subject:"A possible structure isomorphic to reality"

Re: A possible structure isomorphic to reality

2010-12-17 Thread Brian Tenneson

There is evidently a weaker version of the embedding concept.
http://en.wikipedia.org/wiki/Embedding#Universal_algebra_and_model_theory
(No references as far as I can tell for this definition)

I am looking at this definition and the flaw in my proof on page 13
and, while I will have to study it further, preliminarily, it appears
that this weakened concept of embedding will work.

That is to say that the theorem on page 12 will be correct if I simply
remove the word elementary.

The Wiki article is somewhat dubious in lacking references to this
weakened version of embedding.  I don't see this in Chang and Kiesler
(so far).

The definition given seems to, intuitively, say that A is embedded in
B via h if h is 1-1, h preserves the interpretation of function
symbols (I'm not sure how else to state that yet), and h preserves the
truth of relations.  The last bit is significantly weaker than
preserving the truth of all formulas.

In fact, I never needed the embedding to be elementary.

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Re: A possible structure isomorphic to reality

2010-12-10 Thread Bruno Marchal

On 09 Dec 2010, at 20:43, Brian Tenneson wrote:

Is there any first order formula true in only one of R and R*?
I would think that if the answer is NO then R R*.
What I'm exploring is the connection of to [=], with the statement
that implies [=].

The elementary embeddings preserve the truth of all first order
formula. So it should be obvious that if A B, then A [=] B.
In B there might be elements or objects or set of objects obeying
relations which are not consequences of the first order relations.
I think that all standard models of first order theories of finite
structures (like numbers, hereditarily finites sets, rational numbers,
etc.) are elementary equivalent with their non standard models. You
need second order logic to describe what happens in those models.

But I have not invest on model theory since some time.

Are there any other comparitive relations besides elementary embedding
that would fit with what I'm trying to do? What I'm trying to do is
one major leg of my paper: there is a superstructure to all
structures.

But sets and categories have been seen that way. This leads to
reductionism in math, in my opinion. Yet category theory provides
ubiquitous non trivial relations between many mathematical objects.
But Lawvere failed to found mathematics on the category of categories.
And categories with partial objects, like those which populate so much
computer science, are, well, quite close to abstract unintelligibility
(for me, but who knows). Category impresses me the most in knot
theory, and the buildings of models for weak logics (linear logic,
intuitionist logics, quantum linear logic).

What super means could be any comparitive relation. But
what relation is 'good'?

You ask a very difficult question. You might appreciate morphism of
categories (functor), or of morphism of bicategories, or n-categories,
if you want powerful abstractions.

But assuming mechanism, and the 'everything goal': I would insist on
the relations of 'dreaming', or partial emulation between numbers
relatively to universal numbers.
The infinite dynamical mirroring of the universal numbers. That just
exist if we assume the axiom of Robinson arithmetic, and we are
embedded or better: distributed, or multi-dreamed by or in it (with
our richer axioms!) and all, this with notions of neighborhoods and
accessibility between our consistent extensions (that you can extract
from studying what can and cannot prove sound löbian numbers about
themselves. See my papers for more on that, and good basic books are
Boolos 1979, 1993, Smullyan, Rogers, etc).

It depends on what you are searching for. If you want to include
psychology and theology, expect some universal mess diagonalizing
against all complete reductions.

Bruno

On Dec 9, 8:12 am, Bruno Marchal marc...@ulb.ac.be wrote:

On 09 Dec 2010, at 05:12, Brian Tenneson wrote:

On Dec 5, 12:02 pm, Bruno Marchal marc...@ulb.ac.be wrote:

On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

That means that R (standard model of the first order theory of the
reals + archimedian axiom, without the term natural number) is
not

elementary embeddable in R*, given that such an embedding has to
preserve all first order formula (purely first order formula, and
so

without notion like natural number).

I'm a bit confused. Is R R* or not? I thought there was a fairly
natural way to elementarily embed R in R*.

I would say that NOT(R R*).

*You* gave me the counter example. The archimedian axiom. You are
confusing (like me when I read your draft the first time) an
algebraical injective morphism with an elementary embedding. But
elementary embedding conserves the truth of all first order formula,
and then the archimedian axiom (without natural numbers) is true in R
but not in R*.

Elementary embeddings are *terribly* conservator, quite unlike
algebraical monomorphism or categorical arrows, or Turing emulations.

Bruno

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Re: A possible structure isomorphic to reality

2010-12-10 Thread Bruno Marchal

Just to be clear on this:

On 09 Dec 2010, at 20:43, Brian Tenneson wrote:

Is there any first order formula true in only one of R and R*?

So yes, there is one: the weak pure archimedian formula AF:

AF: for all x there is a y such that (xy)

(not your: for all X there is a Y such that (Y is a natural number
and XY), because this is a second order formula. You cannot defined
natural number in first order logic (actually you cannot defined
finite in first order logic).

