On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List < [email protected]> wrote:
> On 5/30/2020 10:44 PM, Bruce Kellett wrote: > > On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <[email protected]> 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 [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAFxXSLTM9g5G3neSA-bn9Td2PqpVW_g%2BBg3hwV%3DbpiToRwYKxw%40mail.gmail.com.
