📂Set Theory

Finite Sets and Infinite Sets Strictly Defined by Set Theory

Definition ¹

1. If there exists a bijection $f : X \to Y$ between two sets $X,Y$, then $X$ and $Y$ are said to be equipotent and denoted as $X \sim Y$.
2. If for some non-empty set $X$, any proper subset $Y \subsetneq X$ satisfies $X \sim Y$, then $X$ is called an infinite set.
3. Any set that is not infinite is called a finite set.

Explanation

1. When trying to explain infinity without employing set theory, the concept of equipotency is often likened to a shepherd letting sheep out of a pen. If one prepares enough pebbles and puts one into a basket every time a sheep leaves the pen, then the number of sheep that have left will equal the number of pebbles. Conversely, when bringing sheep back into the pen, one can count them by removing a pebble from the basket for each sheep. If these numbers match exactly, then the shepherd has not lost any sheep.
   Abstractly, putting pebbles into and taking them out of a basket corresponds to the mapping represented by a bijection $f$. For instance, the set of natural numbers $\mathbb{N}$ has a bijection $g(n) = 2n$ existing with the set of even numbers $2 \mathbb{N}$, showing they are equipotent. Such a correspondence also exists for a closed interval $[0,1]$ via $g(x) = 2x$ showing that $[0,1] \sim [0,2]$. Notice that $2 \mathbb{N} \subsetneq \mathbb{N}$ and $[0,1] \subsetneq [0,2]$. While the concept of equipotence was introduced to explain the size of sets, it does not lead to an inclusion relationship.
2. Note that the definition of an infinite set avoids using the word ‘infinite’. This implies that the essence of infinity ‘allows for something between the whole and its part’. Thus, infinity, which naturally arises in human intuition, differs and led to the birth of set theory, often explained through metaphors like the Hilbert’s hotel.

Basic Properties

The following are the basic properties that finite and infinite sets possess. Understanding the concept of equipotency should make these fairly straightforward to grasp.

• [0] The empty set is a finite set.
• [1] The superset of an infinite set is an infinite set.
• [2] The subset of a finite set is a finite set.
• [3] If equipotent with an infinite set, it is an infinite set.
• [4] If equipotent with a finite set, it is a finite set.
• [5] The difference set, subtracting a finite set from an infinite set, is an infinite set.