TML: The Infinities In Between (1 of 2)

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?

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)

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.

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.

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.

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

This was very exciting, inspiring. Thank you!

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.

thanks, I agree that math is pretty amazing

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