FOM: Church's thesis: reply to Pratt

[Joe Shipman]

Shipman:
>Another way in which we could get at mathematical truth from physics is

>if Church's thesis is false and we could set up an experiment to
>calculate a definable but noncomputable real number; the value of any
>digit beyond a certain finite level of precision would be a
mathematical
>fact independent of ZFC.

Pratt:
>Why do you need the failure of Church's thesis for this?  The truth
>value of Con(ZFC), as the number 0 or 1, is (as a mathematical fact)
>independent of ZFC, yet is an eminently computable number not entailing

>the failure of Church's thesis.

>It's a much taller order to ask nature for a noncomputable real than
>just one bit.

I can't think of a plausible way to get the value of Con(ZFC) from
nature, except as part of a general solution to the halting problem; my
point was that ANY mathematically rigorous physical theory in which
Church's thesis fails will involve the experimental accessibility of a
mathematically definable but noncomputable sequence; and ANY such
sequence has bits which ZFC does not decide the value of.  Given the
dearth of natural examples of intermediate r.e. degrees, if Church's
thesis is false then the halting problem will likely be solvable and we
could theoretically settle Con(ZFC) as well as much more pressing
questions.

-- Joe Shipman