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 isconsistent can be formalized into an axiomatic system, forexample by using one axiom to assert the truth of eachindependent piece of information. Thus, assuming that ourreality has ﬁnite information content, there must be anaxiomatic system that isisomorphic to our reality, where every true thing about realitycan be proved as a theorem from the axioms of that system"Doesn't this thinking contradict Goedel's Incompletenesstheorem for consistent systems because there are true thingsabout consistent systems that cannot be derived from itsaxioms? RichardPresumably those true things would not be 'real'. Only provablethings would be true of reality.Provable depends on the theory. If the theory is unsound, what itproves might well be false.And if you trust the theory, then you know that "the theory isconsistent" is true, yet the theory itself cannot prove it, soreality is larger that what you can prove in that theory.So in any case truth is larger than the theory. Even when truthis restricted to arithmetical propositions. Notably because thestatement "the theory is consistent" can be translated into anarithmetical proposition.BrunoDoes arithmetic have 'finite information content'? Is the axiomof 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 contradictGodel's incompleteness because it refers to "systems with finiteinformation 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 numberseems 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

Brenteven with the infinitely many induction axioms, for they are simpleto 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/--You received this message because you are subscribed to the GoogleGroups "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.

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