On Thursday, November 15, 2018 at 5:15:39 AM UTC-6, Bruno Marchal wrote:
> On 13 Nov 2018, at 11:06, Philip Thrift <cloud...@gmail.com <javascript:>> 
> 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.

The "highest" programming may be higher-type (or higher-order) programming:

examples @ http://www.cs.bham.ac.uk/~mhe/

"Higher-order [programming involves] infinite objects, such as infinite 
strings, real numbers, and even functions themselves, etc. [which 
themselves] are computable. And, more importantly, how to compute them. In 
practice, computation with infinite objects often takes place in languages 
such as ML, Haskell, Agda etc. In theory, some canonical systems are 
Godel’s system T, Platek-Scott-Plotkin PCF, Martin-Lof’s dependent type 
theory, among many others. But how can we (or a computer) compute with 
infinite objects, given that we have a finite amount of time and a finite 
amount of memory and a finite amount of any resource? *Topology comes to 
the rescue* [revolving] around the [finite vs. infinite dichotomy], 
mediated by topology. *We can say that topology is precisely about the 
relation between finiteness and infiniteness that is relevant to 

But there is a new biochemical programming language:

*CRN++: Molecular Programming Language*
(Submitted on 19 Sep 2018)
"We present its syntax and semantics, and build a compiler translating 
CRN++ programs into chemical reactions...laying the foundation of a 
comprehensive framework for molecular programming."

A programming language whose purpose is to create bugs!

So the question becomes: Is bioprogramming > programming? (if biomatter has 
experiential qualities in addition to informational quantities)

- pt

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