Re: The Relativity of Existence

[Bruno Marchal]

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 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 
.




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.

Re: The Relativity of Existence

[meekerdb]

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".




Peano Arithmetic has about 5K of information content, 


Which is just the information in the axioms (actually that number seems high to 
me).

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 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.

Re: The Relativity of Existence

[Bruno Marchal]

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/



--
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.

Re: The Relativity of Existence

[meekerdb]

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

--
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.

Re: The Relativity of Existence

[Bruno Marchal]

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


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.

Re: The Relativity of Existence

[meekerdb]

On 3/1/2012 7:37 PM, Richard Ruquist wrote:



On Thu, Mar 1, 2012 at 7:14 PM, meekerdb > wrote:


On 3/1/2012 9:27 AM, Bob Zannelli wrote:

The Relativity of Existence
Authors: Stuart Heinrich

Subjects: History and Philosophy of Physics (physics.hist-ph); General 
Relativity
and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)

Despite the success of physics in formulating mathematical theories that can
predict the outcome of experiments, we have made remarkably little progress 
towards
answering some of the most basic questions about our existence, such as: 
why does
the universe exist? Why is the universe apparently fine-tuned to be able to 
support
life? Why are the laws of physics so elegant? Why do we have three 
dimensions of
space and one of time? How is it that the universe can be non-local and 
non-causal
at the quantum scale, and why is there quantum randomness? In this paper, 
it is
shown that all of these questions are answered if existence is relative, and
moreover, it seems that we are logically bound to accept it.

http://arxiv.org/pdf/1202.4545.pdf




"To be clear, the idea that our universe is really just a computer 
simulation is
highly controversial and not supported by this paper."
Of course there's no sense in which reality can be a computer 
simulation EXCEPT
if there is a Great Programmer who can fiddle with the program.  Otherwise 
the
simulation and the reality are the same thing.

"By the principle of explosion, in any system that contains a single
contradiction, it becomes possible to prove the truth of any
other statement no matter how nonsensical[34, p.18]. There is
clearly a distinction between truth and falsehood in our reality,
which means that the principle of explosion does not apply to
our reality. In other words, we can be certain that our reality is
consistent."
Hmm? I'd never heard ex falso quodlibet referred to as "the principle of
explosion" before.  But in any case there are ways for preventing a 
contradiction
from implying everything, c.f. Graham Priest's "In Contradiction".  
Contradictions
are between propositions. Heinrich is saying that the lack of 
contradictions in our
propositions describing the world implies the world is consistent.  But at 
the same
time he adopts a MWI which implies that contrary events happen all the time.

"In fact, there are an infinite number of ways to modify an axiomatic 
system while
keeping any particular theorem intact."
This is true if the axioms *and rules of inference* are strong enough 
to satisfy
Godel's incompleteness theorem, something with a rule of finite induction 
(isn't
that technically a schema for an infinite set of axioms?).  Then you are 
guaranteed
infinitely many true propositions which are not provable from your axioms, 
and each
of those can be added as an axiom.  Otherwise I think you only get to add 
infinitely
many axioms by creating arbitrary names, like "aa" and "ab"...

"From the perspective of any self-aware being, something is real if it is 
true,"
A very Platonic and dubious proposition. "True" applies to propositions 
not
things.  2+2=4 is true, but that doesn't imply anything is real.  "Holmes 
friend was
Watson" is true too.

"Recognizing this, the ultimate answer to the question of why our reality 
exists
becomes trivial: because self-awareness can be represented axiomatically, 
any
axiomatic system that can derive self-awareness will be perceived as being 
real
without the need for an objective manifestation."
This is what Bruno Marchal refers to a Lobianity, the provability 
within a
system that there are unprovable true propositions. Marchal formulated this 
idea
before Tegmark and has filled it out and made it more precise (and perhaps 
testable)
by confining it to computation by a univeral dovetailer - not just any 
mathematics.
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html

  
If
you join the everything-list@googlegroups.com
 , he will explain it to you.

