Matthew Dean Foreman is a set theorist

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 at University of California, Irvine

University of California, Irvine

The University of California, Irvine , founded in 1965, is one of the ten campuses of the University of California, located in Irvine, California, USA...

. He has made contributions in widely varying areas of set theory, including descriptive set theory

Descriptive set theory

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...

, forcing

Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...

, and infinitary combinatorics.

Foreman earned his Ph.D.

Ph.D.

A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...

 in 1980 at University of California, Berkeley

University of California, Berkeley

The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...

 under the direction of Robert M. Solovay

Robert M. Solovay

Robert Martin Solovay is an American mathematician specializing in set theory.Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann–Roch theorem...

, 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
  W. Hugh Woodin
  William Hugh Woodin is an American mathematician and set theorist at University of California, Berkeley. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinal, the Woodin cardinal, bears his name.-Biography:Born in Tucson, Arizona, Woodin...
  
  ).
• A Banach-Tarski decomposition
  Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
  
   is possible in which all pieces have the Baire property (with Randall Dougherty
  Randall Dougherty
  Randall Dougherty has made contributions in widely varying areas of mathematics, including set theory,logic, real analysis, discrete mathematics, computational geometry, information theory and coding theory....
  
  ). This problem of Marczewski
  Edward Marczewski
  Edward Marczewski was a Polish mathematician. His surname until 1940 was Szpilrajn.Marczewski was a member of the Warsaw School of Mathematics...
  
   had been open for more than 60 years.
• With Menachem Magidor
  Menachem Magidor
  Menachem Magidor is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as President of the Hebrew University of Jerusalem.- Biography :Menachem Magidor was born in Petah Tikva on January 24, 1946....
  
   and Saharon Shelah
  Saharon Shelah
  Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
  
   he formulated and proved the consistency of Martin's maximum
  Martin's maximum
  In set theory, Martin's maximum, introduced by , is a generalization of the proper forcing axiom, which is in turn a generalization of Martin's axiom....
  
  , a provably maximal form of Martin's axiom
  Martin's axiom
  In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
  
  .
• It is consistent that there exists a -complete, -dense ideal on .
• The collection of distal flows is not Borel
  Borel algebra
  In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
  
  . (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 -dense ideal on uses a huge cardinal.)

With Akihiro Kanamori

Akihiro Kanamori

is a Japan-born American mathematician. He specializes in set theory and is the author of the successful monograph on large cardinals, The Higher Infinite. He wrote several essays on the history of mathematics, especially set theory.Kanamori graduated from California Institute of Technology and...

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

He is also an avid sailor.

Selected publications

link

External links