Reuben Goodstein
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Through Philosophy, Teaching and Mathematics
Education
Goodstein was educated at St Paul's School in London. He received his Master's degree from Magdalene College, Cambridge. After this, he worked at the University of Reading but ultimately spent most of his academic career at the University of Leicester. He earned his PhD from the University of London in 1946 while still working in Reading.
Goodstein also studied under Ludwig Wittgenstein.
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Research
He published many works on finitism and the reconstruction of analysis from a finitistic viewpoint, for example "Constructive Formalism. Essays on the foundations of mathematics." Goodstein's theorem was among the earliest examples of theorems found to be unprovable in Peano arithmetic but provable in stronger logical systems (such as second-order arithmetic). He also introduced a variant of the Ackermann function that is now known as the hyperoperation sequence, together with the naming convention now used for these operations (tetration, pentation, hexation, etc.).
Besides mathematical logic (in which he held the first professorial chair in the U.K.), mathematical analysis, and the philosophy of mathematics, Goodstein was keenly interested in the teaching of mathematics. From 1956 to 1962 he was editor of The Mathematical Gazette. In 1962 he was an invited speaker at the International Congress of Mathematicians (with an address on A recursive lattice) in Stockholm. Among his doctoral students are Martin Löb and Alan Bundy.