Proof of Sklansky's theorem?

[morgan]

"As with Fermat's the four-color theorem is very old. No formal proof exists. It has been proven via computer however."

It is proven. Just not in a very aesthetically appealing way.

Morgan

[morgan]

"It may be a surprise to a lot of people to learn that not all things are either true or false. There are some things that are neither true nor false, but in a third state of "undecideable", and you can prove that."

It certainly surprises me. That any proposition is either true or false is a basic assumption. Without it we couldn't do much. We wouldn't even know if the square-root of 2 were irrational or not.

"Consider the set of all sets that do not contain themselves. Now note that this set contains itself if and only if it does not contain itself. This is a breakdown of so-called "naive set theory" which motivated changes to both set theory and to logic, and indicated the inherent inconsistency of all logical systems."

It did not indicate the inherent inconsistency of all logical systems, just of naive set theory. No one knows if the currently accepted axioms are inconsistent or not, nor will we ever know.

Morgan

[Kim Lee]

Amusing and amazing, mathematicians believed it is self-evident, but they weren't really sure what was true. In other words "it must be true for some space of numbers, possibly the complex numbers, but we just haven't figured out what that space is!"

Actually people have similar confusion about FTO Poker. They believe opponent mistakes are "beneficial" but don't know how to define mistakes. For example, they think you don't benefit from opponent mistakes if your opponent sucks out. It is important to define "mistake" with respect to the same information you use to define "benefits". If you want to assume knowledge of the turn and river then sucking out is never a mistake. If instead you assume only knowledge of the flop then FTOP still holds based on EV.

[Ralle]

That's an interesting link, AceHigh. It's amazing that there is so much disagreement about such a fundamental aspect of the game. I guess it illustrates one of the reasons why the game and the theory behind it so fascinating.

[rkiray]

Oh my,

I find this post very much hard to read. To my mind Godel's Therom is much easier to explain. Any logical system that claim's to be complete cannot handle the statement "This statement is a lie".

Rick

[rkiray]

You need to read the bood "Goedel Escher Bach" it is one of the greatest books ever writen and is easy to read. If you read it, you will see Bruce is correct. I believe it won a Pulziter Prize or something similar.

[tewall]

This is an interesting characterization. Did you think of it yourself?

[rkiray]

I'm actually going to start a string on the Books/Sw board about the book Goedel, Escher, Bach: An etneral golden braid. A wonderful book. That is the summary I have for the main point of the book. Another point is that I believe it was Edward G. Land (the inventor of the polaroid camera) who said that you don't really understand anything if you can't explain to a 10 year old. (I think he said 5 year old, but I think he's brigther than I am). That post would be how I would explain Godel's theroem to a 10 year old. I wish I started a new thread on all this, since I think it's fun stuff to talk about and will be buried.

Rick

[morgan]

I have read it. Bruce is not right. If you doubt me attempt to prove that the root of 2 is irrational without assuming that any proposition is either true or false.

I believe there is some confusion. What Godel proves is not that there are some statements that are neither true nor false. He proves that there are statements which are impossible to prove either true or false. That the statement does have some truth value is not an issue. In fact it is an assumption that it does.

Morgan

[BruceZ]

That's right, I am not right. See the the new thread I have added to address this.