Matthew Foreman

Matthew Dean Foreman (born March 21, 1957) is a set theorist at University of California, Irvine. He has made contributions in widely varying areas of set theory, including descriptive set theory, forcing, and infinitary combinatorics.

Foreman earned his Ph.D. in 1980 at University of California, Berkeley under the direction of Robert M. Solovay, with a dissertation on Large Cardinals and Model Theoretic Transfer Properties.

Among Foreman's theorems are the following:

• It is consistent that for every uncountable regular cardinal κ there is a κ-complete, κ⁺-saturated ideal on κ.
• It is consistent that the Generalized Continuum Hypothesis fails everywhere (with W. Hugh Woodin).
• A Banach-Tarski decomposition is possible in which all pieces have the Baire property (with Randall Dougherty). This problem of Marczewski had been open for more than 60 years.
• With Menachem Magidor and Saharon Shelah he formulated and proved the consistency of Martin's maximum, a provably maximal form of Martin's axiom.
• It is consistent that there exists a σ-complete, $\aleph_1$-dense ideal on $\aleph_2$.
• The collection of distal flows is not Borel. (With his Ph.D. student, Ferenc Beleznay.)

(All the consistency proofs start from the assumption that certain large cardinals are consistent with ZFC; e.g., the $\aleph_1$-dense ideal on $\aleph_2$ uses a huge cardinal.)

With Akihiro Kanamori he is the editor of the monumental Handbook of Set Theory (2010).

He is also an avid sailor.

Selected publications

• Foreman, Matthew (2006). "Has the continuum hypothesis been settled?". Logic Colloquium '03. Assoc. Symbol. Logic, La Jolla, CA. pp. 56–75.  link
• Foreman, Matthew and Menachem Magidor (1995). "Large cardinals and definable counterexamples to the continuum hypothesis". Annals of Pure and Applied Logic 76 (1): 47–97. doi:10.1016/0168-0072(94)00031-W. 
• Dougherty, Randall and Matthew Foreman (1994). "Banach-Tarski decompositions using sets with the property of Baire". Journal of the American Mathematical Society (American Mathematical Society) 7 (1): 75–124. doi:10.2307/2152721. JSTOR 2152721. 
• Foreman, Matthew and Friedrich Wehrung (1991). "The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set". Fundamenta Mathematicae 138 (1): 13–19. 
• Foreman, Matthew and W. Hugh Woodin (1991). "The generalized continuum hypothesis can fail everywhere". Annals of Mathematics (2) (Annals of Mathematics) 133 (1): 1–35. doi:10.2307/2944324. JSTOR 2944324. 
• Foreman, M. and Magidor, M. and Shelah, S. (1988). "Martin's maximum, saturated ideals, and nonregular ultrafilters. I.". Annals of Mathematics 127: 1–47. 

External links

• Matthew Foreman at the Mathematics Genealogy Project.