> On 15 Nov 2018, at 18:13, Philip Thrift <cloudver...@gmail.com> wrote:
> 
> 
> 
> 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 
>> <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 
>> <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:
> 
> http://www.cs.bham.ac.uk/~mhe/papers/introduction-to-higher-order-computation-NLS-2017.pdf
> 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 computation."
> 
> 
> 
> But there is a new biochemical programming language:
> 
> CRN++: Molecular Programming Language
> (Submitted on 19 Sep 2018)
> https://arxiv.org/abs/1809.07430 
> "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)

Assuming some primary matter, and some non mechanist theory, why not. That 
seems to quite speculative, though, and adding difficulties to a subject which 
is already difficult when assuming the “simplifying” assumption of Mechanism. 
With mechanism, the mind-body problem reduced into justifying the existence of 
a canonical measure on all computations “seen from inside” (which admits a 
number of modes, imposed by incompleteness). In case the physics in the head of 
the universal machine/number departs from observation, we get the mean to make 
sense of some non-mechanism, and this might show you right. So let us continue 
the testing/comparison.

What do you think your biomatter do which would be non Turing emulable, nor 
“first person measurable(*) and in what sense would that be relevant with 
respect of consciousness?

I have no doubt chemical computation is a wonderful subject, but with 
“Indexical Digital Mechanism”, the theology and the physics is independent of 
the language and the basic theories as far as they are Turing complete(*), the 
physical appearance, needs to be justified in term of a relative measure 
state/computations "seen from inside” (Incompleteness makes the usual standard 
definition getting sense in those “enough rich” Turing complete(**) theories. 

Bruno


(*) This provides some “free oracle”, like the random oracle and the halting 
oracle, due to the limiting behaviour of the first person indeterminacy).

(**) Turing complete means that for all p sigma_1 (shape ExA(x, y), A 
decidable) we have, with “[]” Gödel’s arithmetical provability predicate,

                   p -> []p

is true. 

Löbian (sufficiently rich) means that for all such p,"p -> []p" is not only 
true, but provable. Put it in another way, this means that

                   [](p -> []p)

is true. (This makes the machine obeying to G and G* and their intensional 
variants).

(See all definitions in the second part of sane04, I recall them in most of my 
papers).










> 
> - pt
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