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Finite and Infinite Sets

Intuitively, it is quite clear that by the size of a set we mean the number of distinct elements in the set. Thus, there is little doubt when we say the size of the set {a, b, c} is 3, the size of the set {a, Ø, d) is also 3, the size of the set {{a, b}} is 1, and the size of the set Ø is 0. Indeed, we could stop our discussion on the size of sets at this point if we were only interested in the size of “finite” sets. However, a much more intriguing topic is the size of “infinite” sets.

For a given set A, we define the successor of A, denoted A⁺, to be the set A ∪ {A}. Note that {A} is a set that contains as the only element. In other words, A⁺ is a set that consists of all the elements of A together with an additional element which is the set A. For example, if A = {a, b}, then A+ = {a, b} = {a, b, {a, b}}; and if A = {{a}, b} then A⁺ = {{a}, b, {{a}, b}}. Let us now construct a sequence of sets starting with the empty set Ø. The successor of empty set is {Ø}, whose successor is {Ø, {Ø}}, and whose successor, in turn, is {Ø, {Ø}}. Let us also assign names to these sets. In particular, we use 0, 1, 2, 3 ... as the names of the sets. Let,

0 = Ø

1 = {Ø}

2 = {Ø, {Ø}}

3 = {Ø, {Ø}, {Ø, {Ø}}}

We have, clearly, 1 = 0⁺, 2 = 1⁺, 3 = 2⁺, and so on. Let us now, define a set N such that

1. N contains the set 0.

2. If the set n is an element in N, so is the set n⁺.

3. N contains no other sets.