overzicht overview
Limit Dynamics
Apart from phrases like for every number $\epsilon > 0$ (i.e. $\forall
\epsilon > 0$), for every number $M > 0$ (i.e. $\forall M > 0$), there
is nothing in the classical definition of a limit that appeals to the notion
of a completed infinity. However, the completed infinity of the real numbers
is a prerequisite with the common definition of a limit. Therefore we have
another proposal. Suppose that the underlying substrate of real numbers can be
regarded instead as a type, like in modern programming languages, meaning
that you can create any real number at will, as soon as you need it, without
having all real numbers being present, as a completed infinity, in a finished
set. Then the limit concept as such is completely finitistic. And $(\forall
\epsilon \in \mathbb{R}+)$ could be regarded as a figure of speech.