On 05 May 2009, at 22:31, Jason Resch wrote:

> On Sun, May 3, 2009 at 10:56 AM, Bruno Marchal <marc...@ulb.ac.be>  
> wrote:
>> With just arithmetic, when we stop to postulate a primitive or
>> ontological material world, all primitive ad-hocness is removed,  
>> given
>> that the existing internal interpretations are all determined, with
>> their relative frequency, by addition and multiplication rules, and
>> physics will be defined by the (absolute) probability of relative
>> computations (here = probability of relative number theoretical
>> relations.
> Bruno,
> In other posts I have seen you mention that the rule of succession is
> not enough, that addition and multiplication are needed.  Why is it
> that it stops at multiplication, and not exponentiation or tetration?
> Is it enough to say some form of iteration + succession are required?
> (e.g. a for loop with succession gives addition, a for loop with
> addition yields multiplication, etc.)
> Jason

It is due to the fact that, when formalized (in first order logic,  
say) Turing Universality begins with addition and multiplication (you  
don't even need succession). Then you can define exponentiation,  
tetration, etc. All partial recursive function can then be defined.

Succession + addition, or succession + multiplication, are not Turing  
Universal, and leads indeed to decidable theories.

For the "ontology" we need no more than a universal system. it  
determines the universal dovetailing.

For the "epistemology" we need succession, addition, multiplication  
and the axioms of induction. This gives a notion of universal system  
together with its internal "self-aware" substructures played by the  
Lobian machine and their consistent extensions (the believer in  
induction), simulated by the universal systems. Those internal  
machines will develop far beyond "simple induction" though. The  
general internal view (the first person plenitude) is not axiomatisable.

Iteration and succession? I don't think so. You need induction. With  
induction it is Turing Universal, but not without, I think. It could  
depend how you formalized the iteration rule, but without induction  
and staying in first order logic, that would astonish me.

The crazy thing, not so simple to prove, is that even without  
induction, addition + multiplication is Turing universal. You bypass  
the role of induction by defining finite sequence through Gödel bata  
function and an ingenuous use of the Chinese Lemma.

Far easier to prove, without induction, is that addition+multiplication 
+exponentiation is Turing universal, but thanks to Godel' beta  
function you can eliminate exponentiation. If you know "Godel's  
original "Godel's numbering" you can guess why.


> >


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