This article uncovers a hidden assumption that the diagonal argument needs, then, explains its implications in matter of infinity. The use of the diagonal digits imposes a condition unnoticed until now. If this assumption were found false, the conclusion of the diagonal argument should be rewritten.

Please read the article at

PDF Hidden assumption of the diagonal argument http://pengkuanonmaths.blogspot.com/2016/01/hidden-assumption-of-diagonal-argument.html
or Word https://www.academia.edu/20805963/Hidden_assumption_of_the_diagonal_argument

The value of a decimal number depends on the number of its digits. For irrational numbers that have infinity of digits, their values seem to be definitive. However, the meaning of infinity is ambiguous because there exist several kinds of infinities. If the infinity used to define the number of digits is not clear, the values of irrational numbers will not be well defined. This is why we have to answer the question of the title.

Please read the article at
PDF Which infinity for irrational numbers? http://pengkuanonmaths.blogspot.com/2016/01/which-infinity-for-irrational-numbers.html
or Word https://www.academia.edu/20147272/Which_infinity_for_irrational_numbers