Reuben Louis Goodstein was an English

England

England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 with a strong interest in the philosophy

Philosophy

Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

 and teaching of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

.

As a boy, he attended St Paul's School in London. He received his Master's degree from the University of Cambridge

University of Cambridge

The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

. After this, he worked at the University of Reading

University of Reading

The University of Reading is a university in the English town of Reading, Berkshire. The University was established in 1892 as University College, Reading and received its Royal Charter in 1926. It is based on several campuses in, and around, the town of Reading.The University has a long tradition...

 but ultimately spent most of his academic career in the University of Leicester

University of Leicester

The University of Leicester is a research-led university based in Leicester, England. The main campus is a mile south of the city centre, adjacent to Victoria Park and Wyggeston and Queen Elizabeth I College....

. He earned his PhD

Doctor of Philosophy

Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...

 from the University of London

University of London

-20th century:Shortly after 6 Burlington Gardens was vacated, the University went through a period of rapid expansion. Bedford College, Royal Holloway and the London School of Economics all joined in 1900, Regent's Park College, which had affiliated in 1841 became an official divinity school of the...

 in 1946 while still working in Reading. Goodstein also studied under Wittgenstein and John Littlewood

John Edensor Littlewood

John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...

.

He published many works on finitism

Finitism

In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...

 and the reconstruction of analysis from a finitistic viewpoint, for example "Constructive Formalism. Essays on the foundations of mathematics." Goodstein's theorem

Goodstein's theorem

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, made by Reuben Goodstein, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic...

 was among the earliest examples of theorems found to be unprovable in Peano arithmetic but provable in stronger logical systems

Formal system

In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

 (such as second order arithmetic). He also introduced a variant of the Ackermann function

Ackermann function

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive...

 that is now known as the hyperoperation sequence

Hyperoperation

In mathematics, the hyperoperation sequenceis an infinite sequence of arithmetic operations that starts with the unary operation of successor, then continues with the binary operations of addition, multiplication and exponentiation, after which the sequence proceeds with further binary operations...

, together with the naming convention now used for these operations (tetration

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

, pentation, etc.).

External links