Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would seem to prove the theory true, but as far as I know it is unprovable. Thus here I am asking for your help.
- Start with the set of all integers $(-\infty,...,-1,0,1,...,\infty)$
- This set can be considered 1-dimensional in that for each position in the set, there is only one possible number
- Consider adding an additional dimension to the set that has only one possible value (0): $(<-\infty;\ 0>,...,<-1;\ 0>, <0;\ 0>, <1;\ 0>,...,<\infty; 0>)$
- Clearly this set has the same cardinality as the integers as there is a clear mapping between them
- Now consider having two values instead of one (0 and 5), and let's represent everything in the second dimension a decimal whose value is added to the first number
- This makes the set $(<-\infty;\ 0;\ 5>,...,<-1;\ 0;\ 5>, <0;\ 0;\ 5>, <1;\ 0;\ 5>,...,<\infty; 0;\ 5>)$, which can be expanded to $(-\infty,...,-1.0, -0.5, 0.0, 0.5, 1.0, 1.5,...,\infty)$
- This new set still has the same cardinality as the set of all integers since each number can be mapped directly to an integer
- This property of the set's cardinality should continue as more numbers are added to the set, up until the point that infinitely many numbers are added
- Once there are infinitely number of numbers added to the second dimension (sequentially), then the mapping to the integers cannot be performed because the infinite size and the maximal density (all integers from 0 to $\infty$)
- This final set would be expanded to be exactly the same as the real numbers, and would seem to indicate that it is impossible to have a cardinality between the integers and real numbers
I guess it comes down to statement #9. It seems to me that it's most important that this set of numbers includes every combination of numbers from 0 to $\infty$ to ensure that the irrational numbers would be included.