On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List < everything-list@googlegroups.com> wrote:
> On 5/30/2020 10:44 PM, Bruce Kellett wrote: > > On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <marc...@ulb.ac.be> wrote: > >> >> Let us write f_n for the function from N to N computed by nth expression. >> >> Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is >> defined on all N. So it is a computable function from N to N. It is >> computable because it each f_n is computable, “+ 1” is computable, and, vy >> our hypothesis it get all and only all computable functions from N to N. >> >> But then, g has have itself an expression in that universal language, of >> course. There there is a number k such that g = f_k. OK? >> >> But then we get that g_k, applied to k has to give f_k(k), as g = f_k, >> and f_k(k) + 1, by definition of g. >> > > > That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, > by definition of g_k. You do not get to change the function from f_n to f_k > in the expression. It is only the argument that changes: in other words, > f_n(n) becomes f_n(k). So you are talking nonsense. > > > No, I think that's OK. It's a straight substitution n->k. The trick is > that g(n) is not some well defined specific function because n has infinite > range. So none of this works in a finite world. But it's not surprising > that there is incompleteness in an infinite theory. > Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAFxXSLTM9g5G3neSA-bn9Td2PqpVW_g%2BBg3hwV%3DbpiToRwYKxw%40mail.gmail.com.
