Information for examiners
The goal of this part is to help instructors or examiners understand a proof which uses ultrasmall
numbers. This can happen when, in our school, final exams have an external juror in addition to
the teacher. It can also be used to see that for the trained mathematician, the translation back
to "classical" methods is quite straightforward.
This part does not explain how to write the proofs, only how to read them.
As in any book about analysis, we do not give the axioms of set theory, but state instead,
properties of real numbers.
Observability
Extra Axioms – called also Principles – are used which allow to make an extra distinction within
the real numbers: observability.
The intuition is that “ordinary numbers” are observable but that there are extremely small
numbers (ultrasmall) which are so tiny that they are not observable. But if one zooms in to
observe these tiny numbers, one can still see zero. And if such a tiny number his aded to 2,
then 2 + his less observable than 2, which remains observable when 2 is observable.
Given xand y, then xis as observable as y, or yis as observable as x. They may have the
same observability. Observability is transitive.
One can consider the metaphor of scales of observation.
Numbers defined without the concept of observability are observable relative to any real
number: they are always observable – or standard.
Ultrasmall
Relative to any real number, there exist ultrasmall numbers, numbers which are less, in absolute
value, than any observable number, yet not zero.
Relative to a, if his ultrasmall, then 1
his ultralarge. But then we also have that aand
a+hare extremely close, written a'a+h, where the only new symbol is "'" which reads
"ultraclose": a difference which is ultrasmall or zero. Note that a+his not as observable as a.
An ultrasmall number has the "flavour" of an infinitesimal.
Context
Given a formula which has some parameters (the context), the concept of ultrasmall, ultralarge
or ultraclose always refer to the whole list of parameters. If his ultrasmall, it must be ultrasmall
relative to each parameter. Thus "'" automatically refers to all of the parameters.
Closure
Non observable numbers do not show up as results of operations if they are not introduced
explicitly. Given fand a, then f(a)is observable (the context being aand the parameters of f).
Observable neighbour
Any number xwhich is not ultralarge can be written in the form x=a+hwhere ais observable
and his ultraclose to zero. Then ais the observable neighbour of x.
The existence of the observable neighbour is equivalent to the completeness of R.
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