Hi John,

On 17 Sep 2009, at 15:14, John Mikes wrote:
>
>
> You went out of your way and did not save efforts to prove how  
> inadequate and wrong (y)our number system is. (ha ha).

Wrong ?



>
> Statement: if square-rooting is right (allegedly, and admittedly)


Well, it is certainly right if we want that to measure by a number  
length of the diagonal of the square unity.




> then THERE IS such a 'quantity' (call it number and by this  
> definition it must be natural)


It cannot be natural number. It has to be strictly bigger than 1, and  
strictly bigger than 2. But there was some hope that it could be the  
ratio of two natural number, so that it can live in arithmetic with  
addition and multiplication.




> we consider as the square root of '2'.
> You gave the plastic elemenary rule, how to get to it. Thankyou.  
> Accepted.
>
> I believe the '1' and the sophistication of Pythagoras. (provided that
> < 1^2 = 1 >  which is also 'funny')  a n d :
> If it is not part of your series of - what you call: - natural  
> numbers, then YOUR series is wrong.


You could as well told something like "My cave is at the level -2  
(minus two) of the building ...



> We need another system (if we really need it).

That is why N (the set of natiral numbers, alias positive integers)  
has been extended into Z, all the integers, itself included in Q (he  
ratio). The my point was that Q was still not enough to define the  
length of the diagonal, we need the real numbers, which are more  
difficult to define in the structure (N, +, *).

 From a logical point of view, N, Z, and Q are roughly equivalent. The  
real numbers are not, most are not definable in the structure N. yet,  
and we will see this (probably), most real numbers that we encounter  
in math and physics can still be defined in the structure (N,+,*). It  
is an open problem in math and physics if there is anything we cannot  
define in (N,+,*), and indeed it is an indirect consequence of COMP  
that we can. This probably why formal set theory is studied only by  
logicians.

Of course Riemann and number theorists, and knot theorists, are used  
to escape from the (N, +, *) structure all the time, and that is why  
we use analysis (based on the real numbers). But we have not yet find  
a theorem which *needs* to escape the structure (N, +, *), except  
those found by logicians to just provide examples. In the mechanical  
theory of mind, we have to escape the structure (N, +, *). Indeed the  
first person notion needs even more than Cantor paradise, from its  
point of view, and that is why and how the epistemology of comp is  
necessarily far bigger than its ontology. I may come back on this, but  
it asks for more model theory and logic to address the question  
technically.

Bruno


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




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