Don't know if this has been done before but:

This statement is false. True or False?

Answer carefully... :D

yeah, done, many times

The classic Epimenides Paradox.

Epimenides was from Crete and made the statement, "All Cretans are liars."

Self-reference and Godel's Incompleteness Theorem. Sweet, sweet stuff.

It is unanswerable, of course. It's grammatically incorrect, as a matter of fact. Such is the foibles of language.

Godel changed it to a different sort of self-reference. Or, rather, he made it more distinct so as to examine it in a better light.

"This well-formed formula is unproveable in this formal system."

This is important. Language is not a formal system and thus is the conundrum raised by your example brushed aside. There are no rules of language that say that self-contradiction is against the rules. Contradiction, in language, is part of the general terrain and something that we, in fact, learn to live with so well that we often fail to see the vast majority of contradictions that grace our presence each and every day.

But, when applied to mathematics, which is what Godel did, then the results are quite different. Mathematics was looked at as a holy grail of consistency. At the first of the 20th century, mathematicians were sure they were homing in on a final and noncontradictory, complete system of formal mathematics. Godel knocked their foundations from under them. For every formal system, there will be a Godel formula which is true but unproveable within the system. Therefore, the system cannot be complete as every true statement expressible within that system is not a theorem of the system. One could try to patch it up by inserting the theorem as an axiom, but then you'd have a new system with its own Godel formula. And the recursion begins....



Hmm. Might have gone too far in describing this to the unprepared. Following what I'm saying?

Let's take a breather and examine the new statement that I've provided.

"This well-formed formula is unproveable within this formal system."

"Well-formed formula" means that the construction of the formula follows all the rules for theorems expressed in the system although its truth or falsehood is not indicated by well-formedness. 2+2=5 is well-formed.

I've already discussed formal systems... So, lets get down to examining the sentence itself.

Let's begin by assuming that it is false. This would mean that the formula is proveable within the system. Which means that the formula is true. Which means that it's unproveable. And we're left with a contradiction.

Next we consider it as being true. This would mean that it's unproveable within the system. Yet true, nonetheless.

The result of both conjectures leaves us with a true statement. A true formula. And one which can't be proved within a formal system. Therefore, the system is incomplete as there will always be true formulas that cannot be proven within that system.

Nifty, eh?
Very tricky to follow some of the twisty logic, but blatantly clear once you've danced the recursions for awhile. (Godel, of course, didn't use plain language as I have done. He, instead, devised a method to convert such statements into numbers and symbols which he could manipulate mathematically and thus translated the epimenides paradox into the heart of the 'consistent' domain of mathematics. Truly amazing.)


I'll leave with a final paradox for you to consider. This one is from Hofstadter's Godel, Escher, Bach (which I recommend everyone and anyone read) and is inspired by Quine.

Rather than using such simple forms of self-reference as 'this', Godel had to devise a means to self-refer in mathematics. He did so by a very difficult to explain method (which I won't even attempt to go into here) which is somewhat demonstrated in the following sentence:

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.


yeah, done, many times

Now that's inspiring.

Now that's inspiring.
what? should we rehash old discussions just because someone new came long? especially ones that have no significants at all? I am sure he could find this discussion on the net, or in our archives. unless something new can be said about the subject, there is really no point in bringing it up.

http://www.sciforums.com/archive/index.php/t-6711.html

what? should we rehash old discussions just because someone new came long?

Why not? If you don't find the subject interesting then don't respond.

especially ones that have no significants at all?

No significance!?
Heh.
If you say so.
I think I know a few million people who might disagree with you there.

Anyway. What if it didn't have any practical significance whatsoever? Does everything have to be 'significant'? Can't a subject be discussed merely for the joy of exploration?

I am sure he could find this discussion on the net, or in our archives. unless something new can be said about the subject, there is really no point in bringing it up.

http://www.sciforums.com/archive/index.php/t-6711.html

Do you really think that reading dead old threads is the same as taking part in an active discussion? Especially with subjects that tend towards such tricky concepts as this?

By the way, the link you've provided is kinda boring and pointless. Here's a better one.

http://www.sciforums.com/showthread.php?t=43398


Anyway. Liven up! This is one of the most interesting and relevant topics in the world and you pop in with a "Ho-hum. Been there done that. Yawn." Your apathy shows clearly that you haven't been there and that you haven't done that.


Consider this, if you will. A teacher teaches the same course year after year after year. The same thing. Over and over and over. Been there. Done that.

Will the students in his class react better to an attitude of *yawn* or to one full of the joy of discovery that is implicit in learning? Do they do better by being in an interactive environment full of vibrance and interest or would they do better by reading transcripts of last year's class?

(Hmm. We dip into recursion with this thought. Because if this were the way of things, then last year's transcript would merely be the transcript of the year before which would be the transcript of the year before. Muaha!!! Recursion is at the heart of the significance of this topic. Recursion is at the heart of your very intellect and awareness of the world around you. And you cry 'insignificant.' Sigh. The jaded youth.)

You do no one favors by being apathetic. You do the world a great disservice...

Don't know if this has been done before but:

This statement is false. True or False?

Answer carefully... :D
true AND false

invert, I just think there are much more interesting things to talk about besides a linguistic paradox.
how about the new white LEDs they came out with.

or the 4x3 nm car, that really drives!

or the oxygen rich molecules on the moon which, some day, may be used to provide oxygen for future moon bases.

or magnetosphereic plasma propulsion, which could lighten our future space ships and make them better able to explore our universe (or at least our small corner of the universe).

or weight increase of a reflective box with light in it over a box with no light.

something, anything, that can have real use.

something, anything, that can have real use.

I guess you just don't see it.

You seem interested only in technology. Bells and whistles. Do you have a psp? Getting the new Xbox?

Godel's theorem goes to the heart of cognition. It is vital to our very existence. I would think that someone who enjoys physics and maths would understand the significance if only as it regards to maths.

Ah well.
If you ever come across GEB, read it. Then see what you have to say on the subject.

no, I do not have a psp, nor will I be getting a xbox. can you tell me what you gain from discussing this, that could not be gained from simply reading material on the subject? seeing the heart of cognition helps very little in the grand scheme of things. discussing chemistry would be much more useful.

the only things I can see coming from this discussion are critical thinking skills, which can be gained in many, more practical, ways.