Event № 908
Nonstandard analysis was first invented by Abraham Robinson in the early 1960s. It allows to prove theorems of “standard mathematics” taking use of infinite and infinitesimal numbers and many other “nonstandard” mathematical objects. In the mid-1970s Edward Nelson developed an axiomatic approach to nonstandard analysis, with the aim of making nonstandard methods available to the working mathematician. Nelson’s axiomatics is called Internal Set Theory (IST); it is an extension of the “usual” axiomatic Zermelo-Fraenkel set theory, ZFC. As Nelson wrote, “All theorems of conventional mathematics remain valid. No change in terminology is required. What is new in internal set theory is only an addition, not a change.”
In the talk I will describe IST axiomatics and show examples of reasoning in it. I will also discuss the following question. Let H be a proper connected subgroup of the additive group R of real numbers. Is it possible to choose one element in every conjugacy class of R by H? If H consists of all infinitesimals, then the answer is positive. Surprisingly, in general case the answer is negative.