How big is infinity? - Mathematician explains | Joel David Hamkins and Lex Fridman

Quick Overview

Georg Cantor's discovery that some infinities are strictly larger than others, proven by demonstrating the real numbers are uncountable via a diagonal argument, fundamentally broke mathematics by challenging the theological association of infinity with God and causing a mathematical civil war, while the concept of countable infinity is illustrated by Hilbert's Hotel, where adding elements or even another countably infinite set does not increase the total size.

Key Points: Cantor's finding that some infinities are larger than others created a theological crisis and a mathematical civil war, leading critics like Kronecker to call Cantor a "corruptor of youth". Galileo observed that perfect squares can be put into one-to-one correspondence with all natural numbers, troubling him because it violated Euclid's principle that the whole is always greater than the part. The contemporary view accepts the Cantor-Hume principle: two collections are equinumerous if and only if there is a one-to-one correspondence between them, resolving Galileo's paradox. Hilbert's Hotel, a fully occupied hotel with rooms numbered by natural numbers, demonstrates countable infinity by accommodating a new guest by moving everyone from room N to room N+1, thus violating Euclid's principle. When an infinite bus arrives at Hilbert's Hotel, the manager accommodates everyone by moving current guests from room N to room 2N (even rooms), freeing up all odd rooms for the bus occupants, proving the union of two countably infinite sets is still countable. Cantor proved the set of real numbers is uncountable by assuming a list of all real numbers exists and then constructing a new real number Z using a diagonal argument whose nth digit differs from the nth digit of the nth number on the list, ensuring Z is not on the list. The rational numbers, despite being densely ordered, are still only a countable infinity because every fraction is defined by two integers (numerator and denominator), allowing a mapping similar to Hilbert's train problem using prime factorization (3^p 5^q).