On 5/28/2012 12:36 AM, Bruno Marchal wrote:

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On 27 May 2012, at 20:59, meekerdb wrote:On 5/27/2012 5:02 AM, Bruno Marchal wrote:As Bruno said, "Provable is always relative to some axioms and rules of inference.It is quite independent of "true of reality". Which is why I'm highly suspicious ofideas like deriving all of reality from arithmetic, which we know only from axiomsand inferences.We don't give axioms and inference rule when teaching arithmetic in high school. Westart from simple examples, like fingers, days of the week, candies in a bag, etc.Children understand "anniversary" before "successor", and the finite/infinitedistinction is as old as humanity.In fact it can be shown that the intuition of numbers, addition and multiplicationincluded, is *needed* to even understand what axioms and inference can be, makingarithmetic necessarily known before any formal machinery is posited.But only a small finite part of arithmetic.I don't think so. Our arithmetical intuition is already not formalizable. If it was, wewould be able to capture it by a finite number of principle, but then we would bepersuade that such finite theory is consistent, and that intuition is not in the theory.I suspect that our intuition is full second order arithmetic, which is notaxiomatizable. In fact it is the very distinction between finite and infinite that wecannot formalize. Like consciousness, we know very well what finite/infinite means, butwe cannot defined it, without using implicitly that distinction. The natural numbers are*the* mystery, and it has to be like that: no machine will ever been able to define whatthey are. Assuming comp, neither will we. Arithmetical truth per se, as no correspondingcomplete TOE. It is inexhaustible.Bruno

`Our intuition is that space is euclidean, the earth is stationary and flat, and that there`

`is only one world. It seems to me that the infinity of arithmetic is just the intuition`

`we should always be able to add one more. But intuition fails us in precisely in`

`questions like Hilbert's hotel. Why should you be so trusting of your intuition is just`

`this particular instance.`

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