I take it for granted Cantor's Diagonal Argument establishes there are sequences of infinitely generable digits not to be extracted from the set of functions that generate all natural numbers. We simply define a number where, for each of its decimal places, the value is unequal to that at the respective decimal place on a grid of rationals (I am here ignoring the "no 0's or 9's" failsafe).
What I cannot understand is how the argument forces the conclusion the set of reals has a greater cardinality than the naturals.
After all, infinite sequences of digits defined by diagonalization are parasitic on digit values on the grid of rationals. If we accept that one-to-one correspondence is the criterion of equal cardinality, can't we correlate any sequence generated by diagonalization with the respective decimal values on the grid used to produce all rational numbers' decimal expression?
I know one-to-one correspondence can be used to show equal cardinality between sequences of apparently (to non-specialists) different cardinalities: x, x+1, x+2... can be correlated with 9x, 9(x+1), 9(x+2)... But were we to choose a base 2 number system, wouldn't Cantor's diagonal be literally one-to-one with diagonal sequence on the list it is produced from? Like, if the grid read 1,0,1,1,0,1,0... then the Cantor diagonal would read 0,1,0,0,1,0,1...
The hypothesis I'm hoping to be corrected on is that diagonalization produces sequences in ways counting cannot. These sequences are uncountable, but not uncountably many, distinct from the natural numbers but not more numerous than them.
I looked around for other posts where the question in the title was answered but found none.