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Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American
List of Fellows of the American Mathematical Society
retrieved 2014-12-17 Martin was the 1992 Tarski lecturer.
List of publicationsUCLA Logic centerPersonal Website at UCLA
1940 births Living people 20th-century American mathematicians 21st-century American mathematicians American logicians UCLA Department of Philosophy faculty American philosophers of mathematics Set theorists Fellows of the American Mathematical Society {{US-mathematician-stub
Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly ...
and philosopher of mathematics at UCLA
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California, United States. Its academic roots were established in 1881 as a normal school then known as the southern branch of the C ...
, where he is an emeritus professor of mathematics and philosophy
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
.
Education and career
Martin received his B.S. from theMassachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
in 1962 and was a Junior Fellow of the Harvard Society of Fellows in 1965–67. In 2014, he became a Fellow of the American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.retrieved 2014-12-17 Martin was the 1992 Tarski lecturer.
Philosophical and mathematical work
Among Martin's most notable works are the proofs ofanalytic
Analytic or analytical may refer to:
Chemistry
* Analytical chemistry, the analysis of material samples to learn their chemical composition and structure
* Analytical technique, a method that is used to determine the concentration of a chemical ...
determinacy
Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "dete ...
(from the existence of a measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', ...
), Borel determinacy
In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. A Gale–Stewart game is a p ...
(from ZFC alone), the proof (with John R. Steel
John Robert Steel (born October 30, 1948) is an American set theorist at University of California, Berkeley (formerly at UCLA). He has made many contributions to the theory of inner models and determinacy. With Donald A. Martin, he proved proje ...
) of projective determinacy In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
The axiom of projective determinacy, abbreviated PD, states that for any two-player infinite game of perfect information ...
(from suitable large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
axioms), and his work on Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consi ...
. The Martin measure In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.
Definition
Let D be the set of T ...
on Turing degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.
Overview
The concept of Turing degree is fund ...
s and the Martin's Conjecture on Turing invariant functions are also named after Martin.
Martin's conjecture
In mathematics, more precisely inrecursion theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
, Martin's conjecture states, in essence, that the only nontrivial definable Turing invariant functions are the Turing jump and its iterates through the transfinite. Martin made this conjecture in the late 1970s; it first appeared in print as item 5 in the list titled “The Victoria Delfino problems” which was published as an appendix to a volume of proceedings of the joint Caltech-UCLA Logic Seminar.
See also
*American philosophy
American philosophy is the activity, corpus, and tradition of philosophers affiliated with the United States. The ''Internet Encyclopedia of Philosophy'' notes that while it lacks a "core of defining features, American Philosophy can neverthe ...
* List of American philosophers
American philosophy is the activity, corpus, and tradition of philosophers affiliated with the United States. The ''Internet Encyclopedia of Philosophy'' notes that while it lacks a "core of defining features, American Philosophy can neverthe ...
References
External links
*List of publications
1940 births Living people 20th-century American mathematicians 21st-century American mathematicians American logicians UCLA Department of Philosophy faculty American philosophers of mathematics Set theorists Fellows of the American Mathematical Society {{US-mathematician-stub