> On 18 Nov 2019, at 15:28, Philip Thrift <[email protected]> wrote: > > > > On Monday, November 18, 2019 at 8:01:12 AM UTC-6, Bruno Marchal wrote: > >> Then a huge technical problem is that the term “model” is used in opposite >> sense by physicists and logicians, and the sense of “model” used by >> logicians is technical and required some good understanding of what is a >> theory as considered in logic (basically a finite machine, actually). > Bruno > > > > I have thought about this almost 50 years, and have come to the conclusion > that 'model' as used in physics to mean a mathematical formulation of a > theory is correct, and that mathematical logicians should have never used > that word for what they are using it for. It should be 'interpretation', > 'semantics', or domain' instead. > > So Peano axioms is a model of arithmetic, and is ℕ a possible interpretation > (or semantics, or domain).
Usually the domain is the set from which the model is built. N is the domain, But the Model is the whole structure set (N, 0, +, *). The interpretation is the function going from the syntactic symbol to diverse object or construction made on the domain. In some more vague context, we can use “interpretation”, “semantic” and “model” as quasi synonym. The term “domain” has acquired a more technical sense in the theory of domain by Scott, but very often is used to described the set used in the model. Logicians use “model" like painters. The naked model is the reality, and the painting is the syntax or theory pointing to that reality. Physicists use model, like in Toy model, a simplification, or a theory, and is used most of the time as both a theory or its interpretation (taken for granted most of the time, although this has evolved a little bit, notably through the difficulties to interpret QM). > > > Mathematical logicians just goofed up, that's all. Logic is mainly the study of proof theory, model theory, and the relations between both. “Model” has acquired a technical meaning. I think the term has been introduced by Löwenheim, probably in his "cornerstone paper” on this subject “Über Möglichkeiten im Relativkalkül” (“On Possibility In the Relative Calculus” in German). A good interesting book on the birth of Model Theory is the book by Calixto Badesa: “The Birth of Model Theory”, 2004, Princeton University Press (translated from Spanish). Bruno > > @philipthrift > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/b68741d7-8a61-4690-919c-8b63fdb7e774%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/b68741d7-8a61-4690-919c-8b63fdb7e774%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/172E1789-8A37-4D7F-85F8-6903694A5020%40ulb.ac.be.
