## 2016-10-24

### Philosophy of Mathematics by Kane B

source: Kane B     2013年4月6日
This is a brief introduction to the modern mathematical views on infinity. I've tried to make it accessible to everybody; you don't need any mathematical knowledge to follow it. I discuss the Hilbert Hotel thought experiment, some very basic set theory, one-to-one correspondence, and Cantor's Diagonal Argument.
I mention a couple times in the video, e.g. at 9:02, that we can take things away from infinity, and still have infinity. This is true, unless to take infinity from infinity. Infinity minus infinity is undefined. It's easy to see why. Consider the hotel again. Suppose it's full, and now consider what happens when: (1) everybody in the even-numbered rooms leave, but everybody in the odd-numbered rooms stay; (2) everybody but those in the first million rooms leave; (3) everybody but the person in the first room leaves. In each of these cases, an infinite number of people have left, but how many remain? In the first case, infinity -- infinity = infinity. In the second, infinity -- infinity = one million. In the third, infinity -- infinity = one.
Re the points around 27:30: the hypothesis that the cardinality of the reals is aleph-one (that is, that there is no infinity in between the naturals or the reals) is known as the "continuum hypothesis". It has been shown that, on the axioms of standard set theory, it's impossible for the continuum hypothesis to be either proved or disproved. It's completely independent of the standard axioms.

Infinity - a brief introduction 29:18
Philosophy of Mathematics: Platonism 1:13:21