Cheap Non-standard Analysis and Computability
This work provides a novel perspective on computability for mathematicians and computer scientists, but it is incremental as it builds on existing cheap non-standard analysis and computability theories.
The paper tackles the problem of presenting computability concepts within cheap non-standard analysis, proving that many ideas from computable analysis and computability can be elegantly reformulated in this framework, offering dual views and proofs to existing statements.
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. This provides constructivity but at some (acceptable) price. We consider computability in cheap non-standard analysis. We prove that many concepts from computable analysis as well as several concepts from computability can be very elegantly and alternatively presented in this framework. It provides a dual view and dual proofs to several statements already known in these fields.