A Century of Controversy Over the Foundations of Mathematics

Absolutely fascinating read. I think I feel the same way prehistoric man would feel looking at nature. Its a daunting task, but it feels like our journey has just begun just like our ancestors; and its getting more beautiful. I wanna hear Bach now.

Thank you, a very interesting lecture. The process physics he is linking to are maybe even more interesting. Everything based on random fluctuations ... hmm I'd better learn the statistics good!

I've been wondering about the Russell Paradox...<p><pre><code> S = the set of all sets not members of themselves x = 1 / 0 </code></pre> Isn't the problem for x the same as for S? Not all mathematical expressions are well-defined, and likewise for all "set expressions".<p>Aside: There was a wonderful quote in Jaynes' <i>Logic of Science</i>, decrying the kind of airy mathematics that Chaitin is doing...<p><i>Should one design a bridge using theory involving infinite sets or the axiom of choice? Might not the bridge collapse?</i>

Great link! I've read about a lot of this stuff in text books etc, but it all fits together so much better when put in its proper historical context like that. If I understand it correctly, his main result in algorithmic complexity theory implies that it is impossible to prove the absence of a pattern in data. Does this thus imply that there is no way for us to know when we know everything?

"So 'this statement is false' is false if and only if it's true, so there's a problem."<p>I don't understand this. This seems to assume that the meaning of a phrase is contained within the phrase itself. But if you instead assume that the phrase is a pointer to meaning that is contained somewhere else then the paradox goes away.

Is this correct in set theory?<p>ω/ω=1<p>1/ω=ε<p>ε*ω=1<p>ω=1/ε<p>If we assume for every positive number there is only one negative number then this must be true: +∞/-∞=1