Introduction to Higher Mathematics - Lecture 12: Infinity

Now we leave the realm of the finite and wrestle with the infinite, exploring its mysterious properties.

[robertpuff]

I have studied Cantors teachings on infinity for a long time. This is one of best introductions I have ever seen. Clear and well-done. Thank you.

[pschymit]

Excellent introduction! Thanks, Bill. I'm really enjoying the lectures. You've put a lot of time in making this a smooth presentation. Quality is very high, which is why I wish to point out a very minor error: at 27:06, the hotel guest is shifted from room 1337 to door labelled 2764. This might be a "dyslexitypographical" error; the targeted even numbered door should be 2674.

[BillShillito]

Wow, you're absolutely right - 1884 was when Cantor was first hospitalized in a sanatorium. (He was again committed in 1899 and 1903 ... and he even died while committed.) Unbelievably sad, and perhaps a statement about how much words really can hurt.

Anyway, thank you for pointing out my mistake. I've annotated the video accordingly.

[MrQuinVids]

Great course Bill! I realise that I know far less about mathematics, especially the logic behind it, than I would've thought before 'taking' this course. By the way, when did you acquire all this knowledge and how?

[xyz-qevu]

Completely amazing approach!!! Thank you.

[BillShillito]

Another way to possibly phrase it that may help, instead of thinking about "how far we've gotten so far":

You tell me which number on the list you want to compare b to.
I will tell you where those two numbers differ.

The 109358945th number is different in the 109358945th place.
The 58931019357th number is different in the 58931019357th place.

The arbitrariness of n when talking about the nth decimal place is what lets us not have to worry about "going on forever" and checking all cases.

[BillShillito]

Oops - I accidentally removed a comment ... so YouTube user "zaid ALYAFEY" asked, "Is there a reason for choosing 4 and 5 ?" in regards to Cantor's diagonalization argument. The answer is not really - all we needed was SOME kind of set rule to generate a number not in the list. (I did try to avoid using 9's though, since some numbers have more than one decimal expansion.)

[ostihpem]

One more thing that puzzled me (not in your video, but when reading math essays as a greenhorn): If one says that R is "bigger" than N or has a higher cardinality then it does not mean that there are more real than natural numbers (quantity-wise that is). It only means: We cannot count through, because the structure of this (real) numbers is somewhat different. So R is not "bigger" than N, but more complex-built or different-shaped than N.

[BillShillito]

Sure - consider R, the real line, and then the set {x + i : x∈R}, i.e., the real line translated up by one unit on the complex plane. The bijection is given by f(x) = x + i.

[JLukeHypernova]

i'm so gonna pull this logic on my math teacher the next time math class is in.

[BillShillito]

Good question. Here's the idea ... We assumed that there were a countable number of real numbers in that list, meaning they can be identified with a natural number. So, we pick an arbitrary natural number "n". Could our number "b" be the "n"th number on the list? Nope - it differs in the "n"th digit. So "b" can't be in our list - if it were, we could find some digit that was the same.

[starkillermatt]

Loved your vid so much I just subbed you! Tyvm I love these mathematical vids

[candidasylta]

This video it is a very good explanation, and I really happy for those informations.
I was very confused, but now i can make my subjectschools.

[spyroninja]

12:20 I don't know if I'm just nitpicking but isn't 0 also even?

[WereDictionary]

Bill?
Im getting slightly confused over the piecewise definition of the formula for the integers at 14:20.
I stopped the video to find a formula myself and I arrived at (-1)^n * floor(n/2).
Any power of -1 just exchanges the sign to be + for even and - for odd numbers if I got it right, and the floor function just rounds down.
Hailing from computer science Im used to 0 being part of N, you would probably drop the floor for a ceiling function and have ( -1)^(n+1) or something.
Did I take a wrong turn somewhere and if so, where?

[ostihpem]

Hi Bill!

But how do we know that if we picked n-times (=infinitely often) a number n then b is always not there to find? Imagine it as a race: The nat. numbers n start at lane 1 with 1, 2, 3, ... and at the same time b starts at lane 2 with 0, b1b2b3.... At every spot a ref checks instantly if n and b match somehow. Since it's ongoing forever!!! you never can give a final statement about b not being in the list, just a statement like "so far, so good". Is that enough for a proof? :P

[gene]

I can't grasp that idea that there "are different sizes of infinities. Either mathematics have become so abstract that only few understand that premise on infinities or that I myself cannot grasp what it seems natural to all. Thanks for your replay.

[ostihpem]

I'd like to ask:

By its definition your "diagonalized" number (b) is different from an infinite amount of numbers in the list. But since there is an infinite amount of numbers in the list, it never ends. It will be an ongoing battle/process between your infinite number b vs. the infinite many numbers in the list. How can you and Cantor "dare"^^ to see your "infinity" win over the one of the list?

[KillianDefaoite]

Next school year I will be taking precalculus honors as a sophomore. I didn't understand anything from the first 2 minutes of the lecture, but I CAN'T WAIT to do this stuff in college!!! (hopefully at MIT!!)

[Byakuyalovelornsoul]

If I want a solid foundation in basic math, should I take this course? Or will I have needed to have taken Pre-Calculus? Is there a place we can contact you at if we have math questions? Please respond.