Am Mar 8, 2012 um 6:23 PM schrieb William Frank:
> I think is the essence of this point is:
>
> "The number of possible conflicts is infinite, and no fixed set
> of universal definitions can anticipate and rule out all of them."
>
> This is true of *logical necessity*; so it is something that we always live
>with
>
> The reason for this truth is that
>
> there is no guarantee that all the models for
> any theory G are consistent with one another. (01)
I not sure what you and John mean by that  consistency is a property of
theories or, more generally, sets of sentences. It doesn't make any clear sense
that I can see to say that two models are inconsistent. I suppose we can make
something up, e.g., models M1 and M2 of theory G are inconsistent with each
other if there is some sentence A in the language of G (obviously not a theorem
of G) to which M1 and M2 assign different truth values. (02)
> More strongly, unless the theory G is complete, which no theory as expressed
>in an application will be (03)
It's not obvious to me that that is so. There are wellknown examples of
complete firstorder theories (e.g., the firstorder real number theory and
Euclidean geometry both have complete axiomatizations). Such theories obviously
cannot contain a lot of arithmetic, but it's not obvious that an interesting
practical ontology has to contain the arithmetic needed to guarantee
incompleteness and hence can't be complete. (04)
> (only theories as bounded and richly expressed as second order arithmetic
>tend to be complete), (05)
Eh? There is no complete axiomatization of secondorder arithmetic. (06)
> there will always be models, describe in theories G' that are extensions of
>G, as expressed in different applications, that ARE inconsistent with each
>other. As in John's example, "an employee must be at least 21 years of age,"
>may be a rule expressed in one applicaton, while in another employees may be
>16, or have no specified age limits. (07)
OK, so if you've got a theory G that doesn't entail anything specific about age
limits, then you could extend G to theories G1 and G2 that specify different
age limits. Hence, a model M1 of G1 and a model M2 of G2 will both be models of
G but will be "inconsistent" in the sense above. Fine, but it seems to me this
is all more easily expressed proof theoretically  it's the simple logical
fact that every consistent incomplete theory has consistent extensions that are
mutually inconsistent. (08)
chris (09)
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