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% (find-angg "LATEX/2008filterp-abs.tex") % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && latex 2008filterp-abs.tex")) % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && pdflatex 2008filterp-abs.tex")) % (eev "cd ~/LATEX/ && Scp 2008filterp-abs.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/") % (find-dvipage "~/LATEX/2008filterp-abs.dvi") % (find-pspage "~/LATEX/2008filterp-abs.pdf") % (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2008filterp-abs.ps 2008filterp-abs.dvi") % (find-pspage "~/LATEX/2008filterp-abs.ps") % (ee-cp "~/LATEX/2008filterp-abs.pdf" (ee-twupfile "LATEX/2008filterp-abs.pdf") 'over) % (ee-cp "~/LATEX/2008filterp-abs.pdf" (ee-twusfile "LATEX/2008filterp-abs.pdf") 'over) \documentclass{book} \usepackage{amsfonts} \begin{document} Natural Infinitesimals in Filter-Powers Eduardo Ochs {\tt http://angg.twu.net/} \bigskip \def\I{{\mathbb{I}}} \def\N{{\mathbb{N}}} \def\R{{\mathbb{R}}} \def\F{{\mathcal{F}}} \def\U{{\mathcal{U}}} \def\Ro{{\mathcal{R}_0}} \def\calN{{\mathcal{N}}} \def\Set{{\mathbf{Set}}} \def\w{{\omega}} \def\o{{\mathbf{o}}} \def\SetN{{\Set^\N}} \def\SetI{{\Set^\I}} \def\SetNN{{\Set^\N/\calN}} \def\SetIF{{\Set^\I/\F}} \def\SetIU{{\Set^\I/\U}} \def\SetRRo{{\Set^\R/\Ro}} Start from the standard universe, $\Set$, and construct the the universe of sequences'', $\SetN$, and then the ``semi-standard universe'', $\SetNN$, in which the quotient by the filter of cofinite sets of naturals, ``$/\calN$'', identifies sequences which differ only on finite sets of indices. Now generalize this a bit: a {\sl filter-power}, $\SetIF$, is a universe of $\I$-indexed sequences modulo a quotient that identifies sequences when they coincide on sets of indices that are ``$\F$-big''. If we substitute the filter $\F$ above by a (non-principal) ultrafilter $\U$ we get a ``non-standard universe'' (or: an ``ultrapower''), $\SetIU$, whose logic is very close to the one of $\Set$ --- it has exactly two truth-values --- but in a $\SetIU$ we have infinitesimals (the equivalence classes of $\I$-sequences tending to 0), and we can use the ``transfer theorems'' of Non-Standard Analysis to move truths back and forth between $\Set$ and $\SetIU$. Non-principal ultrafilters cannot be constructed explicitly, and to show that they exist we need the boolean prime ideal theorem, that is slightly weaker than the axiom of choice; this makes the infinitesimals of NSA quite hard to understand intuitively. On the other hand, the infinitesimals in a semi-standard universe like $\Set^\N/\calN$ or $\SetRRo$, where $\Ro$ is the filter of neighborhoods of $0 \in \R$, are very simple to describe --- but the logic of a filter-power has more than two truth values. We will show how ``strictly calculational'' proofs in NSA involving infinitesimals can be lifted through the quotient $\SetIF \to \SetIU$; and then, by choosing the right $\I$ and $\F$, and by using the ``natural infinitesimals'' --- that are identity maps in disguise, modulo $\F$ --- we get a straightforward translation of these strictly calculational proofs with infinitesimals into standard proofs in terms of limits and continuity. One ``archetypical example'' will be discussed in detail: $\forall \w \sim \infty \; \exists! \o \sim 0 \; (1+\frac{1}{\w})^\w = e^a + \o$, where $\w$ is an infinitely big natural number. The presentation should be accessible to people with basic knowledge of Calculus, Analysis, and Topology. \end{document}