Non-nonstandard analysis: Real infinitesimals

摘要: they seem imposs ib le to nail down. They're hard deal with intellectually. They can mask a fundamenta l lack of unders tanding analysis. There was hope, when Abraham Robinson deve loped nons t anda rd analysis [R], that intui ion and r igor had at last j o ined hands. His work indeed gave infini tesimals founda as member s the se f hyperrea i numbers . But it an awkward foundation, dependen on Axiom Choice. Unlike tandard number systems, there is no canonica way, however , construct ing inf ini natural ly. Ironically, eeds be found in any calculus b k sufficient age. At turn century, typical texts def ine infinitesimal "variable whose limit zero" [C]. That inspirat for p resen approach calculus. Its infmitesimals are sequences tending 0. I call system "non-nonstandard analysis" d raw attent its misfit nature. Having infinitesimals, not "standard." Nor "nonstandard," this term now has well-def 'med meaning. In wha follows, we manipulate real numbers. We rea them (mostly) add them, sub rac put into functions. aren ' numbers, Tr ichotomy fails, example The centra cons ruc ar ic rediscovery. first scovere robably D. Laugwitz. More later.