Euclid flourished about fifty years after Aristotle and was certainly familiar with Aristotle’s Logic. Euclid’s organization of the work of earlier geometers was truly innovative. His results depended upon basic assumptions, called axioms and “common notions”. There are in total 23 definitions, five axioms and five common notions in *The Elements*. The axioms, or postulates, are specific assumptions that may be considered as self-evident, for example “the whole is greater than the part” [TM232 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘The Whole is Greater than the Part — Or is it?’

## Posts Tagged 'Cantor'

### The Whole is Greater than the Part — Or is it?

Published April 21, 2022 Irish Times ClosedTags: Cantor, Geometry, Set Theory

### Cantor’s Theorem and the Unending Hierarchy of Infinities

Published November 11, 2021 Occasional ClosedTags: Cantor, Set Theory

In 1891, Georg Cantor published a seminal paper, *U”ber eine elementare Frage der Mannigfaltigkeitslehren* — On an elementary question of the theory of manifolds — in which his “diagonal argument” first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater than that of the natural numbers. But his theorem is much more general, and it implies that the set of cardinals is without limit: *there is no greatest order of infinity.*

Continue reading ‘Cantor’s Theorem and the Unending Hierarchy of Infinities’