Kurt Godel: The World's Most Incredible Mind (Part 2 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.

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

Category:

Science & Technology

Tags:

• Kurt Godel
• Incompleteness Theorem

[agentredlum]

This is all very nice and mathematics needed a sense of humility which Godel provided, but if you apply kurt's ideas to his own proof you must conclude his proof is also inconsistent. So what now? You have an inconsistent proof that all mathematical systems strong enough to give rise to arithmetic must be inconsistent...essentially we are back to the liars paradox.

[globalbeehive]

@agentredlum Does this mean that the only consistent complete system is Nature (ie God)?

[trumpgeek]

Godel's theorem does not prove arithmetic is inconsistent, rather it is talking about IF you have proven that arithmetic is consistent, then you have proven it not using arithmetic. The speaker has overstated the case in 13:39 mark. The shock to the mathematics community was not that, the shock was that it could not have a consistency proof. The shock is - we do not KNOW IF arithmetic is consistent or WE CAN NOT KNOW if it is. So you are using arithmetic out of faith not out of certainty

[Gevzh]

@agentredlum But inconsistency is not necessarily composite. Godel's proof is metamathematical, and the metamathematics of an inconsistent mathematics need not be inconsistent (if it is inconsistent). For one, metamathematics does not (at least, prima facie) take place in a formal system, which is what Godel's result applied to. If, like Godel, we're not functionalists or computationalists, then our minds, being informal, may have all the properties which do not obtain in a formal system (by G)

[nahaymath]

@globalbeehive

Doesn't the statement "A is provable" really mean either "can prove A true" or "can prove A false"?

Hence, if A = "A is unprovable", can't we also include "A is false and we can prove A false", as well as "A is true but unprovable"?

Did I miss something? (Sorry - I had to skip around.)

[globalbeehive]

@globalbeehive Where "logical systems" are merely "sets"

[globalbeehive]

@Sturkel1292 So, what we have is "Man" developing logical systems that are inconsistent, but that are being presented to the Common Man as consistent. Essentially, "Man" is playing God.

[astroboomboy]

@globalbeehive Well, nature does not seem to be consistent either. Take for instance the fact that anything is rather than not being. Another thing is the peculiarities of QM, like virtual particles that arise and disappear, from nothing to nothing. Maybe it is nature (ie god) that is inconsistent, and that is the reason we are as well as all our formal systems. Inconsistency might be the way everything that can be is. The only thing that is consistent might be nothing, that which does not exist

[erroll9621]

the idea of a god is highly more inconsistent in itself.