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

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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 our realityhas ﬁnite information content, there must be an axiomatic systemthat 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 Incompleteness theoremfor consistent systems because there are true things aboutconsistent systems that cannot be derived from its axioms? 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 truth isrestricted to arithmetical propositions. Notably because thestatement "the theory is consistent" can be translated into anarithmetical proposition.BrunoDoes arithmetic have 'finite information content'? Is the axiom ofsuccession just one or is it a schema of infinitely many axioms?

Arithmetical truth has infinite information content.

`Peano Arithmetic has about 5K of information content, 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/ -- 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.