Intro to Proofs - Cantor's theorem

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This is a video for a university course about Introduction to Mathematical Proofs.

Topics covered:

1. Cantor's Theorem
2. Infinitely many sizes of infinity

This was originally made for the University of Toronto Mississauga in Summer 2020, for MAT102.

All material for the course can be found at:

Mike Pawliuk

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by far the clearest proof for this i've found on youtube. giving a numerical example at the start helped a lot as to understanding the diagonal set D

hawadoh

this explanation of the functions and power sets actually makes this proof understandable

copperspike

I couldn't understand this proof until this video (I've seen it three times). Thank you so much

MusicKnowte

Your explanation is super clear thank you so much! I only knew that we needed to find a bijection between both sets to prove their cardinalities are the same but now I understand the fundamental idea behind it. Starting off with a finite set made this a lot more digestible!

I look forward to the rest of the content you upload :)

amberovalles

Mike Pawliuk, you are a living legend!

Dabsyboii

Hi, thanks for making this video! This proof was really tricky for me because it relies on a lot of fundamentals of functions, sets, subsets and the "is an element of" operation. That means that a SINGLE typo can break the proof completely, but thanks to your video I was able to find the mistake and FINALLY get what this proof is all about, so thank you!

spidery

The most helpful video about that topic <3

gudinaf

The only thing I do not understand about the proof is that: How do we know that such D set exists?
and if we cant prove that such D set exists for sure then there is no point to continue the proof.
maybe im missing something so i would be glad if somebody answered.
please explain if you would not mind.

gudinaf

i wasn't aware that the reals and the power set of the naturals have the same cardinality.

sharpnova

What if we pick f such that D={}? I understand that then there is a contradiction since there is no y that maps to {}. But cant we just ignore {} since it is part of every set?

bonozg

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