Brent Meeker wrote: >> in memory of James, >> > [...] >> ============================== >> >> 2. A paradox ? >> >> I will say that a function f is computable if there is >> a well defined formal language FL in which I can explained >> non ambiguously how to compute f on arbitrary input (number). >> I must be able to recognise if an expression in my language >> is a grammaticaly correct definition of a function, that is >> syntax error must be recoverable. >> All expression in the language must be of finite length, and >> the alphabet (the set of symbols) of the language is asked to >> be finite. >> Now all finite strings, a fortiori the grammatically correct >> one, form a countable set. Just use the lexicographic order >> defined above. >> >> So the set of function computable with FL is countable. >> A synonym of countable is *enumerable*, and is used by >> computer scientist, so I will also use it. >> >> So I can enumerate the functions computable with FL, that is >> I can put them in sequence like f_0, f_1, f_2, f_3, ... >> going through all functions computable in or with FL. >> >> Let us consider the function g defined again by >> >> g(n) = f_n(n) + 1 >> >> Now, g is not only well defined, but is even computable. To >> compute it on n, just search in the enumeration f_i the nth >> function, apply it to n (this gives an answer because the f_i >> are computable) and add 1. >> So, there is a number k such that g = f_k, but then, again >> >> g(k) = f_k(k) = f_k(k) + 1 >> >> But f_k(k) is a well defined number (f_k is computable), and >> no numbers can be equal to "itself + 1". Contradiction. >> >> Could the set of computable functions in FL be uncountable, after >> all, or is g not expressible in FL, or what? >> >> Where is the error? I let you think a little bit. You can >> ask question. (Notably in case of notational problem). > > > > >I think there is a cheat here. "Computable" requires that the function >be defined finitely. While > > g(n) = f_n(n) + 1 > >appears to be a finite description, it implicitly calls an the infinite >list of functions f_n. So it's description is only given in terms of a >enumerable, but infinite string, and hence does not occur in the list >f_k.

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Of course there is a cheat. But I don't see any infinite string appearing. The description of g is only given in terms of an enumerable infinite *list*. And for computing g(n) we need only to generate the n first element of the list, and this for each n. I give a hint. There are two ways to correct the argument, two possible cheats! Correcting one transforms "the paradox" into a wonderful theorem. Correcting the other transforms "the paradox" into another wonderful theorem. It is a deep "paradox". It is at the root of both theoretical computer science and mathematical logic. It is worth to think what is going on here. So I let you think (and perhaps others) a little bit more ... Bruno