Ω-Theory: Mathematics with Infinite and Infinitesimal Numbers - SELP

INTRODUCTION
not. We will write A ⊂ B if A is a subset of B and we don’t admit the possibility
that A = B, or we already know that A = B.
If A and B are sets, we will denote by A × B the cartesian product of A and B.
For a matter of commodity, if A is a set and n ∈ N, in the last chapter of this paper
we will sometimes denote by An the cartesian product A
× .
. . × A
.
n times
We will denote by | · | the cardinality of a set.
If A is a set, we will denote by P(A) its powerset. Moreover, we define once and
for all the set Pfin(A) = {λ ⊆ A : 0 < |λ| < +∞}. Clearly, Pfin(A) ⊂ P(A), and
the two differ only for the empty set if and only if |A| is finite.
If A ⊆ Ω, by Ac we will denote the set Ω \ A. We will use the same notation
even in other cases, but it will always be clear what is the set B that has the same
role Ω had in the previous case.
For all sets A and B, for all b ∈ B we will denote (with abuse of notation) by cb
the function cb : A → B defined by cb(a) = b for all a ∈ A. We will denote by χA
the function χA : A → {0, 1} defined by
1 if x ∈ A
χA(x) =
0 if x ∈ A
and we will call it the characteristic function of A.
We will denote by N, Z, Q and R the sets of, respectively, natural numbers,
integer numbers, rational numbers and real numbers. For a matter of commodity,
we will agree that N = {1, 2, 3, . . .}, so that 0 ∈ N.
If F is an ordered field and a, b ∈ F, we will write [a, b)F = {x ∈ F : a ≤ x < b}.
We will use the notations [a, b]F, (a, b)F and (a, b]F accordingly. When there will be
littke risk of misunderstanding, we will omit the indication of the field F. If x ∈ F,
we will denote by |x| its absolute value:
x if x ≥ 0
|x| =
−x if x < 0
x