Gödel's Incompleteness (extra footage 1) - Numberphile

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Professor Marcus du Sautoy is Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.

Professor du Sautoy's book as mentioned...

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Videos by Brady Haran

[MultiSkidding]

guess you could say the original video was. incomplete

[MMrandomdude]

This guy is really good at explaining stuff.

[twistedsim]

He died because he couldn't prove that the food was poison or not....

[OlafDoschke]

That last bit got me as astonishingly self-referential. The fear of death by poison causing death by starvation. Kind of feels like a made up legend.

[OlafDoschke]

The german word you didn't remember at 10:02 was "wissen": "Wir müssen wissen, wir werden wissen."

[unvergebeneid]

Speaking of infinity and Gödelization, it is also noteworthy that every mathematical statement will map to a natural number and therefore the entirety of mathematics is countable. So whenever one encounters an uncountable set, mathematics can't describe every individual member, only the set itself.

[sevrjukov]

Remarkable series on Gödel! Thank you again, Numberphile, for crunching very hard math topics and making them accessible to regular people.

[PersimmonHurmo]

Godel seemed like a maverick, proving paradoxes and shaking the very fundamentals. Even his death was extraordinary!

[zorrozalai]

Nice video. I have a feeling, that it's a little bit incomplete, that something is missing, but I can't prove it.

[rentzepopoulos]

I have not read Gödel's work and probably I am not in position to do so, but while viewing this video, a question came to my mind: Could Gödle's coding actually be introducing incompletness? I mean, could the outcome of his work is exactly the result of some characteristics of this coding? On the other hand, if this is the case, then the inability of mathematics to describe itself may be a proof of its incompleteness in the first place...

[combodemo]

Kind of surprised the halting problem wasn't mentioned when he talked about going to other fields to see if they had acknowledged "limitations on what they could possibly know." That's essentially what the halting problem amounts to in terms of computation theory(I don't want to stretch too far and say CS) and is something any intro to CS course would at least mention, I think, and probably the easiest example of it in another field as an example of a fundamentally unanswerable question.

[antoniozumpano]

How we know if the system that Godel prove incompleteness is consistent?

[Gunbudder]

stop posting these in the morning brady! i have to go to work!

[sarchie]

The question about what happens if your theorem is undecidable, or how will you know has already been covered to some extent. Euclid's 5th Postulate and the Continuum Hypothesis are both formally undecidable within the mathematical system. It has been proved in each case that they are independent of the remainder of the axiom system. These undecidable propositions then give us options in terms of how we progress (as alluded to in the video). In the case of the 5th Postulate we proceed in one of Euclidean geometry, spherical geometry or hyperbolic geometry. I'm not aware of any work having been done based on different options related to the continuum hypothesis, but there are surely choices that can be made and there must be consequences of those choices.

[telnobynoyator_]

Maths too complex ? Just add more abstractions

[adlsfreund]

Gödel starved himself to death? Wow. What a story, Marc!

[patrickwienhoft]

I'd love a full video of the Axiom of Choice with him!

[ponchout]

its like looking at your eyes with your own eyes (without reflection)

[dnickaroo]

Godel's Theorem is more about the Incompleteness that results when using the Axiomatic Method: he proved there MUST be Theorems which are true but that they cannot be proven to be true using a given set of axioms. Godel did not believe that this was the final word -- humans may agree to loosen the rules of inference and accept the result as a reasonable proof (here he was drawing on Plato's approach to Mathematics).
Goodstein's Theorem illustrates what Godel was talking about. The statement of the Theorem only involves the Integers with Addition and Multiplication. Peano's Axioms gives us this part of mathematics. Goodstein's Theorem was proven about 1944; however it was subsequently shown that it could not be proven using only the Peano Axioms (which does seem to be surprising).
To prove Goodstein's Theorem one needs to use Transfinite Numbers (which need additional Axioms to the Peano Axioms).
Goodstein's Theorem does have some practical uses: it can show that certain Computer Programs will come to a conclusion, rather than continue for ever. Turing had a theorem concerning Computers: that a Computer itself cannot judge whether a reasonably complex program will come to a conclusion or not.

[Bladavia]

I really like that last statement about Gödel's death. Reminds me of Nietzsche who ended up completely psychotic, John Nash, etc. There's definitely a fine line between madness and genius.