Cheap Non-Standard Analysis and Computability: Some Applications

Non standard Analysis is an area of Mathematics dealing with notions of infinitesimal and infinitely large numbers, in which many statements from classical Analysis can be expressed very naturally. Cheap non-standard analysis introduced by Terence Tao in 2012 is based on the idea that considering that a property holds eventually is sufficient to give the essence of many of its statements. Cheap non-standard analysis provides constructivity but at some (acceptable) price. Computable Analysis is a very natural tool for discussing computations over the reals, and more general constructivity in Mathematics. In a recent article, we considered computability in cheap non-standard analysis. We proved that many concepts from computable analysis as well as several concepts from computability can be very elegantly and alternatively presented in this framework. We discuss in the current article several applications of this framework: We provide alternative proofs based on this approach of several statements from computable analysis. This includes intermediate value theorem, and computability of zeros, of maximum points and of a theorem from Rice.