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This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.
Allendoerfer is also known as a proponent of the New Math movement in the 1950s and 1960s, which sought to improve American primary and secondary mathematics education by teaching abstract concepts like set theory early in the curriculum.
He was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time he discovered what came to be called the Burali-Forti paradox of Cantorian set theory.
Based on these properties, the class of combinatorial games may be thought of as having the structure of an Abelian group, although with a proper class of elements rather than (as is more standard for groups) a set of elements.
In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the (ε, δ)-definition of limit and set theory.
Reinhardt, William, "Ackermann's set theory equals ZF" Annals of Mathematical Logic Vol.
During this time he wrote, with G. T. Kneebone, a graduate textbook on set theory, The Theory of Sets and Transfinite Numbers, as well as numerous papers on ordered structures and Boolean algebras, and in 1977 published Jean Piaget: Psychologist of the Real an exposition and critique of the ideas behind the work of the Swiss child psychologist.
In descriptive set theory and general topology, Luzin space or Luzin set, a hypothetical uncountable topological T1 space without isolated points in which every nowhere-dense subset is at most countable
Ω-logic, a deductive system in set theory developed by Hugh Woodin
Gaisi Takeuti, W. M. Zaring, Introduction to Axiomatic Set Theory, Springer GTM 1, 1971, ISBN 978-0-387-90024-7, p.