You can read what this is in the wiki page (https://en.wikipedia.org/wiki/Berry_paradox).

Quote:

Originally Posted by **wikipedia**

Quote:

Originally Posted by **wikipedia**

So, while the expression numbers 57 letters, it states that there is a smallest positive integer which cannot be defined in less than 60 letters. Therefore while it refers to such a number, it goes against the rule that anything which defines such a number must have had at least 3 more letters and furthermore it does not actually present the number itself (although it refers to it). The apparent paradox is about the first part: if such a number exists (ie if the statement is true) then such a number is already referred to by less letters than the statement claims (so the statement is false).

The issue here is that (due to lack of a strict definition of "definable", allowing for use both in and outside the original expression), one can claim that the expression "The smallest positive integer not definable in under sixty letters" is

Ultimately, the "paradox" has to do with the need to identify the statement as using another level of "definition" inside of the statement itself, and outside of the statement (when referring to an actual number).

Apparently some luckier countryman of mine (G. Boolos, of Mit) used this paradox as the basis of a faster/simpler proof of Gödel's incompleteness theorem, which is why I bothered with it. Unfortunately for me I had to first look it up, given it was mentioned already in the second sentence of his paper :P ]]>

why specifically use the diagonal method to show that some sets have larger cardinality than that of the set of natural numbers?

I have to assume that the point was to show that one cannot order the sequence in any way which would correspond to a tie to the natural numbers, eg 1.1, 1.2 etc would still be placed in positions (eg) 1 and 2, but then the full set would take up all of the positions used by natural numbers and still allow room for many more positions for the fractions. But why should one use a diagonal to show there are many more possible positions, instead of any other method? Is this the simplest possible to think of, or does it have any specific use in other things? I mean intuitive it would be self-evident that even a fraction of a fraction of a fraction... of something would go on in a one to one tie to the natural numbers, so is there some use in coming up with a specific and easy to iterate set which can be fed back to the original (different in the diagonal) and still allow for the new set having space for more?

To me the diagonal presentation seemed somewhat similar to the first proof of there being more prime numbers than can be accounted for, so was wondering if it was just one possible way of making the argument or something inherently valuable due to ties to other parts of set theory. ]]>