2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I have added the english equivalent link since this sites' language is english; also added number-theory. You might want to add some background to the question so we can see where you are and where this question comes from. ($\pm 0$) $\endgroup$ – AlexR Aug 6 '14 at 22:55
  • $\begingroup$ Field of fractions might be of interest to you. $\endgroup$ – user98602 Aug 6 '14 at 22:56
  • $\begingroup$ The Enlish Wikipedia article "Cantor's diagonal argument" is not equivalent to the german article "Cantors erstes Diagonalargument". Instead, the English article is equal to the German article "Cantors zweites Diagonalargument" which is about a related but different proof by Cantor. Unfortunately there doesn't seem to be an english version of the German article "Cantors erstes Diagonalargument" which is about Cantors proof that the rational numbers are countable. $\endgroup$ – jimmyorpheus Aug 6 '14 at 23:28
  • 1
    $\begingroup$ Take a look at the surreal numbers. This a framework which you can add, subtract, multiply and divide all sorts of infinite and infinitesimal numbers. $\endgroup$ – Jair Taylor Aug 6 '14 at 23:35
  • $\begingroup$ Take a look at the notion of non-Archimedean fields, one of which is the surreal numbers suggested by @Jair Taylor. $\endgroup$ – Asaf Karagila Aug 6 '14 at 23:40
1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Technically "transfinite" does not mean "non-finite." Transfinite numbers generally refer to cardinal and ordinal numbers only. en.wikipedia.org/wiki/Transfinite_number $\endgroup$ – Thomas Andrews Aug 6 '14 at 23:50
  • $\begingroup$ I find the first example to be lacking. While it's of course true, perhaps it's good to remember from time to time that for most people "numbers come from somewhere" (usually $\Bbb c$ or $\Bbb R$) and to say that $x$ is larger than all the real numbers raises the question "Where did it come from, and how come we didn't know about it before?". $\endgroup$ – Asaf Karagila Aug 6 '14 at 23:50
  • $\begingroup$ @Hurkyl, thanks for a nice post. You should add that the hyperrationals actually provide an answer to the OP's question: since they surject to the reals, the cardinality of the reals is dominated by that of the hyperrationals. This was I think the main thrust of his question (analogy with proof of countability of the rationals) so he might be interested in this. $\endgroup$ – Mikhail Katz Aug 7 '14 at 8:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.