Cantor’s diagonal argument claims that ℝ is uncountable. When we see real numbers as points on the number line, we can put a name on each point and put the names into a list without contravening Cantor’s diagonal argument because we cannot create a diagonal from a list of names.

However, we do not need such a impossible list, but just to split R into two parts, S2 and S10. the members of S2 are real numbers expressed in binary, those of S10 in decimal. We create a list of real numbers by picking one member from S2 and one member from S10 alternately and forever. This list is a composite list whose members are in binary and decimal alternately. The diagonal of this list is a sequence of binary and decimal digits alternately and out-of-the-list-number cannot be constructed from it.

In fact, composite list can be created in splitting R into many subsets in numeral systems of different bases from which no out-of-the-list-number can be created and there is no real number excluded from the composite list. Because the composite list is constructed from the whole R and no real number is found outside, the composite list contains R.

If there is one list that contains R we can already conclude that R is countable. But the permutation of the subsets of R can create a huge number of different composite lists which all contain R. So, we conclude with confidence that R is countable. Then Cantor’s diagonal argument fails.

Cantor’s
diagonal argument expresses real
numbers only** **in one
numeral system, which
restricts the used list. If a binary list is shown
not to contain R, this can be
caused either by “list” or by “binary”. Because Cantor has focused only on “list” overlooking
“binary”, this is the flaw that
breaks Cantor’s diagonal argument which then does not prove ℝ uncountable.

For more detail of this study please read the complete paper here:

«Real numbers and points on the number line with regard to Cantor’s diagonal argument»