"Not many things can be proven objectively true, because
any proof relying on axioms is not objective without proving
that the axioms are also objectively true."
This is confusion bordering on sophistry.  He has introduced a new, 
undefined
concept "objective" and stated that any objectively true statement has an 
objective
proof.  Proof is well defined since it means "following from the axioms by 
the rules
of inference".  Proving something from no axioms just requires more 
powerful rules
of inference.  

Re: The Relativity of Existence

[acw]

On 3/2/2012 03:37, Richard Ruquist wrote:

On Thu, Mar 1, 2012 at 7:14 PM, meekerdb  wrote:


  On 3/1/2012 9:27 AM, Bob Zannelli wrote:

  The Relativity of Existence
Authors: Stuart 
Heinrich
Subjects: History and Philosophy of Physics (physics.hist-ph); General
Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)

Despite the success of physics in formulating mathematical theories that
can predict the outcome of experiments, we have made remarkably little
progress towards answering some of the most basic questions about our
existence, such as: why does the universe exist? Why is the universe
apparently fine-tuned to be able to support life? Why are the laws of
physics so elegant? Why do we have three dimensions of space and one of
time? How is it that the universe can be non-local and non-causal at the
quantum scale, and why is there quantum randomness? In this paper, it is
shown that all of these questions are answered if existence is relative,
and moreover, it seems that we are logically bound to accept it.



http://arxiv.org/pdf/1202.4545.pdf



"To be clear, the idea that our universe is really just a computer
simulation is highly controversial and not supported by this paper."
 Of course there's no sense in which reality can be a computer
simulation EXCEPT if there is a Great Programmer who can fiddle with the
program.  Otherwise the simulation and the reality are the same thing.

"By the principle of explosion, in any system that contains a single
contradiction, it becomes possible to prove the truth of any
other statement no matter how nonsensical[34, p.18]. There is
clearly a distinction between truth and falsehood in our reality,
which means that the principle of explosion does not apply to
our reality. In other words, we can be certain that our reality is
consistent."
 Hmm? I'd never heard ex falso quodlibet referred to as "the principle
of explosion" before.  But in any case there are ways for preventing a
contradiction from implying everything, c.f. Graham Priest's "In
Contradiction".  Contradictions are between propositions. Heinrich is
saying that the lack of contradictions in our propositions describing the
world implies the world is consistent.  But at the same time he adopts a
MWI which implies that contrary events happen all the time.

"In fact, there are an infinite number of ways to modify an axiomatic
system while keeping any particular theorem intact."
 This is true if the axioms *and rules of inference* are strong enough
to satisfy Godel's incompleteness theorem, something with a rule of finite
induction (isn't that technically a schema for an infinite set of
axioms?).  Then you are guaranteed infinitely many true propositions which
are not provable from your axioms, and each of those can be added as an
axiom.  Otherwise I think you only get to add infinitely many axioms by
creating arbitrary names, like "aa" and "ab"...

"From the perspective of any self-aware being, something is real if it is
true,"
 A very Platonic and dubious proposition. "True" applies to
propositions not things.  2+2=4 is true, but that doesn't imply anything is
real.  "Holmes friend was Watson" is true too.

"Recognizing this, the ultimate answer to the question of why our reality
exists becomes trivial: because self-awareness can be represented
axiomatically, any axiomatic system that can derive self-awareness will be
perceived as being real without the need for an objective manifestation."
 This is what Bruno Marchal refers to a Lobianity, the provability
within a system that there are unprovable true propositions. Marchal
formulated this idea before Tegmark and has filled it out and made it more
precise (and perhaps testable) by confining it to computation by a univeral
dovetailer - not just any mathematics.
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html
If you join the everything-list@googlegroups.com , he will explain it to
you.

"Not many things can be proven objectively true, because
any proof relying on axioms is not objective without proving
that the axioms are also objectively true."
 This is confusion bordering on sophistry.  He has introduced a new,
undefined concept "objective" and stated that any objectively true
statement has an objective proof.  Proof is well defined since it means
"following from the axioms by the rules of inference".  Proving something
from no axioms just requires more powerful rules of inference.  There's no
principled distinction between rules of inference and axioms.

