# Matthew Foreman

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For other people named Matt Foreman, see [Matt Foreman (disambiguation)](/source/Matt_Foreman_(disambiguation)).

**Matthew Dean Foreman** is an American mathematician at [University of California, Irvine](/source/University_of_California,_Irvine). He has made notable contributions in [set theory](/source/Set_theory) and in [ergodic theory](/source/Ergodic_theory).

## Biography

Born in [Los Alamos, New Mexico](/source/Los_Alamos,_New_Mexico), Foreman earned his [Ph.D.](/source/Ph.D.) from the University of California, Berkeley in 1980 under [Robert M. Solovay](/source/Robert_M._Solovay). His dissertation title was *Large Cardinals and Strong Model Theoretic Transfer Properties*.[1]

In addition to his mathematical work, Foreman is an avid sailor.

He and his family sailed their sailboat *Veritas* (a [built by C&C Yachts](/source/C%26C_Yachts)) from North America to Europe in 2000. From 2000–2008 they sailed Veritas to the Arctic, the [Shetland Islands](/source/Shetland_Islands), [Scotland](/source/Scotland), [Ireland](/source/Ireland), [England](/source/England), [France](/source/France), [Spain](/source/Spain), [North Africa](/source/North_Africa) and [Italy](/source/Italy).

Notable high points were [Fastnet Rock](/source/Fastnet_Rock), Irish and Celtic seas and many passages including the [Maelstrom](/source/Whirlpool), [Stad](/source/Stad_(peninsula)), [Pentland Firth](/source/Pentland_Firth), [Loch Ness](/source/Loch_Ness), the Corryveckan and the Irish Sea. Further south they sailed through the [Chenal du Four](/source/Chenal_du_Four) and [Raz de Sein](/source/Raz_de_Sein), across the [Bay of Biscay](/source/Bay_of_Biscay) and around [Cape Finisterre](/source/Cape_Finisterre). After entering [Gibraltar](/source/Gibraltar), Foreman and his family circumnavigated the Western Mediterranean. Some notable stops included: [Barcelona](/source/Barcelona), [Morocco](/source/Morocco), [Tunisia](/source/Tunisia), [Sicily](/source/Sicily), [Naples](/source/Naples), [Sardinia](/source/Sardinia) and [Corsica](/source/Corsica). In 2009 Foreman, his son with guest members as crew, circumnavigated Newfoundland.[2]

Foreman has been recognized for his sailing by twice winning the Ullman Trophy.[3]

## Work

Foreman began his career in set theory. His early work with [Hugh Woodin](/source/W._Hugh_Woodin) included showing that it is consistent that the generalized continuum hypothesis (see [continuum hypothesis](/source/Continuum_hypothesis)) fails at every infinite cardinal.[4] In joint work with [Menachem Magidor](/source/Menachem_Magidor) and [Saharon Shelah](/source/Saharon_Shelah) he formulated [Martin's maximum](/source/Martin's_maximum), a provably maximal form of [Martin's axiom](/source/Martin's_axiom) and showed its consistency.[5][6] Foreman's later work in set theory was primarily concerned with developing the consequences of generic large cardinal axioms.[7] He also worked on classical "Hungarian" [partition relations](/source/Infinitary_combinatorics), mostly with [András Hajnal](/source/Andr%C3%A1s_Hajnal).[8]

In the late 1980s Foreman became interested in [measure](/source/Measure_(mathematics)) theory and [ergodic theory](/source/Ergodic_theory). With [Randall Dougherty](/source/Randall_Dougherty) he settled the [Marczewski](/source/Edward_Marczewski) problem (1930) by showing that there is a Banach–Tarski decomposition of the unit ball in which all pieces have the [property of Baire](/source/Property_of_Baire) (see [Banach–Tarski paradox](/source/Banach%E2%80%93Tarski_paradox)).[9] A consequence is the existence of a decomposition of an open dense subset of the unit ball into disjoint open sets that can be rearranged by isometries to form two open dense subsets of the unit ball. With Friedrich Wehrung, Foreman showed that the [Hahn–Banach theorem](/source/Hahn%E2%80%93Banach_theorem) implied the existence of a non-Lebesgue measurable set, even in the absence of any other form of the [axiom of choice](/source/Axiom_of_choice).[10]

This naturally led to attempts to apply the tools of [descriptive set theory](/source/Descriptive_set_theory) to classification problems in [ergodic theory](/source/Ergodic_theory). His first work in this direction, with Ferenc Beleznay,[11] showed that classical collections were beyond the [Borel hierarchy](/source/Borel_hierarchy) in complexity. This was followed shortly by a proof of the analogous results for measure-preserving transformations with generalized discrete spectrum. In a collaboration with [Benjamin Weiss](/source/Benjamin_Weiss)[12] and [Daniel Rudolph](/source/Daniel_Rudolph)[13] Foreman showed that no residual class of measure-preserving transformations can have algebraic invariants and that the isomorphism relation on ergodic measure-preserving transformations is not Borel. This negative result finished a program proposed by von Neumann in 1932.[14] This result was extended by Foreman and Weiss to show that smooth area-preserving diffeomorphisms of the 2-torus are unclassifiable.

