But isn't it critical to show that the example which the diagonal process comes up with is a real number which is not on the list in another form?
No, my argument is this:
- Cantor's diagonal argument proves that there are uncountably many infinite binary strings. The binary string "0.01111.." is a different string than "0.1000..."
- The cardinality of the reals in ##[0,1]## is the same as the cardinality of the infinite binary strings.
- Therefore, there are uncountably many reals in ##[0,1]##