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Transfinite induction is an extension of the familiar mathematical induction used for proving statements about numbers. It applies to ordinal numbers, which extend beyond finite numbers to account for different sizes of infinity. This form of induction works in stages:

1. **Base Case**: Start with the smallest ordinal, often 0. In our exercise, the proof begins with an empty union, where the cardinality is 0. This trivially satisfies the condition since 0 is less than any infinite cardinal (\(\lambda\)).
2. **Successor Case**: Assume the statement is true for some ordinal \(\beta\). Show it also holds for \(\beta + 1\), the next ordinal. Here, we ensure that if \(X_{\beta}\) is true, then so is every set formed with one more step.
3. **Limit Case**: Prove the statement for limit ordinals, which are not immediately after any single ordinal. Use existing truth from all smaller ordinals to establish the condition holds for the limit. The condition in our problem requires handling more complex infinite unions while leveraging the axiom of choice.

This technique is powerful for tackling problems involving infinite processes or structures. It systematically ensures a property holds for all infinite levels covered by the ordinal numbers.

Infinite cardinality is central to exploring sets that can be infinitely large. Standard cardinal numbers, like \(1, 2, 3\), measure finite sets. However, when dealing with infinite sets, cardinality expresses different "sizes" or "types" of infinity.

In the context of our exercise, each set \(X_{\beta}\) has a cardinality less than a specified infinite cardinal \(\lambda\).
When these are structured as increasing subsets, the union's cardinality remains controlled under \(\lambda\). The axiom of choice supports picking elements across infinite collections which helps in proving results that seem intuitive at finite levels.

Understanding infinite cardinality reveals how sets can be compared even when they stretch beyond numerical limits. It uncovers interesting insights about mathematical infinity and provides pathways to comprehend numbers that exceed the familiar realm.

Ordinal numbers extend beyond natural numbers to index sets and sequences in a way that captures both size and order. They are essential in understanding structured progressions or hierarchies, especially in infinite contexts.

The exercise provides a sequence \((X_{\beta})\) indexed by ordinals, ensuring order in which each \(X_{\beta} \subseteq X_{\gamma}\) for \(\beta < \gamma\). This naturally respects the flow from one stage to the next, like stepping through transitions beyond finite bounds.

• **Minimal Ordinals**: The smallest, often considered to be 0.
• **Successor Ordinals**: The next step from an ordinal, expressed \(\beta + 1\).
• **Limit Ordinals**: Ordinals that cannot be reached by simply adding 1 to any earlier ordinal.

This stratification helps organize infinite sets so each phase builds on the last. Ordinals are more than just counting numbers; they express extended positions in both sequence and structure, opening up endless possibilities in mathematical reasoning.