....Thanks to Richard's reference.

 

G. J. Chaitin wrote: "....there are an infinity of positive integers, but at any given time there are only a finite number of words in English. Therefore, if you have a billion words, there's only going to be a finite number of expressions of any given finite length. But there's an infinite number of positive integers. Therefore most positive integers require more than a billion words to describe. So let's just take the first one. But wait a second. By definition this integer is supposed to take a billion words to specify, but I just specified it using much less than a billion words! "

 

Not so fast, Archimedes !! That statement did not begin to specify it. Because if it did, I will be a very happy person when I collect $100,000 tomorrow !! Consider this quote that I found on the Internet:

 

"In addition to the joy of making a mathematical discovery, you might win some cash. The Electronic Frontier Foundation is offering $100,000 to the first person or group to discover a ten million digit prime number!"

 

So I will walk into the office of the Electronic Frontier Foundation tomorrow and say "I have discovered the number you are seeking. It is defined and specified as the least ten million digit prime number...Well, get your checkbook out. I don't have all day."

 

The Electronic Frontier Foundation representative will respond "Who are you kidding ?? You haven't specified or defined it !! You are just playing with words !!!! Get out of here before I call Security !!!!"

 

That is the flaw in the Berry Paradox. It assumes something false: that just verbally pointing to something in a vague manner is the same as defining and specifying it. The great Godel has fallen into a well-known trap: trying to be original in thought without first humbling himself before the superior wisdom of the Ancient Greeks. Zeno of Elea covered this ground long ago. Concerning Zeno's Paradoxes, historian Will Durant said that they depend on people mistaking words for realities. That is exactly what Chaitin and Godel were doing. Only Zeno did it better.