On Monday, November 12, 2018 at 8:35:23 PM UTC-6, Bruno Marchal wrote: > > > A model is a model of a theory. The notion of model of a model can make > sense, by considering non axiomatisable theory, but that can lead to > confusion, so it is better to avoid this. When a model is seen as a theory, > if it contains arithmetic, the theory cannot be axiomatised, proofs cannot > be checked, the set of theorems is not recursively enumerable, etc. > > > Bruno > > > This is why some have mathematical theories (alternatives to ZF) that have finite (i.e. Only a finite number of numbers needed!) models (e.g. *Jan Mycielski,* "Locally Finite Theories" [https://www.jstor.org/stable/2273942 ]). In this approach quantifiers are effectively replaced by typed quantifiers, where the type says "this quantifier ranges over some finite set".
Another approach is to nominalize physical theories theories (*Hartry Field*, *Science Without Numbers,* summary [ http://www.nyu.edu/projects/dorr/teaching/objectivity/Handout.5.10.pdf ]). In this approach the model of the theory is a finite set of (references to) physical objects. This is the best point-of-view to have: *The set of natural numbers simply doesn't exist!* - pt -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to email@example.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.