I recently finished reading Aczel’s The Mystery of the Aleph which is a sort of a mathematical history of infinity and in particular Cantor’s contribution (and mental illness). Having finished, I have a fewbig questions. So, appealing to the higher intellects on the board, here is Q1.
I am quite satisfied that the density of the reals is infinitely greater than the rationals. (I discovered this for myself, but that is another story.) However I do have some problems with Cantor’s diagonalisation proof.
Briefly the proof goes like this: (and I am doing this from memory so that you can see my understanding of it and know where I am going wrong.)
Suppose all the real numbers between 0 and 1 were written out as (infinite) decimal expansions and placed in a long (infinite) list. (This need not necessarily be in numerical order.) You would then have an array of digits countably infinite wide and infinite long.
Suppose you then created a new decimal number by examining the digits on the diagonal and adding 1 to each digit. (9->0)
This new decimal number would be different from any of the listed numbers; therefore demonstarting that there must be more reals than can be listed, therefore the infinity of the reals must be greater than the countable infinity of the digits in the decomal expansion.
Mmmm says me. That last little bit was a bit of a jump. So I get to thinking about it for a while.
If I limited myself to finite decimal expansions, then it is easy to demonstrate that my array of digits is orders of magnitude longer than it is wide, that is, length =10^width. If I then reconsidered infinite decimal expansions then the width is aleph0 and the length is 10^aleph0. In other words, the cardinality of the continuium is n^aleph0.
Ok, I’m happy about that.
Supose I listed the reals in a different order. Suppose I listed all the reals with a one digit decimal expansion, then all thise with a two digit decimal expansion, then three digits, then four and so on. My list would look very similar to a list of whole numbers. In other words, would I not find my listing of the reals was countable and that the cardinality of teh continuium was equal to the cardinality of the whole numbers?
I know this is incorrect, but I can’t spot the error.