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\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}