Foreman's work in set theory continued during this period. He co-edited (with [Kanamori](/source/Akihiro_Kanamori)) the *Handbook of Set Theory* and showed that various combinatorial properties of ω2 and ω3 are equiconsistent with [huge cardinals](/source/Huge_cardinal).[15]

## Recognition

In 1998 Foreman was an Invited Speaker of the [International Congress of Mathematicians](/source/International_Congress_of_Mathematicians) in Berlin.[16]

In 2021, he gave the [Gödel Lecture](/source/G%C3%B6del_Lecture) titled *Gödel Diffeomorphisms.*

He was named to the 2023 class of Fellows of the [American Mathematical Society](/source/American_Mathematical_Society), "for contributions to axioms of mathematics, Banach-Tarski phenomena, and descriptive dynamical systems".[17]

## References

1. Foreman, Matthew (1982). "Large Cardinals and Strong Model Theoretic Transfer Properties". *Transactions of the American Mathematical Society*. **272** (2): 427–463. [doi:10.1090/S0002-9947-1982-0662045-X](https://doi.org/10.1090/S0002-9947-1982-0662045-X). [JSTOR 1998706](https://www.jstor.org/stable/1998706)

1. Foreman, Zachary (2007) "Under Way", Cruising World Magazine, October 2007

1. Tailwind, [Balboa Yacht Club](/source/Balboa_Yacht_Club) "Annual Awards", 2003, 2011

1. Foreman, Matthew & Woodin, W. Hugh (1991). "The generalized continuum hypothesis can fail everywhere". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **133** (1): 1–35. Second Series. [doi:10.2307/2944324](https://doi.org/10.2307/2944324). [JSTOR 2944324](https://www.jstor.org/stable/2944324)

1. Foreman, Matthew; Magidor, Menachem; Shelah, Saharon (1988). "Martin's maximum, saturated ideals, and nonregular ultrafilters. I". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **127** (1): 1–47. Second Series. [doi:10.2307/1971415](https://doi.org/10.2307/1971415). [JSTOR 1971415](https://www.jstor.org/stable/1971415)

1. Foreman, Matthew; Magidor, Menachem; Shelah, Saharon (1988). "Martin's maximum, saturated ideals and nonregular ultrafilters. II". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **127** (3): 521–545. Second Series. [doi:10.2307/2007004](https://doi.org/10.2307/2007004). [JSTOR 2007004](https://www.jstor.org/stable/2007004)

1. Foreman, Matthew (2010). "Ideals and generic elementary embeddings". *Handbook of Set Theory*. Springer. pp. 885–1147. [doi:10.1007/978-1-4020-5764-9_14](https://doi.org/10.1007/978-1-4020-5764-9_14)

1. Foreman, Matthew & Hajnal, András (2003). "A partition relation for successors of large cardinals". *[Mathematische Annalen](/source/Mathematische_Annalen)*. **325** (3): 583–623. [doi:10.1007/s00208-002-0323-7](https://doi.org/10.1007/s00208-002-0323-7)

1. Dougherty, Randall & Foreman, Matthew (1994). "Banach–Tarski decompositions using sets with the property of Baire". *Journal of the American Mathematical Society*. **7** (1): 75–124. [doi:10.1090/S0894-0347-1994-1227475-8](https://doi.org/10.1090/S0894-0347-1994-1227475-8)

1. Foreman, Matthew & Wehrung, Friedrich (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set". *Fundamenta Mathematicae*. **138** (1): 13–19. [doi:10.4064/fm-138-1-13-19](https://doi.org/10.4064/fm-138-1-13-19)

1. Beleznay, Ferenc & Foreman, Matthew (1995). "The collection of distal flows is not Borel". *American Journal of Mathematics*. **117** (1): 203–239. [doi:10.2307/2375041](https://doi.org/10.2307/2375041). [JSTOR 2375041](https://www.jstor.org/stable/2375041)

1. Foreman, Matthew & Weiss, Benjamin (2004). "An anti-classification theorem for ergodic measure-preserving transformations". *[Journal of the European Mathematical Society](/source/Journal_of_the_European_Mathematical_Society)*. **6** (3): 277–292. [doi:10.4171/JEMS/10](https://doi.org/10.4171/JEMS/10)

1. Foreman, Matthew; Rudolph, Daniel; Weiss, Benjamin (May 1, 2011). "The conjugacy problem in ergodic theory". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **173** (3): 1529–1586. Second Series. [doi:10.4007/annals.2011.173.3.7](https://doi.org/10.4007/annals.2011.173.3.7). [ISSN 0003-486X](https://www.worldcat.org/issn/0003-486X)

1. von Neumann, J. (1932). "Zur Operatorenmethode in der klassischen Mechanik". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **33** (3): 587–642. Second Series. [doi:10.2307/1968537](https://doi.org/10.2307/1968537). [JSTOR 1968537](https://www.jstor.org/stable/1968537)

1. Foreman, Matthew (2009). "Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **222** (2): 565–595. [doi:10.1016/j.aim.2009.05.006](https://doi.org/10.1016/j.aim.2009.05.006)

1. Foreman, Matthew (1998). ["Generic large cardinals: New axioms for mathematics?"](https://www.elibm.org/ft/10011637000). *Documenta Mathematica (Bielefeld), Extra Volume ICM Berlin*. Vol. II. pp. 11–21.

1. ["2023 Class of Fellows"](https://web.archive.org/web/20221109214706/https://www.ams.org/cgi-bin/fellows/fellows_by_year.cgi). American Mathematical Society. Archived from [the original](http://www.ams.org/cgi-bin/fellows/fellows_by_year.cgi) on 2022-11-09. Retrieved 2022-11-09.

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