Dear Chris M., (01)
Thanks for adding some precision. (02)
> On Feb 4, 2010, at 12:07 PM, Matthew West wrote:
> > ...
> > Well I agree that anything defined by axioms is affected when those
> axioms change. However, at least some primitives just are - membership
> would be one of them.
>
> I don't understand what it means to say a primitive "just is". A
> primitive of a system is a lexical item in the basic lexicon of the
> language of the system. Ideally, given the axioms of the system, a
> primitive should not be redundant, that is, it should not be definable
> in terms of the other primitives. A prime example of a primitive is
> the membership predicate in a theory of sets -- it is obviously non-
> redundant since it is the only primitive in the system. (03)
MW: I think that is about what I mean by primitives just being. However, the
question is whether using that primitive in say two different
axiomatizations of set theory means that there are two different membership
primitives, which is what I understand Chris P to be suggesting. I can see
you could make an argument either way. (04)
> > Are you therefore arguing that in using membership in the axioms that
> define other things, that that affects the meaning of membership itself
> (as an example)?
>
> I think what he's saying is simply that if you axiomatize the same
> primitive P with different axiom sets A1 and A2 and those axiom sets
> yield different theories T1 and T2, then A1 and A2 provide different,
> possibly incompatible, accounts of the meaning of P.
>
> > I can see that there are clearly subsets of membership that are say
> set membership and type membership, but I'm not sure that that changes
> the meaning of membership.
>
> What, other than the axioms you provide for a primitive, could
> determine its meaning (in a scientifically significant way)? (05)
MW: Well my intuition is that there is some essence of membership in a type
theory that is the same as the essence of membership in a set theory, and at
least not be forced to conclude that they had nothing in common at all. I
might express that as a common supertype with weaker axioms, but I daresay I
am digging a hole for myself... (06)
Regards (07)
Matthew West
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>
> -chris
>
>
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