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/
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to [email protected]
.
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
.
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 [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
