Numbers, Infinities and Infinitesimals

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Numbers, Infinities and Infinitesimals N. J. Wildberger School of Mathematics UNSW Sydney 2052 Australia webpages: October 17, 2006 Problems with infinities and infinitesimals For the last hundred years or so, mathematicians have finally ‘understood the infinite'–or so they think. Despite several thousand years of previous un- certainty, and very stiff opposition from notable mathematicians (Kronecker, Poincaré, Weyl, Brouwer and numerous others), the theory of infinite point sets and hierarchies of cardinal and ordinal numbers introduced by Cantor is now the established orthodoxy, and dominates modern logic, topology and analysis.

• natural functions
• sequence of successors with limit
• modest recovery with the advent of the nonstandard analysis of robinson
• contemplation of infinity reverts from a philosophical pastime to a powerful tool for analysis
• sequences
• limit
• natural numbers
• infinity
• function
• sequence