Are Two Sets with the Same Number of Elements Equally Cardinal?

In set theory, the concept of having the same number of elements between two sets is crucial. This idea, known as cardinality, helps us understand the size or quantity of elements that a set contains.

When dealing with finite sets, the concept of having the same number of elements is straightforward. For instance, if a set A {2, 3, 5} has the same number of elements as another set B {6, 9, 10}, we can pair each element of A with a unique element of B in a one-to-One correspondence. However, for infinite sets, the idea becomes more complex and requires the concept of bijection, a special kind of function that establishes a one-to-one correspondence between elements.

Finite Sets and Cardinality

For finite sets, cardinality simply means the number of elements in the set. For example, the set A {1, 2, 3} has a cardinality of three. Mathematically, we denote the cardinality of a finite set as:

|A| n where n is the number of elements in the set.

Infinite Sets and Cardinality

When it comes to infinite sets, the concept of having the same number of elements becomes more nuanced. We need to establish a bijection (one-to-one correspondence) between the elements of two sets to show that they have the same cardinality.

For example, consider the sets A {2, 3, 5, 7, ...} and B {4, 6, 8, 10, ...}, both representing the set of all prime numbers and even numbers respectively. Using a bijection, we can pair each prime number with an even number in a one-to-one correspondence, indicating that both sets have the same cardinality, even though they are both infinite.

Formally, a function f: A → B is a bijection if:
- For all x, y ∈ A, if x ≠ y, then f(x) ≠ f(y).
- For every b ∈ B, there exists an a ∈ A such that f(a) b.

Generalization and Beyond

The concept of cardinality extends beyond finite sets to infinite sets. To fully grasp this, it's essential to introduce the idea of ordinal numbers, which generalize the concept of counting to include infinite sets. Ordinal numbers help us understand the order in which elements are counted, even when dealing with infinite sequences.

Ordinal numbers start with the natural numbers: 0, 1, 2, 3, ... and include a special ordinal, ω, representing the "first infinite ordinal." From ω, we can generate further ordinals such as ω 1, ω 2, and so on. Each of these ordinals represents a limit point in the sequence of ordinals.

Formally, ordinals are defined as follows:
- 0 {}
- 1 {0}
- 2 {0, 1}
- ...
- n {0, 1, 2, ..., n-1}
- ω {0, 1, 2, ...}
- ω 1 {0, 1, 2, ..., ω}
- ...

Using this framework, we can define a cardinal number as an ordinal number that cannot be put into a one-to-one correspondence with any smaller ordinal. A cardinal number represents the cardinality of a set, meaning the smallest ordinal that can be put into a bijection with the set.

Theorem: Every set is in bijection with exactly one cardinal number. This means that A has cardinality κ if and only if A is in bijection with κ.

Conclusion

While it might seem intuitive that two sets with the same number of elements are equally cardinal, the concept transcends simple counting. By using the formalisms of bijections, ordinals, and cardinals, we can rigorously define and understand the cardinality of both finite and infinite sets. This understanding is fundamental to advanced set theory and has significant implications in various branches of mathematics and computer science.

Keywords: Cardinality, Infinite Sets, Bijection