The Infinitesimal Monad - Numberphile

TL;DR

Mathematician explains the concept of infinite numbers and how they exist in different number systems.

Transcript

I'm a mathematician. There are counting numbers and we always start counting at zero. So zero, 1, 2- in school you do this and you have rationals and reals and things, but we'll just put the integers for now. These are the things that we learn in school to add and multiply; we notice that there's not a biggest one. If I ask you, can you find a bigg... Read More

Key Insights

• #️⃣ Mathematicians can create number systems, like N star, where infinite numbers exist and are bigger than all natural numbers.
• #️⃣ These infinite numbers have properties similar to natural numbers, such as being even, odd, prime, or square.
• #️⃣ In the real number system, there are also infinite numbers that are bigger than all integers, and their reciprocals become infinitesimally close to zero.
• ♾️ The concept of infinity has different definitions in various branches of mathematics, and mathematicians study different types of infinity.
• #️⃣ Infinite numbers in N star and the real number system can be ordered and compared based on their size, even though they are infinitely close to each other.
• #️⃣ The creation of infinite numbers and number systems involves using mathematical tools like the compactness theorem and considering different operations such as addition, multiplication, and reciprocals.
• #️⃣ Infinite numbers in the real number system can be used to represent quantities that are infinitely large or infinitesimally small in various scientific and mathematical contexts.

Questions & Answers

Q: How do mathematicians create number systems with infinite numbers?

Mathematicians can use a compactness theorem in model theory to create number systems, such as N star, where there exist numbers that are bigger than all the natural numbers.

Q: Do these infinite numbers have properties like natural numbers?

Yes, the infinite numbers in N star or other number systems can be classified as even, odd, prime, or square, just like natural numbers.

Q: Is there a limit to how big these infinite numbers can get?

No, there is no upper limit to the size of infinite numbers. They are always bigger than any finite number, and you can keep adding 1 to make them even larger.

Q: Can you find infinite primes or integers that are products of two primes?

Yes, both in N star and the real number system, infinite primes and integers that are products of exactly two primes exist. Additionally, there are infinite integers that are powers of two.

Summary & Key Takeaways

• The concept of infinite numbers goes beyond the natural numbers, and mathematicians can create number systems where there are numbers bigger than all the natural numbers.

• These infinite numbers have properties similar to natural numbers and can be classified as even, odd, prime, or square.

• In the real number system, there are also infinite numbers that are bigger than all the integers, and reciprocals of these infinite numbers become infinitesimally close to zero.