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?
Brent
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