"If the ROE is correct, then reality is defined by the things that
are provably true, and any additional undecidable statements
simply have no bearing on that reality."
 But does he mean provably true from zero axioms plus the usual rules
of first (or second) order logic?  Earlier he argued that the world must be
an axiomatic system because you could just define it

Re: The Relativity of Existence

[Richard Ruquist]

On Thu, Mar 1, 2012 at 7:14 PM, meekerdb  wrote:

>  On 3/1/2012 9:27 AM, Bob Zannelli wrote:
>
>  The Relativity of Existence
> Authors: Stuart 
> Heinrich
> Subjects: History and Philosophy of Physics (physics.hist-ph); General
> Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)
>
> Despite the success of physics in formulating mathematical theories that
> can predict the outcome of experiments, we have made remarkably little
> progress towards answering some of the most basic questions about our
> existence, such as: why does the universe exist? Why is the universe
> apparently fine-tuned to be able to support life? Why are the laws of
> physics so elegant? Why do we have three dimensions of space and one of
> time? How is it that the universe can be non-local and non-causal at the
> quantum scale, and why is there quantum randomness? In this paper, it is
> shown that all of these questions are answered if existence is relative,
> and moreover, it seems that we are logically bound to accept it.
>
>
>
> http://arxiv.org/pdf/1202.4545.pdf
>
>
>
> "To be clear, the idea that our universe is really just a computer
> simulation is highly controversial and not supported by this paper."
> Of course there's no sense in which reality can be a computer
> simulation EXCEPT if there is a Great Programmer who can fiddle with the
> program.  Otherwise the simulation and the reality are the same thing.
>
> "By the principle of explosion, in any system that contains a single
> contradiction, it becomes possible to prove the truth of any
> other statement no matter how nonsensical[34, p.18]. There is
> clearly a distinction between truth and falsehood in our reality,
> which means that the principle of explosion does not apply to
> our reality. In other words, we can be certain that our reality is
> consistent."
> Hmm? I'd never heard ex falso quodlibet referred to as "the principle
> of explosion" before.  But in any case there are ways for preventing a
> contradiction from implying everything, c.f. Graham Priest's "In
> Contradiction".  Contradictions are between propositions. Heinrich is
> saying that the lack of contradictions in our propositions describing the
> world implies the world is consistent.  But at the same time he adopts a
> MWI which implies that contrary events happen all the time.
>
> "In fact, there are an infinite number of ways to modify an axiomatic
> system while keeping any particular theorem intact."
> This is true if the axioms *and rules of inference* are strong enough
> to satisfy Godel's incompleteness theorem, something with a rule of finite
> induction (isn't that technically a schema for an infinite set of
> axioms?).  Then you are guaranteed infinitely many true propositions which
> are not provable from your axioms, and each of those can be added as an
> axiom.  Otherwise I think you only get to add infinitely many axioms by
> creating arbitrary names, like "aa" and "ab"...
>
> "From the perspective of any self-aware being, something is real if it is
> true,"
> A very Platonic and dubious proposition. "True" applies to
> propositions not things.  2+2=4 is true, but that doesn't imply anything is
> real.  "Holmes friend was Watson" is true too.
>
> "Recognizing this, the ultimate answer to the question of why our reality
> exists becomes trivial: because self-awareness can be represented
> axiomatically, any axiomatic system that can derive self-awareness will be
> perceived as being real without the need for an objective manifestation."
> This is what Bruno Marchal refers to a Lobianity, the provability
> within a system that there are unprovable true propositions. Marchal
> formulated this idea before Tegmark and has filled it out and made it more
> precise (and perhaps testable) by confining it to computation by a univeral
> dovetailer - not just any mathematics.
> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html
> If you join the everything-list@googlegroups.com , he will explain it to
> you.
>
> "Not many things can be proven objectively true, because
> any proof relying on axioms is not objective without proving
> that the axioms are also objectively true."
> This is confusion bordering on sophistry.  He has introduced a new,
> undefined concept "objective" and stated that any objectively true
> statement has an objective proof.  Proof is well defined since it means
> "following from the axioms by the rules of inference".  Proving something
> from no axioms just requires more powerful rules of inference.  There's no
> principled distinction between rules of inference and axioms.
>
> "If the ROE is correct, then reality is defined by the things that
> are provably true, and any additional undecidable statements
> simply have no bearing on that reality."
> But does he mean provably true from zero axioms plus the usual rules
> of first (or sec

