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Kurt Gödel: The World's Most Incredible Mind

Part 1:

Fast forward at minute 5 if you want to skip the bio and get to the meat of the matter right away:


http://youtu.be/i2KP1vWkQ6Y

Part 2:

(first incompleteness theorem at 9 mins 40 secs : there is a sentence G such that, if the system is consistent, then G has the form : "I am not provable.")

http://youtu.be/bgvxVUyVdXk

Part 3:

(commentary on the significance of Gödel's results)

http://youtu.be/W4-O7j169D0



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Gödel's Proof

Gödel's Proof Paperback

Ernest Nagel (Author), James Newman (Author), Douglas R. Hofstadter (Editor, Foreword)

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814...

Thomas Jefferson 1796, 1800, 1804; James Madison 1808, 1812; Ron Paul 1988, 2008, 2012; Rand Paul 2016.

Galileo discover the stuff

Galileo discover the stuff 250 years before Hilbert and Cantor discussed here, Galileo understood infinite sets long before anyone else:

http://en.wikipedia.org/wiki/Galileo's_paradox

The Paradox of the Infinite Series of Square Numbers by Galileo
http://www.decodedscience.com/the-paradox-of-the-infinite-se...

Thomas Jefferson 1796, 1800, 1804; James Madison 1808, 1812; Ron Paul 1988, 2008, 2012; Rand Paul 2016.

Top three most brilliant

Top three most brilliant minds:

1. Galileo (1564-1642)

http://en.wikipedia.org/wiki/Galileo_Galilei

2. Evariste D. Galois (1811-1832)

http://en.wikipedia.org/wiki/%C3%89variste_Galois

3. Frank Plumpton Ramsey (1903-1930)

http://en.wikipedia.org/wiki/Frank_P._Ramsey

Thomas Jefferson 1796, 1800, 1804; James Madison 1808, 1812; Ron Paul 1988, 2008, 2012; Rand Paul 2016.

Not really all that impressed

At the 10:04 mark, the narrator brings up a "very, very old philosophical puzzle," the Liar's Paradox, which simply states, "This statement is false."

He should have talked to a seventh grade English teacher (yeah, I know he was Austrian, but you can make the analogy). The word "This" can only rightly refer to a previous or subsequent statement. It cannot refer to itself. And since there is no other statement provided, it is an exercise in futility.
The narrator completely misses that and says that there are only "two possibilities, either it's true or it's false."

These guys think they're smart? Ha, he even admits around the 10:58 mark that "Philosophers have worried about this for a couple o' thousand years; a-and to be quite honest, we're no closer to a solution than we were 2,000 ye..."
(he interrupts himself) He then follows that statement with: "But we've had a lot of fun!"

This is an example of one of the troubling things I see on occasion here on the Daily Paul - people who like to argue for the fun of it. They like a mental brawl, and have no interest in truth, much less, absolute truth. In fact, I'm afraid there are many who do not even believe in absolute truth. Just because someone makes a claim of absolute truth that is later proven wrong does not mean there is no absolute truth. It just means they have not found it yet. But the arguing for no purpose other than to rile someone or to call someone names, or to give a retort that in their mind "scores" in the minds of others of their mindset, is unproductive to our goals of restoring liberty.

The more I listen to this guy, the more this comes to mind:

Romans 1:22 Professing themselves to be wise, they became fools,

Freedom is the ability to do what you want to do.
Liberty is the ability to do what you ought to do.
"Where the Spirit of the Lord is, there is liberty." 2 Corinthians 3:17

Numbers are man-made...

all corresponding theories, proofs, analysis and formulae are consistent with man-made numbers/factors. Before the capacity of a species to harness numbers, mathematics was complete by the expressions of nature itself. For all out thinking and contemplation on the subject, man/woman created the symbolism that are numbers and thus the gateway to what this man conveys. We prove what we have created.

Father - Husband - Son - Spirit - Consciousness

"man-made numbers/factors"

Interesting phrase. Just yesterday I was sitting with a dog who just had eight puppies, and as I counted to make sure they were all there and none had wandered off, I wondered, "How do dogs count and keep up with their puppies?"
I doubt they count like we do. I think they just know. But, what do I know?

All I know is that there is more knowledge out there than we can know; therefore, we need to focus on what we can know. That's where our responsibilities lie.

Freedom is the ability to do what you want to do.
Liberty is the ability to do what you ought to do.
"Where the Spirit of the Lord is, there is liberty." 2 Corinthians 3:17

This series needs to include...

http://youtu.be/9qDc22UfKy8 "Carl Sagan and Kurt Godel" to be complete.

My take on Godel's Essential Incompleteness Theorem is that no finite set of axioms can capture all the truthful statements that can be made about any given infinite set. Stated like that, it doesn't really seem all that remarkable or surprising to me. The truth about infinite sets is simply incompressible into a finite set of axioms. Seems perfectly reasonable to me.

The way to cut a lot of the Gordian Knots that get entangled with these questions is to ask yourself,... is infinity real or merely an artifact of our all too human imagination? If you subscribe to an absolutely literal materialism then there exists no infinities, in the sense of being "real", so the paradoxes and problems all go away. Maybe in raw physical reality nothing larger than say 10^300 is actually "real". Of course, this would also mean Physicists would have to modify their theories of quantum mechanics and such, in response.

