FOM: Re: RE: FOM Arbitrary Objects
silver_1 at mindspring.com
Tue Feb 5 07:27:57 EST 2002
> Re (Charles Silver): "The trick is to make reasonable mathematical/logical
> sense of this notion [arbitrary object] using just the resources of
> first-order logic."
> Hilbert used the epsilon symbol to elucidate the notion of an arbitrary
> selection, which he axiomatized in the epsilon-calculus. Is this not
> nowadays considered a useful and adequate treatment of 'arbitrary'? In
> respects is it deficient?
In my opinion, it is deficicient because 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. One deficiency, mentioned
early by Leisenring and credited to Carnap is that if F happens to be empty,
some *extra* object, outside the domain of all-objects, must be pressed into
service (F could turn out to be empty even though ExFx may appear as a line
of a proof). I regard all this as "unnatural" and "unintuitive."
Further, the theorems established on behalf of the epsilon calculus seem
overly technical and artificial--not natural at all. Moreover, and this I
think is the most relevant point to the questions I raised, there is no
indication that the element plucked out either of the range of Fs or
elsewhere is at all "arbitrary." It's just one of many, having no claim
whatever to arbitrariness. I know that there are other developments of H's
epsilon calculus, but to my knowledge the working out of it provided by
Leisenring is the most prominent. I am not that it isn't interesting, for
I think it is. I'm saying only that it doesn't seem to provide answers to
the questions I raised. 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.
I appreciate your reminding me of this. I know a person who's active
in Hilbert's epsilon research and have read a couple of papers on the topic
(besides looking at Leisenring's book).
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