Finitely definable real numbers are endless sequences of digits (plus a decimal point) that can be specified in a finite number of words, e.g. “two plus two” defines 4.000... (the 3 dots signifying an endless sequence of zeros, as usual). Sounds simple, but there is a paradox here (published by Richards in 1905), which inspired Gödel to worry about the definability of arithmetic truth (and was recently discussed in an open access issue of Philosophy), which are the subject of this week’s In Our Time.
......Such finite strings of words as specify real numbers can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. And given that list one can, it seems, define a number (between 0 and 1 and) not in that list using the diagonal procedure made famous by Cantor: The digit in its nth decimal place is the final digit of 1 plus the digit in the nth decimal place of the nth number on the list. The paradox is that such a number, say R, has a finite description (as above) and so it should be in our list. (Indeed, as with all the finitely definable reals, R will occur in the list infinitely many times, as it can be specified by arbitrarily longer descriptions too.)
......Consider such a description of R as the above, occupying the Nth place on our list of such descriptions (for some particular positive integer N). What does it say that R’s Nth digit is? That R’s Nth digit is different from what it is! So then, that descrpition could hardly be defining a number after all. So it should not be on our list in the first place. But then, if it’s not on our list we should after all be able to get a new number from that diagonal procedure, which therefore means that R should be on our list.
......This paradox is therefore a bit like the paradox of the club for all those who are not in a club. There is, of course, no problem setting up a club for all those who are not in any other club (which was surely what was intended). Similarly a barber could shave everyone else who doesn’t shave himself unremarkably, and we would (were it not for the other set-theoretical paradoxes) be able to have a well-founded set of all the other well-founded sets; or we could have a class of all and only the well-founded sets (which is little more than terminologically different) of course, or a self-membered set of all the sets, and so forth.
......So maybe the description of R above included an implicit exclusion of the Nth description from the diagonal procedure (an exclusion that would be explicit on the Nth line of our list).
......That is counter-intuitive (since the imagined diagonal cuts across the whole list) but any resolution must be, and note that the list included already—whether or not we noticed it—anything that anyone could possibly say (finitely) that would define a definite real number. So why should the diagonal procedure without such an exclusion not fail? Or, to put it another way, insofar as we think that that procedure won’t fail, because all those numbers exist before the procedure, maybe the aforementioned exclusion was implicit (in that intention), for all that it was unnoticed.
......I find it hard to make any realistic sense of (the alternative resolution of Richards’ paradox, the one that inspired Gödel’s syntactical results) a sentence that neither defines a real nor fails to define one. Maybe that does make sense (there is something philosophically obscure about the nature of mathematical truth), but if the description is so unclear that it fails objectively to define a real, how then could it also be failing to fail to define one? Some philosophers mention mathematical creativity in this context, but surely any mathematical object that could be created by us is already described in our original list.
......In my defence (of my sticking with the simple resolution, of R’s description containing, if R exists, an implicit exclusion of the Nth description) I can make realistic sense of the indefinite extensibility of the simply infinite. Consider a geometrical line of points, with two points labelled ‘0’ and ‘1’ (to define a metric) and another point between them. That third point can be surrounded by nested intervals, focussing in upon it, yielding its decimal expansion. Such an expansion may, clearly, be generated endlessly by the point in such a way that it fails to encapsulate all the information contained in that point’s precise position (relative to the points 0 and 1)—for details see here.