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. (01)
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. A good
example of a redundant primitive is found in the system PA^, which is the
result of adding an exponentiation operator to the language of Peano Arithmetic
(PA) (whose primitives are 0, +, •, and ' (the successor operator)) and
axiomatizing it as usual. That operator is redundant since (as Gödel showed in
a beautiful lemma in his proof of his incompleteness theorem) it can be defined
in terms of addition and multiplication. (02)
> 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)? (03)
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. (04)
> 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. (05)
What, other than the axioms you provide for a primitive, could determine its
meaning (in a scientifically significant way)? (06)
-chris (07)
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