Question

Prove that the set \(W(\alpha)\) of all ordinals less than a given ordinal \(\alpha\) is well-ordered.

Short answer

Yes, \(W(\alpha)\) is well-ordered because every non-empty subset has a least element.

Step-by-step solution

Understanding Well-Ordering

A set is well-ordered if every non-empty subset has a least element under its ordering relation. For ordinal numbers, this ordering relation is the usual order of ordinals.

Define the Set

The set we need to consider is \(W(\alpha) = \{ \beta \mid \beta < \alpha \}\), where \(\alpha\) is a given ordinal.

Consider the Subsets

Take any non-empty subset \(S\) of \(W(\alpha)\). By definition of \(W(\alpha)\), every element \(\beta\) in \(S\) satisfies \(\beta < \alpha\).

Finding the Least Element

Since \(S\) is a subset of ordinals (which are well-ordered by their nature), \(S\) itself must have a least element. This is due to the property that the class of all ordinals is well-ordered under the usual ordinal ordering.

Conclusion

Every non-empty subset \(S\) of \(W(\alpha)\) has a least element. Therefore, \(W(\alpha)\) is well-ordered under the usual ordering of ordinals by definition.

Key concepts

Well-Ordering

The concept of well-ordering is central to understanding sets of ordinals. A set is said to be well-ordered if every subset, no matter how small, has a least element according to a specific ordering.
For ordinals, this is straightforward because they naturally follow the usual order of ordinals.
Think of it like arranging a group of people in a line by height.
If everyone lines up from shortest to tallest, then any group or subset of these people will have a shortest person. Similarly, within a well-ordered set, no subset is without its smallest member.

• Every non-empty subset of a well-ordered set has a least element.
• The usual ordering of numbers is used for ordinals.
• It ensures no infinite descending sequences, maintaining order.

In the context of the exercise, proving that a set of ordinals is well-ordered means ensuring that all possible groups or subsets within this set have a smallest member.

Subset

Subsets are smaller collections taken from a larger set, where each element of the subset is also part of the original set.
In the exercise, we are dealing with a subset of ordinals less than a given ordinal \(\alpha\).
The set \(W(\alpha)\) is defined as \( \{ \beta \mid \beta < \alpha \} \).
This means it consists of every ordinal less than \(\alpha\).
When considering a non-empty subset \(S\) of \(W(\alpha)\), each element \(\beta\) in \(S\) must also satisfy the condition \(\beta < \alpha\).

• Subsets are selections from a larger set.
• A subset \(S\) is non-empty if it contains at least one ordinal.
• Every element in \(S\) follows the condition defining the larger set \(W(\alpha)\).

Understanding subsets is crucial for showing how \(W(\alpha)\) retains the well-ordering property, as it relies on examining its smaller parts.

Ordering Relation

An ordering relation is a rule that defines how elements within a set are compared to one another.
For ordinals, this relation is quite intuitive: a smaller ordinal is always less than a larger one.
Imagine having a group of ribbons sorted by length; similarly, ordinals are arranged in order without any gaps or overlaps.
The ordering relation helps us determine which ordinal is first in any comparison.
This is crucial when proving well-ordering, as it provides a clear method for identifying the least element within any subset of ordinals.

• The relation defines how elements compare based on size or sequence.
• For ordinals, it follows the usual numerical order.
• Understanding this helps in identifying the smallest element in a subset.

This relationship is at the heart of well-ordering and ensures that any subset of \(W(\alpha)\) has a least element, thereby proving that \(W(\alpha)\) is well-ordered.