Kurt Godel: The World's Most Incredible Mind (Part 1 of 3)

Kurt Godel: The World's Most Incredible Mind.

"Either mathematics is too big for the human mind or the human mind is more than a machine" ~ Godel

Kurt Godel (1931) proved two important things about any axiomatic system rich enough to include all of number theory.

1) You'll never be able to prove every true result..... you'll never be able to prove every result that is true in your system.

2) Godel also proved that one of the results that you can never prove is the result that says that the system is consistent. More precisely: You cannot prove the consistency of any mathematical system rich enough to include the known theory of numbers.

Hence, any consistent mathematical system that is rich enough to include number theory is inherently incomplete.

Second, one of the propositions whose truth or falsity cannot be proved within the system is precisely the proposition that states that the system is consistent. "

What Godel's proof means, then, is that we can't prove that arithmetic—let alone any more-complicated system—is consistent.

For 2000 years, mathematics has been the model—the subject—that convinces us that certainty is possible. Yet Now there's no certainty anywhere—not even in mathematics.

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http://teachingcompany.12.forumer.com/viewtopic.php?t=2327

Category:

Science & Technology

Tags:

• Kurt Godel
• Incompleteness Theorem

[NewInfinityRecursion]

@Vikster93 Why you think you arent a computer computer can simulate phisics you work on phisics and so :) you can be simulated in computer :) just big computer fast computer :) so you brains are the cpu of the computer :) computer can simulate the computer :) try to write someting that isn't so stupid :)

[Vikster93]

@NewInfinityRecursion no u, dumb ass.

[suitabledude]

@NewInfinityRecursion Clearly, Godel showed we're not.

[rotweissrot100]

"Bruce here's in charge of the sheep dip, he's also teaching Gödelian philosophy."

Australia! Australia!

[DarkwingScooter]

@HebaruSan If you start out with pure logic you cannot do things like a=a+1. Pure logic doesn't admit the concept of infinite procedural truths.

This goes back to something called the paradox of analysis. Either it is true or it tells you something useful, not both.

a is a

a is a+1

then a+1 is a

If a is 1 then a+1 is 1

Unless you want to claim that 1=2 you cannot hold the proposition a+1=2. This is prohibited by the law of the excluded middle.

[NewInfinityRecursion]

we are computers.

[HebaruSan]

@o0thisismyusername0o I don't know how that would work--I haven't taken math classes where infinity can be assigned to a variable. But it isn't a problem for the example I gave, because you can just add some extra qualifiers to the definition to rule it out ("let a be a finite integer such that a=a+1").

[o0thisismyusername0o]

@HebaruSan but what if a = infinity?

[wmcoach]

dude there is no paradox..its very clear.This sentence is false.(lets say statement x) .if its true then statement x is true that is " this sentence is false " is correct in whatever it implies. if statement x is false then whatever statmnt x says is false , which means " this sentence is false "is incorrect in whatever it implies.Its all about the syntax of the language of statmt .try this with the following statement: "This Sentence, that i am god is false" .And you should be understand better

[33tracker]

Or put another way: ask yourself the question "Who am I"? (while simultaneously remembering that the subject cannot be it's own object).