Full-Name: Joshua D. Guttman
Posted-Date: Fri, 26 Jul 91 14:51:53 -0400
Date: Fri, 26 Jul 91 14:51:53 -0400
From: guttman@org.mitre.linus
Message-Id: <9107261851.AA12984@malabar.mitre.org>
To: info-hol@edu.uidaho.cs.ted
Subject: Re: More on the choice function
In-Reply-To: R.B.Jones@win0103.icl.icl.gold-400.gb's message of Fri, 26 Jul 1991 17:41:17 GMT
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Return-Receipt-To: guttman@linus.mitre.org

I clearly used the wrong word when I said non-deterministic.  After all, I
certainly don't mean that the same epsilon-term could have different
denotations at different occurrences.  Certainly @x . P x = @x . P x.

Rather, my point is this: Given an ordinary structure, there's no canonical way
to determine the values of epsilon-terms.  That is, it's arbitrarily chosen.
The fact that you have a choice function or universal well-ordering in your
background structure doesn't really make it any the less arbitrary, because of
course there's nothing canonical about its being one choice function rather
than any other.

Hence, as we pass from one model (with a particular choice function in place)
to another, the values of the epsilon-terms change in a somewhat arbitrary way.

And this is true even if the models, restricted to the signature of the theory,
are actually *isomorphic* -- because the language gives us no way of referring
to the underlying choice function.

The axiom of choice is not something that bothers me, since it is semantically
just like any other axiom.

        Josh