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Fotografii | Monede | Timbre | Schite | Cautare |
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Zorn emigrated to the United States and was appointed a Sterling Fellow at Yale University. He worked there from 1934 to 1936 and it was during this period that he proposed "Zorn's Lemma" for which he is best known. We describe below the form in which Zorn originally stated this result. Following his years at Yale, he moved to the University of California at Los Angeles where he remained until 1946. During this time Herstein was one of his doctoral students. He left the University of California to become professor at Indiana University, holding this position from 1946 until he retired in 1971. Since Zorn is best known for "Zorn's Lemma" it is perhaps appropriate that we should begin a discussion of his mathematical achievements by considering this contribution. Of course Zorn did not call his result "Zorn's Lemma", rather it was given by him as a "maximum principle" in a short paper entitled A remark on method in transfinite algebra which he published in the Bulletin of the American Mathematical Society in 1935. Perhaps in passing we should note that the name "Zorn's Lemma" was due to John Tukey . Zorn's aim in this paper was to study field theory and in particular to improve on the method used for obtaining results in the subject. Methods used up to that time had depended heavily on the well-ordering principle which Zermelo had proposed in 1904, namely that every set can be well-ordered. What Zorn proposed in the 1935 paper was to develop field theory from the standard axioms of set theory, together with his maximum principle rather than Zermelo 's well-ordering principle. The form in which Zorn stated his maximum principle was as follows. The principle involved chains of sets. A chain is a collection of sets with the property that for any two sets in the chain, one of the two sets is a subset of the other. Zorn defined a collection of sets to be closed if the union of every chain is in the collection. His maximum principle asserted that if a collection of sets is closed, then it must contain a maximal member, that is a set which is not a proper subset of some other in the collection. The paper then indicated how the maximum principle could be used to prove the standard field theory results. Today we know that the Axiom of Choice, the well-ordering principle, and Zorn's Lemma (the name now given to Zorn's maximum principle by Tukey and now the standard name) are equivalent. Did Zorn know this when he wrote his 1935 paper? Well at the end of the 1935 paper he did say that these three are all equivalent and promised a proof in a future paper. Was Zorn's idea entirely new? Well similar maximum principles had been proposed earlier in different contexts by several mathematicians, for example Hausdorff , Kuratowski and Brouwer . Paul Campbell examines this question in and:
Zorn made other contributions to set theory, such as his 1944 paper Idempotency of infinite cardinals in which he proved that an infinite cardinal number is equal to its square. His proof uses his maximum principle rather than using ordinal numbers as had been done in previous proofs of the result. In addition to his well known work in infinite set theory, Zorn worked on topology and algebra. As we mentioned above his doctoral thesis was on alternative algebras. These are algebras in which (xy)z - x(yz) is an alternating function in the sense that it is zero whenever any two of x, y, z are equal or, put another way, any two dimensional subalgebra is associative. Zorn went on to publish four papers on alternative algebras. He proved the uniqueness of the Cayley numbers (or octonians) in 1933 by showing that it was the only alternative, quadratic, real nonassociative algebra without zero divisors. He studied the structure of semisimple alternative rings in 1932, proving that such a ring is a direct sum of simple alternative algebras which he classified. In Alternative rings and related questions I: existence of the radical published in 1941 Zorn considered the theory of the radical of an alternative ring. He also published results on algebras which were fundamental in the study of algebraic number fields. We have looked briefly at Zorn's contributions to algebra and to set theory. Let us now take a brief look at his contributions to analysis. In 1945 he published the paper Characterization of analytic functions in Banach space in the Annals of Mathematics. We quote from the introduction to that paper since it both states the type of problems that Zorn was examining very clearly, and also because it illustrates his clear style of writing mathematics:
After 1947 Zorn stopped publishing mathematical papers. This does not mean that he gave up mathematics. As Haile said at the Memorial Symposium held for Zorn at Indiana University in June 1993 (see ):
Ewing writes :
Halmos also describes the colloquia:
Returning to Ewing :
Perhaps the reference to the Piccayune Sentinel deserves comment. Zorn did spell this with two c's but it is named after the newspaper the New Orleans Picayune. Halmos in gives more details:
Max Zorn married Alice Schlottau and they had one son Jens and one daughter Liz.
Source:School of Mathematics and Statistics University of St Andrews, Scotland |