User blog comment:Wythagoras/All my stuff/@comment-10429372-20130718073726/@comment-7484840-20130718104237

@littlePeng9: If the z(n) function does not name values, then neither does ¥(n). They were supposed to be the same bar one trivial difference, which shouldn't change their number-naming abilities. The point was there is no limit at all to how large the number named by, say, ¥(1000) is.

@Ikosaract: my point was that, using your analogy, if you just ask for a million desires with the original 3 desires, then you have 3 million of them, and then you could use them to get anything you wanted. But if you used them all to ask for a million desires again before continuing to what you actually wanted, you could have 3 trillion desires to use for anything. This can continue and you can have as many desires as you want before continuing. Similarly with the ¥ function, you could use only one level of referral to another function and get a finite output, or use two and get a far larger output. The point is you could name any number at all between 0 and infinity (I.e. all of them)

As for banning it from using languages, mathematics is a language in itself, so that would stop it using any functions that use math(s). If you take a less strict approach, then it is practically impossible to define a firm boundary between a function using a language or not as it is possible to make a definability function using things that really don't seem to use language at all. (I'm working on an example)