"The Principle of Shrinkage"

[franktaw at netscape.net]

Really, the incompleteness theorem means that this can't be true.

It isn't very hard to show that the general question of whether a
function converges is undecideable.

This means that the domains of functions where the principle
applies are all trivial (e.g., rational functions), or else membership
in them is itself undecideable - which isn't very useful.

The phrase "naturally occurring function" seems to be trying to
avoid this, but it doesn't work that  way.  You can't rule out
the exceptions without also excluding the interesting cases.

In this case "He challenged me to say I really believed that the
error might suddenly explode outside such bounds after such
extensive evidence of shrinkage."  Suddenly explode?  No.
Gradually diverge?  Unlikely, but possible.

Franklin T. Adams-Watters


-----Original Message-----
From: eclark at math.usf.edu

... In the proof GSB used
a new axiom ("the principle of shrinkage") which Tim Chow
formulated as follows:

  If f:R -> R or f:N -> R is a "naturally occurring function" and
  lim(x->oo) f(x) appears to exist, then it does.

Then Tim asks the question:

Can anyone give an example of a conjecture of the form "lim(x->oo) f(x)
exists" for which there was extensive numerical evidence and which








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