Cantor also proved that the number of real integers is equal to the number of naturals, but it's smaller than the number of reals, the demostration is here: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument[^]Also the first Hilbert problem (of his 23 famous set) is whatever or not exists a transfinite number in between the number of integers and the number of reals. This problem is considered solved, but the answer is "it depends...". It depends on the axioms in which you base the number theory (that has to do with gödel's work), but for the street walking people, no there is not.
paradoxes and infinite, this is my favorite:
paradoxes and infinite, this is my favorite: Say you have the list of all possible words wrote in any given language with a finite set of symbols. Now you sort them by it's length, starting by the language defined by 0 characters, and then those with a single symbol,then those with two, and so on. Now you can define a language by saying whatever or not, it contains any given word in the list, that way the language is represented by a sequence of bits, being 1 = it has the word, and 0 = it hasn't the word. Now Let's say we have the list of definition of all possible languages. Each definition is also a sentence wrote in an standard language (BNF, English, Binary, C#, you say... I'll take English), again you sort the list, and you can give each of then a natural number (1, 2, 3...). Now you pair the definition with the binary sequence of the language.
The next step is to use the Cantor diagonal argument: you define a language that differs in the Nth bit to the Nth language. So you get a language that is proven that is not in the list, so you has the proof that there are more languages than natural number, that's not a paradox. But you can define that language that's not in the set with a finite number of symbols in your standard language, In fact I just defined it above in English, and it took me a finite sentence to do it, so it should be in the list, that's the paradox, it should be, but it isn't.
I'll extract the definition in a more formal way: "The language that differs in the Nth bit to the Nth language in the list of all language that uses English alphabet, sorted by the length of it's definition wrote in English, where each bit says whatever or not the language contains the word that is in the position of the bit in the list of all possible words that can be wrote with English alphabet sorted by it's length"
I found the above paradox in a book, I don't remember which book, neither the exact text. What is above is a reconstruction from my understanding of the paradox. Anyway, I've learned that this paradox is a variation of Kleene–Rosser paradox[^] and that one is in turn based on Richard's paradox[^].
modified on Friday, November 5, 2010 3:40 PM
modified 27-Jan-13 2:48am.
Last Visit: 31-Dec-99 19:00 Last Update: 25-May-15 22:22