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I'm puzzled by a sentence in an old paper on the history of the notions of categoricity and completeness in mathematical logic.

In a paragraph near the end, the author, John Corcoran, takes some writers to task for failing to appreciate the relation between the two notions. I won't bother to give the author's definition of "complete," because it is irrelevant:

Kline 1972, 1015 gives an argument for "categoricity implies completeness" which is similar to arguments found in Forder 1927, 6, Wilder 1952, 36, and Wylie, 1964, 26. The argument here reproduced is closed to Wilder in detail but similar in spirit to the rest. Let us say that a postulate set $P$ is definite if all of its models are truth-value equivalent. By contrapositive reasoning: it is easy to see that non-completeness and non-definiteness are coextensive and, thus, completeness and definiteness are coextensive. All of the above writers presuppose (or assert without proof) that isomorphism implies truth-value equivalence, from which it follows that categoricity implies definiteness, and so also completeness. None of the above writers shows any awareness of the need for qualifications in the statement that isomorphism implies equivalence.

I'm at a loss to imagine what the "qualifications" referred to in the bolded sentence might be; Corcoran doesn't say. Earlier in the paper he writes:

Loosely speaking, one of the important things about isomorphism is that, given a suitable language, isomorphism implies truth-value equivalence.

So I suppose the "qualifications" might have something to do with constraints on the language. Obviously, to get elementary equivalence, one needs the apparatus of interpretation to interact well with the map giving an isomorphism. But I've never seen in my (admittedly limited) exposure to model theory examples where we adopt such a pathological language that isomorphism doesn't imply elementary equivalence.

So, in short, is Corcoran's objection interesting or merely pedantic?

symplectomorphic
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  • I don't have access to the full text of the article, but: is it possible he is referring to "sentence" in a very general context? E.g. let's say we're living in a model $V$ of ZFC; then "contains ${{{},{{}}}}$" is a property which (the underlying set of) a structure may or may not have; but it's obviously not isomorphism-invariant, since what it's "actually about" is the external nature of the elements of the structure. Intuitively, such properties shouldn't be considered part of our language, but we need to say so. – Noah Schweber Dec 08 '17 at 05:59
  • @Noah Schweber: I guess something like what you say is right. Earlier he writes, in reference to Dedekind and using some notation I don't have space to explain: "it is obvious that if, for example, $\langle G, +\rangle$ and $\langle G', +'\rangle$ are isomorphic the proposition that the former is a group of permutations may be true without the latter being a group of permutations." I missed this the first time through, but your comment helped me pinpoint it. I guess the property "is a group of permutations" is like your example. – symplectomorphic Dec 08 '17 at 06:21
  • Judging from this quote, Corcoran seems to have a different concept of equivalence than I do. I conjecture that his notion of equivalence is, at least in part, based on some Platonist view. I'd certainly consider a group that is isomorphic to a group of permutations to be a group of permutations - although it may not look like one on the surface. But then again, I consider ${ {}, {{}} }$ to be a perfectly good natural number. – Stefan Mesken Dec 08 '17 at 09:32

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