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.
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/
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