We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.
LC On Sunday, November 3, 2019 at 6:39:53 AM UTC-6, Philip Thrift wrote: > > > *Leibniz's Infinitesimals: Their Fictionality, Their Modern > Implementations, And Their Foes From Berkeley To Russell And Beyond* > https://arxiv.org/abs/1205.0174 > > *Infinitesimals, Imaginaries, Ideals, and Fictions* > https://arxiv.org/abs/1304.2137 > > *Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania* > https://arxiv.org/abs/1603.07209 > > Leibniz frequently writes that his infinitesimals are useful fictions, and > we agree; but we shall show that it is best not to understand them as > logical fictions; instead, they are better understood as pure fictions. > > @philipthrift > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/bf376129-a933-4d79-9134-8568795df2a4%40googlegroups.com.
