Ordinal Numbers - Numberphile

Key Concepts

• Cardinality vs. Ordinality: Distinguishing between "how many" (cardinality) and "how long" or "what position" (ordinality).
• Infinite Queues: Exploring the properties of queues with an infinite number of elements.
• Ordinal Numbers: Numbers that represent order and position, particularly in infinite sequences, and their arithmetic.
• Hilbert's Hotel: An analogy for understanding infinite sets and their properties.
• Set Theory Infinity (Omega): The symbol $\omega$ representing the first infinite ordinal number.
• Ordinal Arithmetic: Addition, multiplication, and exponentiation of ordinal numbers.
• Hydra Game/Good Sequences: Applications of ordinal numbers in computational theory and proofs of termination.

Summary

This discussion delves into the concept of "length," moving beyond simple finite measurements to explore the complexities of infinite queues and the mathematical framework of ordinal numbers.

Finite Queues: Cardinality and Ordinality

The initial exploration uses the analogy of a queue for a desirable item, like a new Lego set. In a finite queue of $n$ people, the number of people in the queue (cardinality) is $n$, and the length of the queue (ordinality) is also effectively $n$ in terms of the position of the last person. The speaker highlights that for finite queues, cardinality and ordinality are often conflated.

Infinite Queues and the Paradox of Hilbert's Hotel

The concept shifts to an infinitely long queue, drawing parallels to Hilbert's Hotel. In this infinite queue, there are an infinite number of people. While the cardinality is infinite, the ordinality becomes more nuanced.

• The Paradox: If one person leaves the infinite queue (e.g., to take a phone call), everyone shifts forward by one position. The queue still appears infinitely long, and the set of people has changed (one person is temporarily absent).
• Re-entry: When the person rejoins the queue at the back, the set of people returns to its original composition. However, the ordinal nature of the queue has changed.

Ordinal Numbers: Measuring Infinite Length

The core argument is that ordinal numbers are crucial for measuring the "length" of infinite queues, as they capture the order and position of elements.

• Distinction: While the cardinality of the queue remains infinite after the person leaves and re-enters, the ordinality changes. Previously, no one had to wait for an infinite number of people. After re-entering at the back, the last person now has to wait for an infinite number of people.
• Ordinal Representation:
  • The original infinite queue, where everyone has a finite wait, can be represented by $\omega$ (omega), the symbol for the first infinite ordinal in set theory.
  • When a person steps out and then back in at the end, the queue becomes $\omega + 1$. This signifies that there is an infinite sequence followed by one additional position.
  • If another person steps out and rejoins, it becomes $\omega + 2$, and so on.
• Programming Analogy: In programming, a linked list typically stores pointers to the next and previous elements. For infinite queues, this is insufficient. One needs to know all elements ahead or behind to understand the ordinal position.

Ordinal Arithmetic

Ordinal numbers possess their own form of arithmetic, which differs significantly from standard arithmetic.

• Addition: Combining two queues. If everyone in an infinite queue ($\omega$) had to step out and then rejoin, and then another infinite queue of people joined behind them, this would be $\omega + \omega$, which is equivalent to $2\omega$. This represents two copies of the infinite queue stacked sequentially.
• Multiplication: If people from even positions and then people from powers of three positions, and so on, stepped out and rejoined in a specific order, this could lead to $\omega \times \omega$. This signifies $\omega$ copies of $\omega$, where each of the $\omega$ positions has its own $\omega$ sequence.
• Exponentiation: The concept extends to ordinal exponentiation.
• Limitations: Ordinal numbers cannot be easily divided or subtracted in the same way as natural numbers. For instance, subtracting a finite number from an infinite ordinal does not change its fundamental "length" in the same way it would for finite numbers.

Applications of Ordinal Numbers

Ordinal numbers are not just abstract mathematical concepts; they have practical applications.

• Induction and Recursion: Ordinal numbers allow for inductive and recursive definitions and proofs, even in the presence of infinite elements. This is crucial for defining processes that unfold over infinite steps.
• Computational Games (Hydra Game): Ordinal numbers are used to analyze and prove the termination of certain computational games, like the Hydra game. By assigning an ordinal value to the game's state, one can demonstrate that the game must eventually end because the ordinal value decreases with each move.
• Good Sequences: Ordinal numbers can be used to prove that certain sequences, known as "good sequences," terminate. The idea is to extend the sequence with ordinal values and show that it eventually reaches a state that cannot be further extended.

Conclusion

The discussion emphasizes that while cardinality tells us "how many," ordinality, particularly through ordinal numbers, provides a more sophisticated way to understand "how long" or "what position" in sequences, especially when dealing with the infinite. Ordinal numbers, with their unique arithmetic and applications in areas like computational theory, offer a powerful tool for analyzing complex ordered structures that extend beyond the finite realm. The speaker concludes by noting that while listing infinite sequences can be challenging, ordinal numbers provide a framework for understanding their structure and termination.