People are answering this question based on the climate at the time. I was thinking more along the lines of how likely is it that someone else could have come up with his technique. Put differently, if there were a 100 million dollar prize to do this work offered in 1930 and you could not collaborate, how many people, working independently, could have claimed the prize in 1935?

Quit a lot would be my guess.

It may be a famous problem but it was relatively straightforward to solve. Quite unlike Fermat's last theorem in that respect.

At least 1935.

Inducement is subjective. Shrug. I do see what you're trying to gauge though. It is a reasonably level playing field coming out of the womb.

Words to live by.

If it is the case that you left off an e on the first word, I protest.

If I touch this without a thesis I will be discredited by proxy in the areas I am focusing on this run.

Suffice to say.

If the brain processes like a computer, which I believe it does;

Processing the probabilities of reality staying the same each time I blink would no doubt shut me down, cannot know it will, just kinda trust my heart that it does.

An extension of this on the everything happening for a reason line is to propose that in order to stimulate work instead of stagnating due to the 'oh it's all happened before' line a piece needed to be released that would act as safety net of deniability for the heavily maths focused runs.

I suggest that the work is only released on stagnated lines as a universal defensive mechanism to prevent technology not being in the range it needs to be to prevent incidental genocide.

That's enough forums for me today. Good day.

I

Chelsea

Yeah. The Halting Problem is just a rehash of incompleteness and its almost inconceivable to imagine a universe were we have cell phones etc and don't understanding the Halting Problem.

It seems shocking to me that incompleteness wasn't discovered sooner. Its really just a fancy version of Cantor's diagnolization and its really really simple from a technical standpoint.

Humans consider themselves unique so they've rooted there whole theory of existence on their uniqueness. One is their unit of measure, but it's not. All social systems we've put into place are a mere sketch. One plus one equals two. That's all we've learned, but one plus one has never equaled two. There are, in fact, no numbers and no letters. We've codified our existence to bring it down to human size to make it comprehensible. We've created a scale so that we can forget its unfathomable scale.

As long as you mention it, isn't it even more shocking that Cantor's proof didn't come a lot sooner?

I dont find it all that shocking since diagnolization seems pretty dam original to me. I guess you could say the ancient greeks should have got there from knowing that root 2 isnt rational....but that doesnt seem reasonable to me.

But the all time leader for shocking it didnt happen sooner is arabic style counting/numerals . Its so intuitive thats its one of the first things we teach young children....but somehow people like Euclid and Archimedes didnt figure it out

Grothendieck has a great quote on this

I thought of this also when it was mentioned. Did not the discovery of irrationals cause the Greeks some mathematical/mental crisis? or philosophical disconnect and dissonance? Or at least that was the opinion of old Bertrand Russell. Perfect symmetry being a holy and sacred thing to the Greeks I think.

Yeah....i think it was especially infuriating that the hypotenuse of a right triangle with 2 sides equal to 1 cant be expressed in the rationals! Such a natural number that really cuts to the heart of the smallness of the countable when compared to the continuum. But ithink there are almost superhman leaps of imagination needed to get from irrationals to uncountabiliy. Way more than required to get fron uncountability to incompleteness

Not necessarily. Sqrt(2) is computable, and computable numbers are countable.

Fair enough...you can group root 2 along with the integers consistently and call those numbers countable. But that wasnt how it happened historically and i dont think thats what people mean when they say countable

Sure, and I presume that some of the Greek fury you mentioned persists to this day via the use of the word "irrational". But the only difference between the sequences 0.3333333333... and 1.4142135623... is that the rule for generating the second is more complicated than that for the first.

Sure...but more complicated looks more arbitrary than expressible by integers a and b in the form a/b

I think you could take 100 bright math undergrads who've studied up to that point and be a favorite to get something equivalent to the diagonalization proof if you gave them all a week to do it as extra credit. It's just glorified contradiction-by-failed-pigeonholing.

Any math concept that can be understood quickly by intelligent ten year olds should not be considered as impressive as the tougher stuff just because it took cleverness. If there was a gun to everyone's head I'm fairly certain that far more people would have independently thought of proofs of the infinity of primes, the irrationality of the square root of two, the divergence of the harmonic series, the uncountability of real numbers etc than would have come up with Godel's proof during the same years that these things were done.

Understanding something once explained is different than coming up with that something independently.

By "different" I mean "only similar in the way that elephants and poker are similar."

I'm fairly certain you think this because you understand the irrationality of root 2 etc but not Godel's theorem.

And the whole gun to the head thing misses the psychological difficulty of solving certain problems. Hilbert didn't prove incompleteness because he assumed the opposite was true and got stuck trying to prove that, he didn't prove the Poincare Conjecture even though he understood it was true because it was too difficult.

There are six year olds that understand the mathematical concept of primes. Therefore, primes are unimpressive objects to consider.

I meantt "proofs", not concepts.

I still don't get why you think this. Do you think incompleteness was some sort of technical tour de force that only research level students can prove as Godel did? It isn't.....it's routinely taught to freshman a few months from high school.