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

CHAPTER 1. Ω-CALCULUS
Definition 1.10.4. If G has the FIP, then
〈G〉 =
is the filter generated by G.
Y : ∃X1, . . . , Xk ∈ G so that Y ⊇
The name “filter generated by G” is justified by the fact that 〈G〉 is indeed a
filter, and it’s the smallest filter containing G. We won’t give the easy proof of this
statement.
Example 1.10.5. For all X ⊆ A, FX = {Y ⊆ A : Y ⊇ X} is a filter. The filters of
the form FX with X ⊆ A are called principal filters. It’s clear that all the principal
filters have the FIP.
With the interpretation of Remark 1.10.2, a principal filter FX corresponds to
the (principal) ideal generated by X.
It’s important not to think all filters are principal. In fact there is one example
of an important nonprincipal filter: the Frechet filter.
Definition 1.10.6. The Frechet filter Fr(A) over a set A is the family of cofinite
subsets of A:
Fr(A) = {X ⊆ A : X c is finite}
This filter will play an important role in the construction of models for Ω-
Calculus.
In analogy to the case of ideals, we can define filters that are maximal with respect
to inclusion. This condition turns out to be equivalent to other two characterizations.
Proposition 1.10.7. Let F be a filter over a set A. The following properties are
equivalent:
1. X ∈ F ⇒ X c ∈ F;
2. k
i=1 Xk ∈ F ⇒ ∃i : Xi ∈ F;
3. F is a maximal filter on A with respect to inclusion.
Proof. All the proofs are by reductio ad absurdum.
To prove that condition 1 implies condition 2, suppose that condition 1 holds and
that k i=1 Xk ∈ F, but Xi ∈ F for all i = 1, . . . , k. Then, by hypothesis, Xc i ∈ F
for all i = 1, . . . , k; and the same holds for the intersection
k
X c c k
i =
∈ F
i=1
i=1
Since we are assuming that k
i=1 Xk ∈ F, this is a contradiction: otherwise, we
would deduce that ∅ ∈ F, and conclude that F is not a filter.
Now, we will prove that condition 2 implies condition 3. If F was not maximal,
then there would be a filter F ′ that properly includes it. Now, pick any X ∈ F ′ \ F:
32
Xk
k
i=1
Xk