"The dearest ambition of a slave is not liberty, but to have a slave of his own."
Sir Richard Burton

"None are more hopelessly enslaved than those who falsely believe they are free."
Johann Wolfgang von Goethe

Time is weird

Is it not always now? Now is always giving rise to now. We say there is a past, but that is only residue/memory left on the brain by experience. Time is movement, it is a measurement only made by having a starting point. All physical things are moving, noticeably or not. But all movement is current, for there is only 'what is', is there not the 'universal moment'. As I blink is not the furthest existence of the universe also moving at the same time. Is not everything moving in the moment? Again, is it that the moment gives rise to the moment? And if time is only a measurement of movement, is there an activity that is not 'moving', if there is it could not be physical, which means is could not consist of sound; it would be silent, it could not consist of time: it would be timeless, but it does not exclude space, for space does not necessarily imply movement. If one is to find such a state of being, must it not in some way be connected to brain? Is thinking the movement of the mind? And what is the space of the mind that is absent of thought?

Enonesoch

Godel proved you cannot prove

Godel proved you cannot prove everything. For example, it is impossible to prove Goldbach's Conjecture.

Thomas Jefferson 1796, 1800, 1804; James Madison 1808, 1812; Ron Paul 1988, 2008, 2012; Rand Paul 2016.

On a related note

Titus 1:12 One of themselves, even a prophet of their own, said, the Cretians are alway liars, evil beasts, slow bellies.
13 This witness is true. Wherefore rebuke them sharply, that they may be sound in the faith;

Thanks again Cyril.

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.

Mathematics

"the science of that which is or is not"

Is it fair to say then that number one is "that which is"?

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.

About half way through part 2 it dawned on me

that a component lacking from the liar's paradox, "this statement is unprovable", assuming mathematics is consistent, infinity, and any fundamental difference between minds and machines is a function of time.

Cyril's picture

Well, as I see it, that's the

Well, as I see it, that's the very most humbling thing for hard sciences and what's usually regarded as the purest of all, namely maths:

even if one would bother (or, in some sense, dare) to explicitly model time itself as an orthogonal dimension of a logical framework to gauge about their statements' truth, would it have any invalidating impact on Godel's findings?

The quite amazing thing to me is that does seem unlikely, since even Cantor's orders of infinity haven't proven to be tough enough to resist Godel's theorems; which don't even need to rely upon anything stronger than the simplest infinity (of the natural numbers).

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

one of the most interesting

one of the most interesting courses I took in university was Computability. We covered Godel, Turing etc... Very cool stuff. Going through the proofs and the reasoning was tough but the exams were easy :)

"My theories explain, but cannot slow the decline of a great civilization. I set out to be a reformer, but only became the historian of decline."
- Ludwig Von Mises

bump for Godel!

One of the most significant achievements of the 20th century. Godel killed Leibniz's dream. Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion. Alas, the universe disagreed. To generalize his conclusions to outside of mathematics,Godel showed that no system can fully understand itself from within itself.

Also relevant to this is Alan Turing's discovery that the halting problem is undecidable over Turing machines. In essence, this showed that there is an infinite number of undecidable problems that cannot be solved with a single algorithm.

“The welfare of the people in particular has always been the alibi of tyrants.” — Albert Camus

Cyril's picture

Yes, indeed. Ditto on the

Yes, indeed. Ditto on the usability and intrinsic limits of any axiomatic, by very construction/definition.

And for whoever is interested to learn more about this outstanding yet little known (still) achievement of the human mind, Godel's, I also warmly recommend to dig into how Godel proceeded in his proof, and in particular, his clever Godel's numbering scheme.

Oh, and also relevant, beyond Godel and Turing:

Church's works, too, and Curry, with the Curry-Howard correspondence.

And keep being introspective, too. ;)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

So even the libertarian

So even the libertarian "system" cannot understand itself within itself.

of course

The libertarian "system" is based on some axioms. The axioms cannot be evaluated within the system. We have to agree on the axioms first, before we discuss the consequences of the axioms.

“The welfare of the people in particular has always been the alibi of tyrants.” — Albert Camus

Bump

I was taught to close my mouth when chewing. So until I get a few more hours under my belt I'll do just that. But I will say this interests me because I read in an old(ancient really) book about the "beginning of knowledge" or the "key of knowledge" or the "greatest commandment," as well as the fact that "man cannot find out the work that is done under the sun."

That book hit the nail on the head if you ask me.

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.

I am sorry, but why did you post this?

Why did you post an intro to discrete mathematics? While I do appreciate higher level math, I do not see what this has to do with on these forums.


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http://youtu.be/_ePTZezN_Iw

I'd rather have a bottle in front o' me than a frontal lobotomy
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Cyril's picture

I posted this because while

I posted this because while everybody (or almost) heard about Einstein's works, very few people have had about Godel's and even fewer only remotely understand the full extent of implications of his discovery.

His legacy truly deserves better recognition, IMO.

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

He Did Post It...

...in the Science & Technology forum category, which seems appropriate. After all, that category is an option on the DP for folks who are interested in Science & Technology. Frankly, the information talked about here is extremely useful to the pursuit of truth and reality -- values that most libertarians and Ron Paul supporters hold dear. Surely we are not confined, as a community, to the political and popular in our search for truth and reality. I actually appreciated this post as a contribution to the "big picture," hope of freedom and understanding among us all. But then, I am one who enjoys science & technology. Different strokes for different folks.

Thanks Cyril -- good stuff.

Bump

..

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.