Re: The Relativity of Existence

[meekerdb]

On 3/1/2012 9:27 AM, Bob Zannelli wrote:

The Relativity of Existence
Authors: Stuart Heinrich 

Subjects: History and Philosophy of Physics (physics.hist-ph); General Relativity and 
Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)


Despite the success of physics in formulating mathematical theories that can predict the 
outcome of experiments, we have made remarkably little progress towards answering some 
of the most basic questions about our existence, such as: why does the universe exist? 
Why is the universe apparently fine-tuned to be able to support life? Why are the laws 
of physics so elegant? Why do we have three dimensions of space and one of time? How is 
it that the universe can be non-local and non-causal at the quantum scale, and why is 
there quantum randomness? In this paper, it is shown that all of these questions are 
answered if existence is relative, and moreover, it seems that we are logically bound to 
accept it.


http://arxiv.org/pdf/1202.4545.pdf




"To be clear, the idea that our universe is really just a computer simulation is highly 
controversial and not supported by this paper."
Of course there's no sense in which reality can be a computer simulation EXCEPT if 
there is a Great Programmer who can fiddle with the program.  Otherwise the simulation and 
the reality are the same thing.


"By the principle of explosion, in any system that contains a single
contradiction, it becomes possible to prove the truth of any
other statement no matter how nonsensical[34, p.18]. There is
clearly a distinction between truth and falsehood in our reality,
which means that the principle of explosion does not apply to
our reality. In other words, we can be certain that our reality is
consistent."
Hmm? I'd never heard ex falso quodlibet referred to as "the principle of explosion" 
before.  But in any case there are ways for preventing a contradiction from implying 
everything, c.f. Graham Priest's "In Contradiction".  Contradictions are between 
propositions. Heinrich is saying that the lack of contradictions in our propositions 
describing the world implies the world is consistent.  But at the same time he adopts a 
MWI which implies that contrary events happen all the time.


"In fact, there are an infinite number of ways to modify an axiomatic system while keeping 
any particular theorem intact."
This is true if the axioms *and rules of inference* are strong enough to satisfy 
Godel's incompleteness theorem, something with a rule of finite induction (isn't that 
technically a schema for an infinite set of axioms?).  Then you are guaranteed infinitely 
many true propositions which are not provable from your axioms, and each of those can be 
added as an axiom.  Otherwise I think you only get to add infinitely many axioms by 
creating arbitrary names, like "aa" and "ab"...


"From the perspective of any self-aware being, something is real if it is true,"
A very Platonic and dubious proposition. "True" applies to propositions not things.  
2+2=4 is true, but that doesn't imply anything is real.  "Holmes friend was Watson" is 
true too.


"Recognizing this, the ultimate answer to the question of why our reality exists becomes 
trivial: because self-awareness can be represented axiomatically, any axiomatic system 
that can derive self-awareness will be perceived as being real without the need for an 
objective manifestation."
This is what Bruno Marchal refers to a Lobianity, the provability within a system 
that there are unprovable true propositions. Marchal formulated this idea before Tegmark 
and has filled it out and made it more precise (and perhaps testable) by confining it to 
computation by a univeral dovetailer - not just any mathematics. 
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html  If you join 
the everything-list@googlegroups.com , he will explain it to you.


"Not many things can be proven objectively true, because
any proof relying on axioms is not objective without proving
that the axioms are also objectively true."
This is confusion bordering on sophistry.  He has introduced a new, undefined concept 
"objective" and stated that any objectively true statement has an objective proof.  Proof 
is well defined since it means "following from the axioms by the rules of inference".  
Proving something from no axioms just requires more powerful rules of inference.  There's 
no principled distinction between rules of inference and axioms.


"If the ROE is correct, then reality is defined by the things that
are provably true, and any additional undecidable statements
simply have no bearing on that reality."
But does he mean provably true from zero axioms plus the usual rules of first (or 
second) order logic?  Earlier he argued that the world must be an axiomatic system because 
you could just define it by one axiom for each fact.  Which would make the 'axiom