FOM: Re: RE: FOM Arbitrary Objects

[William Tait]

On Tuesday, February 5, 2002, at 06:27 AM, charles silver wrote:

for the epsilon-intantiation to
work, some particular element must always be plucked out of the range of
objects over which ExFx ranges.    That is, an element having F must be
somehow obtained ("chosen") from all the F's.

Hilbert regarded the epsilon terms as denoting `ideal objects'; and his 
hope (probably one should say:  conviction) was that in any formal 
deduction of a variable-free formula, they could be eliminated. But if 
one were to give a `semantics' for the epsilon-calculus,  it would not 
be necessary to interpret epsilon x F(x) as denoting an F---for example, 
if there are no F's, it could denote anything. Likewise, in eliminating 
this epsilon term in a deduction, if there are no critical formulas F(t) 
arrow F(epsilon x F(x) in the deduction, then epsilon x F(x) can be 
replaced by any term.

On the other hand, Charles was responding to  Jay Halcomb who wrote

  Hilbert used the epsilon symbol to elucidate the notion of an arbitrary
selection, which he axiomatized in the epsilon-calculus.

This is surely wrong, too: Hilbert introduced the epsilon terms as a 
tool in his proof theory. I do believe that the notion of an arbitrary 
number has a place in his philosophy---in understanding his finitism; 
but I see no place for it in his conception of the epsilon-calculus.

> Of course, one has to admit that some of its
> popularity is that it's attached to a famous person.   If Joe Blow had
> invented it, we'd have no "Blow's epsilon calculus" at all.

Is Joe Blow an arbitrary person? If so, perhaps we _should_ pay more 
attention to Hilbert. But, if we do, then it won't lead us down the path 
being discussed by Charles and Jay.

Incidentally, for a very nice discussion of the history of the 
epsilon-calculus  and substitution method, see Jeremy Avigad and Richard 
Zach's

http://www.ucalgary.ca/~rzach/epsilon-calculus/

Jeremy posted a notice about this yesterday---perhaps as a hint.


Respectfully,

Bill Tait