Well, I hit upon another beauty in Godel's theorem. I am wondering whether I should name this blog as 'Mind and Matter' or "The beauty of Godel's theorem'. Anyway, here it goes.
Any article (that people write) can be looked as a formal system having a content (sequence of sentences, each of which has meanings) and a message (the meaning of the whole article). Now, I define the following:
Definition 1 : An article is consistent as long as it conveys a message.
Definition 2 : An article is complete as long as each sentence in the article has some meaning associated with it.
The terms 'conveys a message' and 'meaning associated' can differ from person to person but they cannot be radically different. Naturally, we may raise the question whether, in an article we'll have both completeness and consistency. Godel's theorem asserts that - No, we cannot have both completeness and consistency for an article, provided the article is 'rich' enough. One may question, what is 'rich' enough. I would say 'rich enough' here is that the article makes someone 'ponder'.
One must understand that Godel's theorem doesn't say we cannot have any axiomatic system which is both complete and consistent. For example, a mathematical system that allows addition on two numbers and just that alone, but doesn't have any other rules, is both complete and consistent. Similarly, an article that has for instance, "Man is a human being. And humans have life", is both complete and consistent, because one doesn't 'ponder' much over these facts.
So, my claim is that "any sufficiently rich article which is consistent is bound to be incomplete". That is, we cannot search for meanings for all the statements if we are looking for a message in the article. In other words, a rich article should allow some exaggerations if we want to convey some message through the article. Well we can have a bland article without any exaggerations, for after all it is our choice to be either inconsistent or incomplete. But, it is a fact that the world prefers incompleteness over inconsistency !
Any article (that people write) can be looked as a formal system having a content (sequence of sentences, each of which has meanings) and a message (the meaning of the whole article). Now, I define the following:
Definition 1 : An article is consistent as long as it conveys a message.
Definition 2 : An article is complete as long as each sentence in the article has some meaning associated with it.
The terms 'conveys a message' and 'meaning associated' can differ from person to person but they cannot be radically different. Naturally, we may raise the question whether, in an article we'll have both completeness and consistency. Godel's theorem asserts that - No, we cannot have both completeness and consistency for an article, provided the article is 'rich' enough. One may question, what is 'rich' enough. I would say 'rich enough' here is that the article makes someone 'ponder'.
One must understand that Godel's theorem doesn't say we cannot have any axiomatic system which is both complete and consistent. For example, a mathematical system that allows addition on two numbers and just that alone, but doesn't have any other rules, is both complete and consistent. Similarly, an article that has for instance, "Man is a human being. And humans have life", is both complete and consistent, because one doesn't 'ponder' much over these facts.
So, my claim is that "any sufficiently rich article which is consistent is bound to be incomplete". That is, we cannot search for meanings for all the statements if we are looking for a message in the article. In other words, a rich article should allow some exaggerations if we want to convey some message through the article. Well we can have a bland article without any exaggerations, for after all it is our choice to be either inconsistent or incomplete. But, it is a fact that the world prefers incompleteness over inconsistency !
5 comments:
Hmm. Very interesting.
After all, article has 'art' in it, which has some amount of abstratction and illogic associated with it. :)
Where as a group of statements is consistent and complete. [but obviously there is nothing to ponder]
It would be very nice if you could make the definition of 'exaggeration' clear! Simply put exaggeration means overstating the facts.
There is a difference between stating the obvious and stating what one 'feels' is the obvious. One would be fact and the other exaggeration!
And, personally, I diagree with your claim "any rich article should be either incomplete or inconsistent".
Ramya, by exaggeration I mean something that is "overstated" to address the point. For instance, Barathiyar songs have exaggeration in them to invoke the strong sense of patriotism. Actually, I should have stated "any sufficiently rich article is bound to be incomplete" rather than "any rich article should be either incomplete or inconsistent". I have changed it now.
Wow tj, nice observation !
Very interesting though it appeared to go above my head when I read it first.
Thanks for dropping by.
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