On Mon, 29 Oct 2012 09:14:02 -0700 (PDT), Arturo Magidin <magidin@member.ams.org> wrote:
>> >The government doesn't like it when I read minds without a >> >warrant, so I try not to do it, you see. >> >> There are constructivist objections to the whole enterprise. If >> nothing else, I was hoping to see something like these addressed >> against the diagonal argument piecemeal. > >In a constructivist universe, you cannot have a function defined >on the natural numbers, and you cannot have a set of real numbers at all. >The very premise ("Suppose f:N-->(0,1) is a function") is considered >nonsensical in the constructivist point of view. > >You can only talk about a "rule" that will allow you, in principle, >to compute f(n) up to any degree of exactness that you care to specify. >You can encode the procedure in the proof given assuming that the >function f is given by Turing Machine which, when given n as an input, >will give you the number of a Turing machine that computes f(n). >Then you can establish the existence of a Turing machine r which, >when given as input the number of a Turing machine that computes >a function f as described above, will produce as an output the >sequence b_n which is not equal to the output of any Turing machine >whose number is produced by the Turing machine corresponding to f. >That much is allowable within the computational universe.
Doesn't this sound rather like what some of the "cranks" hereabouts are trying to say?
