http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German)
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English)
While looking at Cantors method of proof, which he used to show that the set of the rational numbers is countable and that it has got the same cardinality (Aleph-naught) as the set of the natural numbers, I recognized that if there were fractions that used transfinite numbers as their numerators and denominators, then those infinitely precisely defined fractions could be used within Cantors zizag-counting-grid to address not only all the rational numbers but all the real numbers (of course only in theory because transfinite numbers usually cannot be written down or spoken out very easily).
So my question is as stated above: Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators? If yes, what is the name for these kind of fractions? Or is there a reason why one shouldn't use something like this.
A simple example of such a fraction would be a fraction where the numerator is an infinte sequence of 1s and the denominator is an infinte sequence of 2s.
A more complex example would be a fraction where the numerator would consist of the decimal places of Pi and the denominator would consist of the decimal places of 2^0.5.
It's not hard to construct examples.
For example, you could consider the ring of all polynomials in $x$ with real coefficients such that $x$ is greater than every real number, and thus transfinite. Then the fractions -- the rational functions -- would be of the form you ask for.
Similarly, the hyperrational numbers from nonstandard analysis would be another example: each has a numerator and denominator that is a hyperinteger, and those can be transfinite. This is probably closer to what you have in mind.
Guessing at how it applies to your motivation, the problem is that the hyperrational numbers are too precisely defined: to every irrational real number, there are hyperrational numbers that are infinitesimally close, but none of them are actually equal to the real number. However, you can always round one to its 'standard part'.
That said, externally, the hyperintegers (and the hyperrationals) are uncountable too, so you can't have a (countable) list that contains all of them.
Also, it is important to note that the transfinite numbers that appear in examples like the above have absolutely nothing to do with set theory; they have no relation to the sizes of sets.