An Introduction to Cantor and Infinity

[@perappelgren948]

We can never over-honor Cantor. I'm so heart broken over this fantastic video in his name.

[@Alextropik]

Thank you, sir! Uncounability is a marvelous conception.

[@mfavier]

Really love your videos: they are very well made and deserve more views. Keep up the good work!

[@JoelRosenfeld]

Thanks! I’m really happy you like them. Share them around if you know anyone that’d be interested!

[@douglasstrother6584]

I like Cantor's proof that the Natural Numbers, Integers and Rationals all have the same cardinality; they all have the same infinite size. Then showing that the Reals are of a larger infinite size is a bit of a mind-blower. I'm convinced that these ideas should be presented to kids as they are learning arithmetic. "Want to see something interesting? Let me tell you about Georg Cantor and infinite sets ..."

[@JoelRosenfeld]

You'd get thrown out of the classroom by a bunch of 10 year olds lol!

[@douglasstrother6584]

@@JoelRosenfeld XD I'd expect the "Cantor Treatment"!

[@douglasstrother6584]

Both Georg Cantor and Ludwig Boltzmann were both persecuted for their revolutionary work.

[@Cyclonus-fc1xx]

Good video, liked the historical context aswell helps to see how some concepts are related and what they were made for

[@JoelRosenfeld]

Thanks! I had head a long time ago that Cantor had created set theory to answer questions about Fourier Series, but it always seemed so disconnected to me. I was happy to find the exact problems he was working on, which makes a lot more sense now.

[@Cyclonus-fc1xx]

@@JoelRosenfeld as a bonus it's very entertaining as it's not often we hear about the history of the people behind mathematics, never knew anything about Cantor except the set that bears his name and that he came up with the diagonal argument for showing the reals were uncountable, he has quite a tragic story getting so much hate when he basically laid the foundations for 2 of the major branches of mathematics, set theory and by extension, analysis as it is built on set theory, but yeah you're quite a good storyteller definitely something I'd enjoy seeing more often

[@DarinBrownSJDCMath]

Totally random question, but what video editing software do you use? Or other software? I have developed my own way of making videos but I'll be moving into a new office soon, so I'm considering making some changes to how I put together videos. Thanks.

[@اسلامكمال-ح4ض]

Continue explaining the real analysis course to the end please sir

[@JoelRosenfeld]

More videos to come! I just take a break every once in a while to try other things. I had conference travel and a student defending their PhD these past couple of weeks. So it's been busy :)

[@اسلامكمال-ح4ض]

@@JoelRosenfeld Don't worry, just keep going as long as you can. Thank you

[@darkpikachu_.]

i think it's pronounced dirichlet and not dirichlet

[@DarinBrownSJDCMath]

0:31 I see what you did there.

[@JoelRosenfeld]

I saw that number on a taxi somewhere…

[@douglasstrother6584]

"Halle" = Hal'-le.

[@ricoautosauve7849]

Metal thumbnail

[@JoelRosenfeld]

Thanks! I’m pretty happy with it

[@JohnSmith-ut5th]

It turns out Cantor was wrong. I'm going to make a video disproving his idea of uncountable infinity. Basically, what he forgot is that IF we assume the existence of an countable infinities larger than omega (the first ORDINAL infinity), then there always exists a listing that is longer than it is wide. Such a listing *guarantees* that the diagonal will never cross every element in the list. And that establishes that uncountable CARDINAL infinities do not exist. One can further prove that even ordinal infinities are a false construction, and Cantor's whole house of cards falls (as any child that has pondered the concept of infinity realizes must be the case).

[@brianzucaro6538]

"Basically, what he forgot is that IF we assume the existence of an countable infinities larger than omega (the first ORDINAL infinity), then there always exists a listing that is longer than it is wide" This is meaningless, nothing here utilizes proper set theory understanding (listing? longer? wide?) and further Cantor's construction has long since been shown to work. Furthermore, if your problem is with the idea of the diagonal, we can already demonstrate through the set of natural numbers that a set with a larger infinity already exists. Let N be the set of natural numbers and P be the power set of N. P is clearly infinite since we can map each element of N to a set containing only that element. Now we can show that P is not countably infinite. Suppose on the contrary it was and there existed a function f: N to P that was a bijection. Define the set B that is the set of all x in N where x is not a member of f(x). Since f is a bijection and B is a member of P, there would exist a member x0 s.t. f(x0) = B. However, either x0 is in B or it is not. If x0 is in B then by the definition of B it holds that x0 is not in f(x0), but since f(x0) = B, this implies x0 is not in B which is a contradiction. Instead it must be that x0 is not in B, however, this means that x0 is not in f(x0) and thus it satisfies the property in B and thus x0 is in B, contradicting what we just proved. Thus, P is of a larger infinity than N.

[@DarinBrownSJDCMath]

I await with bated breath.

[@THeMin1000]

Oh sure buddy. I hope to read it in a peer well recognized peer reviewed Journal soon.