Hi Mohsen,

Le 22-mai-07, à 12:20, Mohsen Ravanbakhsh a écrit :


> Hi Bruno,
>  
> My sixth sens says you're talking about something important :) but I 
> don't get it.


Note that it could help me if you could be a little more specific. OK I 
see another post of you.




> It could have been of much more interest, if you could elaborate, or 
> provide us with some references for each part of your


So you are able to make sense of the fact that   
[LOGIC+ADDITION+MULTIPLICATION] gives already a Universal Turing 
Machine. This is no more astosnishing than the fact that the K and S 
combinators provides already turing-universality, or that the Conway 
Game of Life is already turing universal.
The advantage of [LOGIC+ADDITION+MULTIPLICATION]  is that (universal) 
computability is seen as a particular case of provability.

What is more long to explain in details is that 
[LOGIC+ADDITION+MULTIPLICATION + INDUCTION] is already lobian.  But I 
will first look to your other post which title refer to incompleteness.



> argument.(Beginning from the 'OBVIOUS IMPORTANT QUESTION' it 
> becomes vague for me)


The key point consists in understanding the difference between 
computability/simulability and provability. I will come back on this, 
but the idea is that, assuming comp, I can simulate Einstein's brain 
exactly, and still not share his beliefs. Similarly the very non 
powerful Little-Robinson-arithmetic can simulate rich theories like 
PEANO or ZF, but cannot prove the theorem of PA or ZF.

For example PA can prove that ZF can prove the consistency of PA, yet, 
PA cannot prove the consistency of PA.

Bruno



http://iridia.ulb.ac.be/~marchal/

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