Structural relativity and informal rigour
This article deals with a basic difficulty in studying the foundation of mathematics. People want to find a system of axioms for the intended structure, but once such a system found, one gets whole classes of unintended models for the given system. In those models, all the propositions proven using the axioms are true, but a lot of other properties are model-dependent. This introduces a kind of relativism in mathematics, feature which is not agreed by most of the scientists. The author suggests ``that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic''. By a theorem of Zermelo discussed by Kreisel, the second order set theory \(\text{ZFC}_2\) has only models \(M\) and \(N\) which are either isomorphic, or one of them is isomorphic with an initial segment \(V_\alpha\) of the other one, for some inaccessible \(\alpha\). For me, one of the most interesting discussions in this article is the enumeration of possibilities presented below. (I display this paragraph with the exact wording of the author.) \begin{itemize} \item[1.] We will not, as a matter of fact, go astray in justifying new axioms extending \(\text{ZFC}_2\). Challenge: If we take this horn of the quadrilemma, we then have to explain why we will not go astray in justifying new axioms. This looks like a difficult task and has the whiff of mysticism about it. \item[2.] We accept \(\text{ZFC}_2\), but also hold that we cannot justify axioms extending it (except perhaps large cardinals). \(\text{ZFC}_2\) (possibly with large cardinals added) is the limit of our possible justifications. Challenge: This option essentially gives up on trying to resolve any sentences that are not consequences of large cardinals (e.g. CH). \item[3.] An agent can become less precise by refining their set concept (if they pick an axiom with some false consequences). Challenge: This response seems counter-intuitive; conceptual refinements should result in more rather than less precision. \item[4.] One rejects the principle of charity, and accepts that in coming to justify new axioms, we might just say false things about the structure of sets, rather than true things about a range of structures. Challenge: This likely ascribes a widespread error theory to many attempts to extend \(\text{ZFC}_2\). In particular (given the pessimistic probabilistic argument) it is likely that we will be in error in futures in which we accept a theory resolving questions independent of \(\text{ZFC}_2\) (plus large cardinals). \end{itemize} Finally, the author gives convincing arguments for the thesis that a good response to different problems put by our wishes in balance to the inherent difficulties of finding the best foundation could be the quasi-weak axiomatisation \(\text{ZFC}_{2QW}\). For the entire collection see [Zbl 1483.00027].
scientific article; zbMATH DE number 7598438
scientific article