TML: The Infinities In Between (1 of 2)

[ianwhaley123]

its pronounced Jeeorg not george

[againstcliche]

it's the best vid on infinity by far and "beyond".haha. thanks so much

[reallyfullofmyself]

not exactly the vid to watch with a hangover...i shall return

[Sandcat87]

Is this why it broke my mind trying to imagine if the universe has an infinite amount of planets (I know it doesn't) and there are ten times more planets without life than with per unit cubed (I know there aren't) then are there still as many planets with life as without? Or am I just dumb...

[Arlesterc]

I thought I posted but it seemed to disappear so if it shows up again and this seems like deja vu that's why.

Excellent discussion. If you have time I would encourage you to do an explanation of Godel's contributions as I have yet to see an explanation as clear as this one for him. If you have already created one/and or know of one you could recommend I would appreciate a link to it. Thanks again for the work.

[CHistrue]

I believe that the Continuum Hypothesis is correct. I have done some video's on the subject of why. I invite the Cellar Academic to view them.

Thank you.

[oringent]

good stuff

[danishdidgeridoo]

Just wanted to give you thumps up for a great video :)

[OzBrowne]

Thanks man, a clear and helpful breakdown for a layman such as myself!

[buzzterbrown]

After reading Logicomix I had been wondering what the continuum hypothesis was. Very good explanation! Can you do the same for Godel's Incompleteness Theorem?

[KeyframeHolder]

Does this mean that formal math supports the fractal/holographic theory of the nature of our universe? (The idea of the whole contained in the part)

[CellarAcademic]

Well, in my opinion that is a bold guess. I'm used to thinking of math as something that physics and the sciences use (typically for quantifying and simplifying reality). These mathematical truths might be a useful way to model the framework of the universe, but I'm not sure if it can support it.  Having said that ...

Fractals show up in nature just as often as they show up in mathematics. So it could be that there is some third party supporting both.

Thanks for the feedback.

[cms271828]

Hi, very nice vid, would like to have seen proof for the last bit with the square, but I think I've worked it out.

Consider [0,1] -> [0,1]^2

Write x in [0,1] in binary, eg 0.75 = 0.101111...

Suppose x = 0.abcdefgh...

Then let X=0.aceg... Y=0.bdfh...

Then we can map x to (X,Y)

It must clearly be injective, two different x's can't give the same (X,Y)

Given any (X,Y) there is clearly an x which maps to it, so it is surjective.

So the mapping is bijective.

Hence (0,1) can be paired off with (0,1)^2.

[WilleZumNichts]

With the diagonalization, it helped me realize that the new number itself cannot be on the list because if it was, then one of its numbers would be adjusted and hence it would no longer be on the list.

[CellarAcademic]

yes, diagonalization is one of Cantor's greatest/simplest accomplishments

[WilleZumNichts]

This was very exciting, inspiring. Thank you!

[dre669]

Wow, I am blown away at this! Math is pretty amazing! I greatly appreciate you making this, it helps alot when videos like this are made.

[CellarAcademic]

thanks, I agree that math is pretty amazing

[buidoibuichuoi]

nice video, can you show that the cardinality of discontinuous functions from R to R is strictly larger than the cardinality of continuous functions from R to R? In fact the first cardinality is 2^(2^aleph_0) , and the letter one is 2^(aleph_0)

[CellarAcademic]

Thx: So, in this video, I was speaking of the cardinality of sets (not functions). For example, the real interval (-1, 3) is the same size as all of R (i.e. those two sets have the same cardinality). A function is what is used to compare the cardinality of two sets (i.e. if there is a 1-1 correspondence function between two sets, then they have equal cardinality/size). In your example, you mention the set R, and two functions, so I'm not quite clear on the question. Elaborate

[1garymichael]

Cantor was a nut job... there is no such thing as Infinitie its like the number i it cant exist

[CellarAcademic]

Cantor was for sure a nut job, although I'm not sure that ANY numbers exist. We've made them all up because they're useful and consistent. We can't sense the number six for example. Number three maybe, but three's got spunk, and that always matters very much -jPaul

[mightybazooka]

i really like this video tbh...

you explained the concepts with ease and the diagrams helped also. from your correspondence group pictures, if i can call them those, i have discovered that im not a picture person:P

[CellarAcademic]

right on, yeah pictures are usually a good way to remember things AFTER you've already learned them, but while learning ... well I'm not sure how that works, thanks for the comment Bazooka

[kautilya33]

I just want to say that I enjoyed your movie enormously. You have taken up the most fundamental subject of mathematics and explained it in a very entertaining way. Is this not the most sophisticated use of youtube? Two questions: How did you produce the beautiful white pictures on the grey background? How long did it take to make the movie? Thanks again for the spectacular work.

[CellarAcademic]

This video is made up of a bunch of Power Point slides. The picture of Cantor I got from Google.  I then imported the slides into Windows Movie Maker, along with the video of me.

I spent 2 evenings on the slides, and another evening on the movie. I'm glad you enjoyed my ten minute lecture. Your videos are wonderful as well. I believe I learned an equation from you:

Mathematicians + Ghandi = Mentor

[kautilya33]

My procedure for making the movie is the same as yours except that I use Keynote and iMovie instead of PowerPoint and Movie Maker. I am hoping you will come out with a lot more movies of the type you are making. I feel that education should be much more entertaining than what we have today. Here is my strong, but honest, view: There is no good mathematics a good high school student cannot understand. I believe that you have proved this by your movie. Thanks for the detailed reply.

[lizthar]

Awesome video!! My brain hurts now, I think it's bleeding. Honestly I didn't understand a single thing you said, but it was interesting. What is it you're always drinking in your videos??

[CellarAcademic]

I was drinking Scotch while making the video so that my brain wouldn't bleed, thanks for watching (and for the positive comment)