DescriptionLeibniz and Newton invented infinitesimal calculus in the 1660s in course of their differential and integral calculus. Unfortunately, despite their intuitive appeal, these early attempts were proven to be non-rigorous, and were fiercely criticized by many authors. Only much later, in the early 18th century, the standard epsilon-delta calculus was invented which allowed for a way to avoid notions of infinitely small quantities, and the foundations of differential and integral calculus were made firm. Yet, the idea of infinitesimals has a strong intuitive appeal, and the search to formalize them went on. In the 1960s, Robinson created a theory of hyperreal numbers, based on ultrafilters, essentially leading to a rigorous formal treatment of infinitesimal calculus. Robinson's construction is rather technical, and requires advanced mathematical concepts. A simpler approach, which goes by the name internal set theory was invented by Nelson. Internal set theory simply adds a few simple axioms to the usual ZFC axioms of mathematics, and thereby recovers the key features of Robinson's theory, in a way that is easier to understand and to work with. The aim of this project is to study the basics of nonstandard analysis via internal set theory, how it allows one to avoid tricky epsilon-delta arguments and thereby simplify concepts such as derivatives and integrals, and how it leads to simpler proofs of various traditional theorems. PrerequisitesAlgebra II or Probability II An interest in logic and in theorem proving. Resources
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