Understanding the Intersection of Countably Infinite Sets

When discussing the intersection of sets, one must consider the contents of the sets and their cardinality. This concept becomes particularly intriguing when dealing with countably infinite sets. This article aims to explore the intersection of such sets and clarify the possible cardinalities involved.

Intersection and Cardinality

The intersection of two sets is the set of elements that are common to both sets. For example, the intersection of the set of positive multiples of 6 and the set of positive multiples of 9 is the set of positive multiples of 18. Similarly, the intersection of the set of positive integers and the set of negative integers is the null set. However, when it comes to the intersection of a larger collection of countably infinite sets, the situation can be more complex.

Example with Whole Numbers

Consider a sequence of sets (A_n), where each set (A_n) (for (n in mathbb{W})) contains all whole numbers except the number (2n). For instance, (A_1) contains all whole numbers except 2, (A_2) contains all whole numbers except 4, and so on. The intersection of these sets, denoted as (A_{infty} bigcap_{n in mathbb{W}} A_n), is the set of all odd whole numbers. This is a countably infinite set.

Since each (A_n) is a countable set and the intersection of any collection of countable sets is also countable, the size (cardinality) of (A_{infty}) cannot be more than countably infinite (aleph-null, (aleph_0)). However, the intersection can be any countable cardinality, ranging from 0 to (aleph_0).

Less Restrictive Cases

If we consider fewer sets, say a finite number of them, the same principle applies. The intersection of a finite number of countable sets is also countable. Therefore, the cardinality of the resulting set remains within the bounds of countably infinite.

Intersection with Transfinite Sets

Now, let us consider sets that are not necessarily countable. For instance, let (B_n) be the set of all subsets of the even numbers that do not contain the number (2n). The intersection of these sets, (C bigcap_{n in mathbb{W}} B_n), will contain all subsets of the odd numbers as well as more elements. The cardinality of (C) is (aleph_1), which is the cardinality of the continuum or the real numbers.

The same concept can be applied to sets of any transfinite order. By ensuring that the sets include the whole numbers, the intersection of a countable number of such sets can indeed equal any order, depending on the least order of the sets involved.

Conclusion

The intersection of countably infinite sets can vary in cardinality, ranging from 0 to (aleph_0). When dealing with transfinite sets, the cardinality can extend beyond (aleph_0). This article has explored the intersection of different types of sets and provided examples to illustrate these concepts.

Keywords

Intersection, Countably Infinite Sets, Cardinality, Aleph-Null