Godel's incompleteness theorum

I was recently listening to a podcast about Godel's incompleteness theory. I don't claim to fully understand it, but I do find it very interesting. As i understand it, his theory states that mathematics is not perfect and has contradictions. One such contradiction is the shaving problem. I don't know if we have any mathamaticians here, but we have a lot of people who know a lot about shaving. So I thought some of you might like to have a crack at the shaving problem.
So here it is.
There is an island and every man on this island shaves, there are no beards.
There is a barber on the island who shaves some of the men.
Some of the men do not shave themselves.
The barber shaves all the men who do not shave themselves.
The barber does not shave any of the men who shave themselves.
Because he is a man on this island the barber shaves himself.
But as the barber he does not shave any man who shaves themselves.
And so the contradiction

So can anyone make sense os this, or does it make your head hurt?

For me it is clear where the contradiction is, if that was the question?

"This paradox is often attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to him as an alternate form of Russell's paradox,[1] which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber's paradox was an instance of his own" - wikipedia

Quote:

---

Originally Posted by Bertrand Russell, The Philosophy of Logical Atomism

That contradiction [Russell's paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.

---

and now I could use a coffee

Is there a mathematical principle that the barber's paradox represents?

As it is, it seems to me that were asked to accept a false premise, then be confused when the rest of what's presented doesn't fit.
Or... It is only a paradox in the way its presented. If we accept how it is presented then we accept a non existent paradox.
I could spend all day writing puzzles like these, that are only paradoxical within the constraints written into the puzzle.

Right, and so, the question then is "How many on the island are SRP members?".

Very interesting stuff, thanks!

Although, it reminds me of those "deserted island threads."

You know what I mean.

"if you could take one soap to your deserted island what would it be???? lololL!!!"

"DUH MdC!!! LOLOLllllllloooll"

You calling me stupid ???? :thinking:

I am the barber!
Or the walrus, I sometimes confuse them.

Quote:

---

Originally Posted by gugi

I am the barber!Or the walrus, I sometimes confuse them.

---

Goo goo gajoob!

I don't see any problem.

The contradiction is there because someone put it there. If you say "the barber only shaves men who don't shave themselves" and "The barber shaves himself" then you've created the paradox arbitrarily. If the former statement isn't true (which it can't be if the latter is true) then the problem is solved, and as far as I can see it's an arbitrary, independent statement.

So basically the "problem" says "The barber doesn't shave himself, but the barber shaves himself."

*Edit: never mind. I missed the point and inadvertently agreed with the theorem.

Maybe the barber has extremely low testosterone and can't grow a beard or he uses Nair???or being a man maybe he just sets his beard on fire? If he uses Nair then he technically is not shaving! Lol

Do these guys have to send their blades out for honing?

This is not a paradox. This is an example of ill-defined sets.

James.

I would also like to see an example of a mathematical theory being contradictory

How about this one.
3 guys want to rent a boat. They each have 10 dollars a piece. So the get to the boat rental and ask the guy how much is it to rent a boat. He responds that it's 30 dollars to rent a boat. Well that's cool, they each have 10 bucks a piece x 3 guys = 30 dollars. So they give the guy the money and he goes into the boss. He tells the boss that here's 30 dollars for a boat rental. The boss says hey, there's a special today it's only 25 dollars to rent a boat today so here is 5 bucks give them back their change. So the guy is trying to figure how he is going to split 5 dollars between 3 guys so he decides to keep 2 dollars as a tip and gives each guy a dollar back. So each guy gets a dollar back so they really paid 9 dollars each. 9 dollars x 3 guys = 27 dollars plus the 2 bucks the kid kept = 29 dollars!! What happened to the other dollar????

Quote:

---

Originally Posted by Tsunami

How about this one. 3 guys want to rent a boat. They each have 10 dollars a piece. So the get to the boat rental and ask the guy how much is it to rent a boat. He responds that it's 30 dollars to rent a boat. Well that's cool, they each have 10 bucks a piece x 3 guys = 30 dollars. So they give the guy the money and he goes into the boss. He tells the boss that here's 30 dollars for a boat rental. The boss says hey, there's a special today it's only 25 dollars to rent a boat today so here is 5 bucks give them back their change. So the guy is trying to figure how he is going to split 5 dollars between 3 guys so he decides to keep 2 dollars as a tip and gives each guy a dollar back. So each guy gets a dollar back so they really paid 9 dollars each. 9 dollars x 3 guys = 27 dollars plus the 2 bucks the kid kept = 29 dollars!! What happened to the other dollar????

---

There's not an extra dollar. You were supposed to subtract the 2 bucks the kid kept to equal the 25 dollars it cost to rent the boat. Adding 2 bucks just adds an unnecessary and irrelevant element to the story!

True but you can not solve the problem with multiplication. Only with addition or subtraction

Quote:

---

Originally Posted by Jimbo

This is not a paradox. This is an example of ill-defined sets.

James.

---

It certainly (in my understanding) does not have to do with Godel's incompleteness theorem.

Quote:

---

Originally Posted by TwistedOak

I would also like to see an example of a mathematical theory being contradictory

---

If this is related to Godel's incompleteness theory, it doesn't talk about contradictions.
Instead, it says (proves) that within each non-trivial axiomatic system, there will be propositions which cannot be proven or disproven either way. In other words, the axiomatic system is incomplete. And indeed, in mathematics, there are propositions which can be identified as unprovable. The math is beyond me, but there are classes of problems that within a certain context, can be proven to be unprovable either way.

My head hurts.