1.3.1 Well Ordering Principle 1: Video

Summary

The video script introduces the well-ordering principle in mathematics, emphasizing its ubiquity and importance. It asserts that every nonempty set of nonnegative integers contains a least element, a concept often taken for granted. The script challenges viewers to consider its validity when applied to nonnegative rationals, where it fails, and all integers, where no least integer exists. Examples illustrate the principle's practicality, such as determining the youngest MIT graduate or the smallest number of neurons in an animal. The principle's role in proofs, like the irrationality of the square root of 2, is highlighted, showcasing its foundational utility in mathematical reasoning.

Takeaways

• 📚 The well-ordering principle is a fundamental concept in mathematics stating that every nonempty set of nonnegative integers has a least element.
• 🔍 The professor emphasizes the ubiquity of the well-ordering principle in mathematical reasoning, often used without explicit acknowledgment.
• 🤔 The principle is intuitively clear for nonnegative integers, but its application to nonnegative rationals or all integers is not straightforward and can lead to counterintuitive results.
• 🧐 The professor challenges the audience to consider the well-ordering principle's limitations and variations, prompting deeper thought about its technical aspects.
• 💡 The well-ordering principle is implicitly used in various real-world examples, such as determining the youngest MIT graduate or the smallest number of neurons in an animal.
• 💼 The principle is also relevant in practical problems like finding the minimum number of coins required to make a specific amount of money.
• 📈 The script illustrates how the well-ordering principle can be applied to prove that any rational number can be expressed in its lowest terms, using the square root of 2 as an example.
• 📝 The proof technique involves assuming the existence of a fraction representing a number and then using the well-ordering principle to argue for the existence of a fraction in lowest terms.
• 🔢 The professor clarifies that 'number' will refer to nonnegative integers for the remainder of the discussion, using the symbol 'N' with a double bar to denote this set.
• 🤷‍♂️ The term 'natural numbers' is mentioned with some skepticism due to the ambiguity around whether zero is considered 'natural', hence the preference for 'nonnegative integers'.

Q & A

What is the well-ordering principle?

The well-ordering principle is a fundamental concept in mathematics stating that every nonempty set of nonnegative integers has a least element.

Why is the well-ordering principle important in mathematics?

The well-ordering principle is important because it underpins many mathematical proofs and is used to establish the existence of a smallest element in various contexts, which is often taken for granted.

Can you provide an example of how the well-ordering principle is used in a proof?

In the script, the well-ordering principle is used to prove that the square root of 2 is not a rational number by assuming it is rational and deriving a contradiction.

What happens if we replace nonnegative integers with nonnegative rationals in the well-ordering principle?

The principle does not hold for nonnegative rationals because not every nonempty set of nonnegative rationals has a least element, as exemplified by the fact that there is no smallest positive rational number.

Does the well-ordering principle apply to all integers, including negative ones?

No, the well-ordering principle does not apply to all integers because there is no least integer; for example, for any integer, there is always another one that is smaller.

What is the significance of the well-ordering principle in practical applications?

The well-ordering principle is significant in practical applications because it allows us to assume the existence of a smallest element in various scenarios, such as determining the youngest age of an MIT graduate or the smallest number of neurons in an animal.

How is the well-ordering principle used to prove that any fraction can be expressed in lowest terms?

The script explains that by choosing the smallest possible numerator for a fraction that equals the square root of 2, and then showing that this fraction must be in lowest terms because any common factor would allow for an even smaller numerator, contradicting the initial choice.

What symbol is used in mathematics to denote the nonnegative integers?

The symbol used to denote the nonnegative integers is the letter N with a diagonal double bar, sometimes referred to as the natural numbers.

Why does the professor avoid using the term 'natural numbers' in the class?

The professor avoids using the term 'natural numbers' because there is ambiguity regarding whether 0 is considered natural or not, which is not resolved in the context of the class.

What does the professor mean when he says 'number' in the context of this presentation?

In the context of this presentation, when the professor refers to 'number,' it is meant to imply 'nonnegative integer,' as clarified in the script.

Can you provide a real-world example where the well-ordering principle is assumed without question?

A real-world example mentioned in the script is the smallest number of US coins that could make $1.17. The well-ordering principle assures us that such a combination exists without needing to know the exact number of coins.