INFINITE AND INFINITESIMAL NUMBERS:
FROM NUMEROSITIES TO NONSTANDARD ANALYSIS

Mauro Di Nasso - University of Pisa, Italy

	We give an elementary introduction to nonstandard analysis, 
starting from the theory of numerosities. No logical background is needed.

Part I.
We discuss the notion of a "counting system" and briefly review the notions 
of cardinal and ordinal number. By means of three elementary axioms, we introduce 
a new counting system to count the size of infinite sets. 
The resulting "numerosity theory" extends the validity of the ancient 
Aristotle's principle: "The whole is larger than the part", from finite sets to 
a large class of infinite sets, namely the "labelled sets". 
The "numerosities" generalize the natural numbers, in that they satisfy 
the same "first-order" properties.

Part II.
Developing the theory, one obtains that every sequence f:N-->N of natural numbers 
can be naturally extended to take an "ideal value" f(a), where $a=num(N)$ 
is the numerosity of the set of natural numbers. We postulate five elementary 
properties that generalize the behavior of "ideal values" f(a) to ideal values 
of arbitrary N-sequences. The resulting theory is called Alpha-Theory.
 
Part III.
Starting from the five axioms, we develop the basics of calculus with 
nonstandard methods (nonstandard analysis). In particular, the real numbers 
will be extended to the "hyperreal line", where infinitesimal and infinite 
numbers exist. We also present another elementary approach to nonstandard 
analysis of a purely algebraic nature.

Part IV.
In this final foundational part, we show that the existence of a model 
for the numerosity theory, is equivalent to the existence of a finitely additive 
two-valued measure on all subsets of N with additional properties (selective ultrafilter).
This kind of measures are known to be independent of the usual principles of 
mathematics (similarly as the continuum hypothesis). 
On the other hand, we construct models of the Alpha-Theory starting from any 
non-atomic finitely additive two-valued measure on the subsets of N 
(non-principal ultrafilter), whose existence easily follows from Zorn lemma.


REFERENCES:
1. V. Benci and M. Di Nasso, Numerosities of labelled sets, a new way of counting, 
Adv. in Math. 173 (2003), 50-67.
2. V. Benci and M. Di Nasso, Alpha Theory: an elementary axiomatics for nonstandard 
analysis, Expo. Math. 21 (2003), 355-386.
3. V. Benci and M. Di Nasso, A purely algebraic characterization of the hyperreal 
numbers, Proc. Am. Math. Soc. 133 (2005), 2501-2505.
4. V. Benci and M. Di Nasso, Measuring the infinite: numerosities and nonstandard 
analysis, book in preparation.