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

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 suspicious of ideas like deriving all of reality from arithmetic, which we know only from axioms and inferences.

We don't give axioms and inference rule when teaching arithmetic in high school. We start from simple examples, like fingers, days of the week, candies in a bag, etc. Children understand "anniversary" before "successor", and the finite/infinite distinction is as old as humanity. In fact it can be shown that the intuition of numbers, addition and multiplication included, is *needed* to even understand what axioms and inference can be, making arithmetic 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 be persuade 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 not axiomatizable. In fact it is the very distinction between finite and infinite that we cannot formalize. Like consciousness, we know very well what finite/infinite means, but we 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 what they are. Assuming comp, neither will we. Arithmetical truth per se, as no corresponding complete TOE. It is inexhaustible.


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.

Not really. I think there are complete theories of (N, successor). But we have an intuition of adding and multiplying and this makes that intuition inexhaustible. Intuition is not entirely a given, it is something which develop with the familiarity and life working. It is different for all of us, so it nice that we can share some big initial segment of the arithmetical truth. Comp does not need more than the sigma_1 intuition, 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 finite things.

Why should you be so trusting of your intuition is just this particular 

Do you doubt elementary arithmetic?

I doubt infinities.


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