If mathematics can prove that a question cannot be answered from its own axioms… is that question truly unsolvable — or are we simply using the wrong axioms?

What do you think? 👇

Thank you. I took a Set Theory class from Paul Cohen in summer, 1995. It was like free climbing a vertical polished granite wall. Every couple of days, I felt that I understood a little.
I was playing chess with friends in the campus bar and he sat with us. I asked if he wanted to play in and he said :"I don't play games". Makes sense; with that mind, playing recreational chess would be like taking a Formula 1 car on a grocery run.

A little bit loose on the wording sometimes, but a good story.  Cohen did not prove it could not be proved, he proved that the negative of the proof could be consistent with a set of axioms that are not inconsistent with ZFC.  Yes, that is tangled.  Of course Cantor never used ZFC, it came after to try to formalize the question, at least for limited purposes.  It would improve the story to tighten up the language and talk to current mathematicians and what they now do with forcing - and how that reflects back on Cohen, Gödel, Cantor, and the continuum hypothesis, and maybe even on the continuum itself.  There will likely be more to this story, someday.

Thank you so much Sir... The content was mind boggling and very inspirational...

Really well done. Got me to subscribe. Keep up the good work!

I wish I had the framework to understand this. However I do get how profound it is.

I think I read the outline of a proof that Kurt Gödel wrote.  I have to say my head was spinning.  I only read through it once and quickly but my immediate thoughts were, “Either this guy is completely full of s—t OR I’m not qualified to give an opinion on it.”  I mean the logic was very tricky to follow and it was quite complex and following it was nearly impossible for me.  But I had the feeling that unfortunately, the latter option seemed to be the most likely.  If so he was, well, kind of a smart guy.  Maybe in a Good-Will-Hunting kind of way.

The story of "forcing". Cool!

He is my hero!!!!

Keep it up.

I suppose if you take enough math(s) classes you’ll eventually realize that a proof is sometimes more valuable than someone’s solution.  The movie, “The Man Who Knew Infinity” (MKI) — I give it a mild recommendation as it has its flaws (I read the book first) — touches on this idea a bit.  But the funny thing is, I guess a lot of the time people who write a proof for a solution often spend more time than the guy who offered the candidate solution.  For example, I guess that today mathematicians are still working on proofs for a lot of Ramanujan’s equations and Ramanujan died over a hundred years ago.

These questions are completely similar to the question of provability of Euclid's Parallel Postulate from the first four axioms of Euclid. The construction of Euclidean geometry establishes that you cannot prove the Fifth Postulate is false and the construction of non-Euclidean Geometry establishes that you cannot prove the Fifth Postulate is true. This is not a conundrum, it simply says the axioms are insufficient to establish every model must have said property (which is equivalent to provability). Godel's deep interest (11:50) in whether the continuum hypothesis was "really" true meant he wanted to know if basic ZF axioms were missing a fundamental, obvious "truth" (that could be added to the axioms) that would allow the continuum hypothesis to be proved. This is entirely in line with the thinking of Euclid - is there an "obvious" geometric axiom that could be added so that he could prove the Parallel Postulate. Euclid was thinking of "Truth" as a physical model - the Universe - which Einstein seems to have settled as the real Universe is non-Euclidean. Godel was thinking of the abstract mathematical universe - was there an "obvious" mathematical truth about sets that could be added to ZF that would allow the continuum hypothesis to be proved. Cohen, interestingly, thought of it as meaningless - at least in the case of the mathematical universe - since mathematics was a game, not a physical reality.

fascinating agin!

Cohen was a prodigy, like Feynman he was also a Bronx Science alum.

配信を、ありがとうございます。

Godel is an absurd proof finder. Gotta be the best ever

Part of the problem here is the insistence that it be either true or false. Forced binary choice is a fallacy, and possibly a crime against humanity. Gödel himself responded to his own incompleteness theorems by creating a 3 valued logic (we was close to RM3 but missed). Three-valued logic solves most paradoxes. It solves the Liar Paradox (the answer is both true and false). Cantor found numbers like 0.1001010101B1001001... where B is on the Cantor diagonal, it's both 0 and 1. People run screaming from inconsistency, yet, you can make a completely usable logic that DEALS with it, without exploding like binary logic

I never completed my PhD in math but I had begun my dissertation on using Gromov’s ideas on manifold convergence to use on optimal design of curvature and volume constrained compact manifolds. As an undergraduate I purchased Paul Cohen’s paperback book. Strange, I am near positive I still have that book titled “The Independence of the Continuum Hypothesis” and when I tried to find a reference, all I found was a seven page article in PNAS. Anyway, I was typical of most grad students in that topology, analysis, and geometry(except algebraic geometry) were easy but set theory, logic, and algebra made my eyes bleed. I really appreciate this video because I kind of understood what was said. I am very interested in the philosophy question as to whether formal mathematics is a dead end in addressing ontological questions like the measurement problem in quantum field theory. Physicists, for the most part despise philosophy so that isn’t going to raise any eye brows there. It took over twenty years for mathematicians to prove that Dirac’s delta function was valid. This would be relevant for the analytic vs synthetic divide in philosophy where mathematical truth is analytic and unassailable truth.

This is pretty good, but Cohen's method is remarkably intuitive and can be described for lay audiences.

I understand sets, but the fact is infinity is not a number.