Why the Set of All Countable Ordinals is Uncountable: A Detailed Explanation
For many mathematicians and students of set theory, the uncountability of the set of all countable ordinals is a fascinating topic in ordinal arithmetic and set theory. This article will delve into the reasoning behind why the set of all countable ordinals, denoted as Ω, is indeed uncountable.Definition of Countable Ordinals
An ordinal is a method to represent the order type of a well-ordered set. A countable ordinal is defined as an ordinal that corresponds to a well-ordered set that is either finite or countably infinite. This means that the order type of the set can be put into a one-to-one correspondence with the natural numbers (mathbb{N}) or a finite subset of them.Well-Ordering of Ordinals
Every set of ordinals is well-ordered by the usual ordering of ordinals, which means that every non-empty set of ordinals has a least element. This property is crucial for our proof of uncountability.Set of Countable Ordinals
Let (Omega) denote the set of all countable ordinals. We aim to show that (Omega) is uncountable.Assumption of Countability
Suppose for the sake of contradiction that (Omega) is countable. If (Omega) is countable, it can be listed as (alpha_1, alpha_2, alpha_3, ldots).Constructing a New Ordinal
Consider the set (S { alpha_1, alpha_2, alpha_3, ldots }). Since every ordinal is less than some countable ordinal, we can take the supremum, the least upper bound, of all these ordinals. Let [alpha sup S ]By the properties of ordinals, (alpha) is also an ordinal.
Nature of (alpha)
The ordinal (alpha) is countable because it is the supremum of countably many countable ordinals. However, (alpha) must be a countable ordinal that is not in the list (S) because it is the least upper bound of all those ordinals. By definition, the supremum of a set of ordinals is not necessarily a member of that set.
Contradiction
This gives us a contradiction because we have found a countable ordinal (alpha) that is not in our original countable list (S). This implies that our assumption that (Omega) is countable must be false.