] M. Goldstern, J. Kellner, S. Shelah and W. Wohofsky: Borel Conjecture and Dual Borel Conjecture. preprint, arXiv: math/1105.0823.

[3] J. Kellner, S. Shelah: Creature forcing and large continuum: The joy of halving. to appear in Arch. Math. Logic, arXiv: 1003.3425.

[4] J. Kellner, S. Shelah: Decisive creatures and large continuum. J. Symbolic Logic 74 (2009), No. 1, 73--104, arXiv: math/0601083.

[5] J. Kellner, S. Shelah: Saccharinity. J. Symbolic Logic, 74 (2011), No. 4, 1153--1183. arXiv: math/0511330.

[6] J. Kellner, S. Shelah: More on the pressing down game. Arch. Math. Logic 50 (2011), No. 3, 477--501, arXiv: math/0905.3913.

[7] J. Kellner, S. Shelah: A Sacks Real out of Nowhere. J. Symbolic Logic 75 (2010), No. 1, 51--76, arXiv: math/0703302.

All of these papers are about set theory, and forcing theory plays a central part in all of them.

Due to space restrictions, I only briefly comment on the result of [2].

A set of reals is called 'strong measure zero' (smz), if for all natural functions f the set is covered by some intervals I_n of length 1/f (n).

Equivalently, a set is smz if it can be translated away from any meager set.

A set is strongly meager (sm) if it can be translated away from every Lebesgue null set.

The Borec Conjecture (BC) is the statement that all smz sets are countable, the dual Borel Conjecture (dBC) that all sm sets are countable.

It is known that CH implies that BC and dBC fail, Laver showed the consistency of BC and Carlson of dBC is consistent. It was open for quite some time whether BC+dBC are consistent, which was proved in [2]. The obvious attempt is to combine Laver's and Carlson's constructions. However, Laver uses a countable support and Carlson a finite support construction, and none of either constructions seem to work for BC+dBC. Our solution was to mix Laver's and Carlson's constructions 'generically': we start with a preparatory forcing R. A condition consists of a countable (but not transitive) model M and in M iteration (basically finite or countable support) of length omega2. The iteration of a weaker condition is canonically and M-completely embedded into the stronger one. R adds as generic element an iteration of length omega2 (in some way a generic mixture of finite and countable support).

(B) Additional achievements related to the goals listed in the grant proposal:

- Kellner acquired Habilitation in 2010.

- Kellner acquired new grants as principal investigator: two Austrian Science Funds (FWF) 'Einzelprojekt' grants (volumes 75K EUR and 300K EUR, respectively), and an FWF and Japan Society for the Promotion of Science (JSPS) joint seminar grant (for a seminar in Kobe, Japan, in January 2012, Austrian volume 10K EUR, Japanese P.I. Sakae Fuchino)

- Kellner finished supervising a Diploma student in 2011/08, and started supervising a PhD student 2011/10 (funded by one of the grants mentioned above).

- The grant allowed Kellner to continue his successful collaborator with his Marie Curie EIF host, S. Shelah, resulting in several publications.

- The grant funded several visits of world leading researchers to the KGRC (including J. Brendle, M. Gitik, G. Hjorth, M. Magidor).

- The grant partially funded a very successful set theory meeting, the Erwin Schrödinger Institute (ESI) workshop June 2009 (more than 130 participants).

(A) Mathematical results:

The following publications were created with support by the ERG grant:

[1] J. Kellner. Non elementary proper forcing. preprint, arXiv: math/0910.2132.

[2] M. Goldstern, J. Kellner, S. Shelah and W. Wohofsky: Borel Conjecture and Dual Borel Conjecture. preprint, arXiv: math/1105.0823.

[3] J. Kellner, S. Shelah: Creature forcing and large continuum: The joy of halving. to appear in Arch. Math. Logic, arXiv: 1003.3425.

[4] J. Kellner, S. Shelah: Decisive creatures and large continuum. J. Symbolic Logic 74 (2009), No. 1, 73--104, arXiv: math/0601083.

[5] J. Kellner, S. Shelah: Saccharinity. J. Symbolic Logic, 74 (2011), No. 4, 1153--1183. arXiv: math/0511330.

[6] J. Kellner, S. Shelah: More on the pressing down game. Arch. Math. Logic 50 (2011), No. 3, 477--501, arXiv: math/0905.3913.

[7] J. Kellner, S. Shelah: A Sacks Real out of Nowhere. J. Symbolic Logic 75 (2010), No. 1, 51--76, arXiv: math/0703302.

All of these papers are about set theory, and forcing theory plays a central part in all of them.

Due to space restrictions, I only briefly comment on the result of [2].

A set of reals is called 'strong measure zero' (smz), if for all natural functions f the set is covered by some intervals I_n of length 1/f (n).

Equivalently, a set is smz if it can be translated away from any meager set.

A set is strongly meager (sm) if it can be translated away from every Lebesgue null set.

The Borec Conjecture (BC) is the statement that all smz sets are countable, the dual Borel Conjecture (dBC) that all sm sets are countable.

It is known that CH implies that BC and dBC fail, Laver showed the consistency of BC and Carlson of dBC is consistent. It was open for quite some time whether BC+dBC are consistent, which was proved in [2]. The obvious attempt is to combine Laver's and Carlson's constructions. However, Laver uses a countable support and Carlson a finite support construction, and none of either constructions seem to work for BC+dBC. Our solution was to mix Laver's and Carlson's constructions 'generically': we start with a preparatory forcing R. A condition consists of a countable (but not transitive) model M and in M iteration (basically finite or countable support) of length omega2. The iteration of a weaker condition is canonically and M-completely embedded into the stronger one. R adds as generic element an iteration of length omega2 (in some way a generic mixture of finite and countable support).

(B) Additional achievements related to the goals listed in the grant proposal:

- Kellner acquired Habilitation in 2010.

- Kellner acquired new grants as principal investigator: two Austrian Science Funds (FWF) 'Einzelprojekt' grants (volumes 75K EUR and 300K EUR, respectively), and an FWF and Japan Society for the Promotion of Science (JSPS) joint seminar grant (for a seminar in Kobe, Japan, in January 2012, Austrian volume 10K EUR, Japanese P.I. Sakae Fuchino)

- Kellner finished supervising a Diploma student in 2011/08, and started supervising a PhD student 2011/10 (funded by one of the grants mentioned above).

- The grant allowed Kellner to continue his successful collaborator with his Marie Curie EIF host, S. Shelah, resulting in several publications.

- The grant funded several visits of world leading researchers to the KGRC (including J. Brendle, M. Gitik, G. Hjorth, M. Magidor).

- The grant partially funded a very successful set theory meeting, the Erwin Schrödinger Institute (ESI) workshop June 2009 (more than 130 participants).