Dan Says - Introductory Computability Theory - Turing Machine Introduction 2

[eggistential]

I love your videos...I was wondering if you could make a video that explains the "incompleteness theorem," and why you can't prove everything in arithmetic? That REALLY intrigues me!!! Could you also explain the "internal contradictions" that you describe? Of course, could you explain it so I (a Calculus I student) could understand : )

It is alot to ask to make a video but I had to ask.

Also, do you know of any great books that deal with the philosophy of mathematics?

Thanks

[Entertainmentwf]

Aww thank you I really appreciate that :). I'm sure eventually I will get to making a video describing such paradoxes in such things as that which give us these bizarre results in mathematics (which give us the very limits of mathematics).

I would recommend if you want a book that relates to the 'philosophy of mathematics', "Godel Escher Bach" would do a good read to kind of introduce you to the concepts. I'm sure when I get time I can make a one on the I. Theorem by Kurt Godel :).

[Entertainmentwf]

A basic example to 'internal contradictions' would be how does one prove the basic rules of arithmetic (addition, subtraction, and so on). How would one go about doing something like this. Godel mapped the arithmetic in number theory to something called Godel Numbers and using these he basically applied a rule that essentially said 'this statement cannot be proven' and how would one try to prove this? If you assume it is 'true', the statement can't be proven making it incomplete.

[Entertainmentwf]

That's it in a basic sense but, if you want some good resources I would highly recommend getting used to mathematical logic and set theory then trying to look up books on Godel, and any book you can find on the Limits of Mathematics (some basic ones I find that are handy are ones by Gregory Chaitin on the subject (his books mostly are his lectures written down)). This field branches off into really deep stuff. If you want a good documentary on this look up "Dangerous Knowledge" (it's a BBC one)