> On 13 Nov 2018, at 11:06, Philip Thrift <[email protected]> wrote: > > > > 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!
I agree. It is actually a consequence of mechanism. The set of natural numbers does not exist, nor any infinite set. But that does not make a physical universe into something existing. Analysis, physics, sets, … belongs to the numbers “dreams” (a highly structured set, which has no ontology, but a rich and complex phenomenological accounts). I gave my axioms (Arithmetic, or Kxy = x, Sxyz = xz(yz)). As you can see, there is no axiom of infinity. Bruno PS Sorry for the delay. > > - 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
