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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
