[FOM] Re: Henk's Characerization of Recusivity

[Ali Enayat]

This is a response to Henk's comments (July 10) on my most recent attempt
at explaining his charcterization of recursivity. in my July 1 posting.

I believe that Henk and I are very close to "converging". Indeed the whole
issue at this point hinges on whether Henk can convincingly defend the
following claim of his, when he writes:

>But it's quite possible that you can end extend F(q) such that phi(q)
>will be valid in the extension -- so, in this pathological extension
>"there exists t, t' such that psi_e(t,0,x) and psi_e(t',1,x)"
>will hold.

Here F(q) is a sufficiently large finite portion of the hereditarily finite
sets, in which one can recognize a specific Turing machine (in this case
the e-th "DECISIVE TM") to have reached a halting state and produced either
a 0 or a 1. This latter statement is a bounded statement, and will remain
true in any end extension of F(q), no matter how pathological. To put it
dramatically: A dead Turing machine cannot be resurrected, even in a
nonstandard universe.

If I am wrong, and such a counterexample can be convincingly exhibited,
then I for one would gladly welocome Henk's characterization as a new one.

Regards,

Ali Enayat