> On 13 Nov 2018, at 11:06, Philip Thrift <cloudver...@gmail.com> 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
> 
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