On 5/28/2012 1:13 PM, Bruno Marchal wrote:

## Advertising

On 28 May 2012, at 18:02, meekerdb wrote:On 5/28/2012 12:36 AM, Bruno Marchal wrote: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 suspiciousof ideas like deriving all of reality from arithmetic, which we know only fromaxioms and 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,we would 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,but we cannot defined it, without using implicitly that distinction. The naturalnumbers are *the* mystery, and it has to be like that: no machine will ever been ableto define what they are. Assuming comp, neither will we. Arithmetical truth per se, asno corresponding complete TOE. It is inexhaustible.BrunoOur intuition is that space is euclidean, the earth is stationary and flat, and thatthere is only one world. It seems to me that the infinity of arithmetic is just theintuition we should always be able to add one more.Not really. I think there are complete theories of (N, successor). But we have anintuition of adding and multiplying and this makes that intuition inexhaustible.Intuition is not entirely a given, it is something which develop with the familiarityand life working. It is different for all of us, so it nice that we can share some biginitial segment of the arithmetical truth. Comp does not need more than the sigma_1intuition, at the ontic level.But intuition fails us in precisely in questions like Hilbert's hotel.Why? Not sure, but it does not concern us, as comp builds on the intuition of the finitethings.Why should you be so trusting of your intuition is just this particular instance.Do you doubt elementary arithmetic?

I doubt infinities. Brent -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.