On 27 May 2012, at 01:41, meekerdb wrote:

On 5/26/2012 12:11 PM, Bruno Marchal wrote:

On 26 May 2012, at 17:56, meekerdb wrote:

On 5/26/2012 2:16 AM, Bruno Marchal wrote:

On 02 Mar 2012, at 06:18, meekerdb wrote (two month agao):

On 3/1/2012 7:37 PM, Richard Ruquist wrote:


Excerpt: "Any system with finite information content that is consistent can be formalized into an axiomatic system, for example by using one axiom to assert the truth of each independent piece of information. Thus, assuming that our reality has finite information content, there must be an axiomatic system that is isomorphic to our reality, where every true thing about reality can be proved as a theorem from the axioms of that system"

Doesn't this thinking contradict Goedel's Incompleteness theorem for consistent systems because there are true things about consistent systems that cannot be derived from its axioms? Richard

Presumably those true things would not be 'real'. Only provable things would be true of reality.

Provable depends on the theory. If the theory is unsound, what it proves might well be false.

And if you trust the theory, then you know that "the theory is consistent" is true, yet the theory itself cannot prove it, so reality is larger that what you can prove in that theory.

So in any case truth is larger than the theory. Even when truth is restricted to arithmetical propositions. Notably because the statement "the theory is consistent" can be translated into an arithmetical proposition.

Bruno

Does arithmetic have 'finite information content'? Is the axiom of succession just one or is it a schema of infinitely many axioms?

Arithmetical truth has infinite information content.

That's what I thought. So the above Excerpt does not contradict Godel's incompleteness because it refers to "systems with finite information content".

Gödel's theorem applies also to many systems with infinite information content. Even arithmetical truth itself is undecided on many second order arithmetical propositions, and some occurs naturally like in the G* (first order) modal logic.

Arithmetic has few information content, but "arithmetic seen from inside" as an infinite (and beyond!) information content. This should be the case for any proposed TOE.




Peano Arithmetic has about 5K of information content,

Which is just the information in the axioms (actually that number seems high to me).

OK. (I said 5K to imply it is very little, but 5K is much too much indeed). Note that my computer already uses 4K for an empty document, but that kind of thing is very contingent.

Bruno



Brent

even with the infinitely many induction axioms, for they are simple to generate. There are two succession axioms (0 ≠ s(x), and s(x) = s(y) .-> x = y. Those are not scheme of axioms.

Bruno



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




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