I would think that if the answer is NO then R R*.

You would be right. But AF is true in R, and false in R*

In R* there is an object infinity which is such that there is no y
such that infinity y, making AF false.

Bruno

What I'm exploring is the connection of to [=], with the statement
that implies [=].

On Dec 9, 8:12 am, Bruno Marchal marc...@ulb.ac.be wrote:

On 09 Dec 2010, at 05:12, Brian Tenneson wrote:

On Dec 5, 12:02 pm, Bruno Marchal marc...@ulb.ac.be wrote:

On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

That means that R (standard model of the first order theory of the
reals + archimedian axiom, without the term natural number) is
not

elementary embeddable in R*, given that such an embedding has to
preserve all first order formula (purely first order formula, and
so

without notion like natural number).

I'm a bit confused. Is R R* or not? I thought there was a fairly
natural way to elementarily embed R in R*.

I would say that NOT(R R*).

Elementary embeddings are *terribly* conservator, quite unlike
algebraical monomorphism or categorical arrows, or Turing emulations.

Bruno

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Re: A possible structure isomorphic to reality

2010-12-09 Thread Bruno Marchal

On 09 Dec 2010, at 05:12, Brian Tenneson wrote:

On Dec 5, 12:02 pm, Bruno Marchal marc...@ulb.ac.be wrote:

On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

That means that R (standard model of the first order theory of the
reals + archimedian axiom, without the term natural number) is not
elementary embeddable in R*, given that such an embedding has to
preserve all first order formula (purely first order formula, and so
without notion like natural number).

I'm a bit confused. Is R R* or not? I thought there was a fairly
natural way to elementarily embed R in R*.

I would say that NOT(R R*).

Elementary embeddings are *terribly* conservator, quite unlike
algebraical monomorphism or categorical arrows, or Turing emulations.

Bruno

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Re: A possible structure isomorphic to reality

2010-12-09 Thread Brian Tenneson

Is there any first order formula true in only one of R and R*?
I would think that if the answer is NO then R  R*.
What I'm exploring is the connection of  to [=], with the statement
that  implies [=].

Are there any other comparitive relations besides elementary embedding
that would fit with what I'm trying to do?  What I'm trying to do is
one major leg of my paper: there is a superstructure to all
structures.  What super means could be any comparitive relation.  But
what relation is 'good'?

On Dec 9, 8:12 am, Bruno Marchal marc...@ulb.ac.be wrote:
 On 09 Dec 2010, at 05:12, Brian Tenneson wrote:



  On Dec 5, 12:02 pm, Bruno Marchal marc...@ulb.ac.be wrote:
  On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

  That means that R (standard model of the first order theory of the
  reals + archimedian axiom, without the term natural number) is not
  elementary embeddable in R*, given that such an embedding has to
  preserve all first order formula (purely first order formula, and so
  without notion like natural number).

  I'm a bit confused.  Is R  R* or not?  I thought there was a fairly
  natural way to elementarily embed R in R*.

 I would say that NOT(R  R*).

 *You* gave me the counter example. The archimedian axiom. You are  
 confusing (like me when I read your draft the first time) an  
 algebraical injective morphism with an elementary embedding. But  
 elementary embedding conserves the truth of all first order formula,  
 and then the archimedian axiom (without natural numbers) is true in R  
 but not in R*.

 Elementary embeddings are *terribly* conservator, quite unlike  
 algebraical monomorphism or categorical arrows, or Turing emulations.

 Bruno



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Re: A possible structure isomorphic to reality

2010-12-08 Thread Brian Tenneson



On Dec 5, 12:02 pm, Bruno Marchal marc...@ulb.ac.be wrote:
 On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

 That means that R (standard model of the first order theory of the
 reals + archimedian axiom, without the term natural number) is not
 elementary embeddable in R*, given that such an embedding has to
 preserve all first order formula (purely first order formula, and so
 without notion like natural number).



I'm a bit confused.  Is R  R* or not?  I thought there was a fairly
natural way to elementarily embed R in R*.

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Re: A possible structure isomorphic to reality

2010-12-05 Thread Bruno Marchal

On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

On Dec 4, 2:52 am, Bruno Marchal marc...@ulb.ac.be wrote:

I just said that if M1 M2, then M1 [=] M2. This means that M2 needs
higher order logical formula to be distinguished from M1.
Elementary embeddings () are a too much strong notion of model
theory. It is used in context where we want use non standard notions,
like in Robinson analysis.

Doesn't the archemedian property show that R is not elementarily
equivalent to R*? I mean the following 1st order formula true in only
one of R and R*:
for all X there is a Y such that (Y is a natural number and XY)

Note that you cannot define natural number in a first order theory
of the reals. In the reals, natural numbers are second order notions,
or you have to add a first order axiomatic of the sinusoïdal function.

This is true in R but not in R*. This would appear to me to be an
example of why R is not [=] to R*.

Bruno

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Re: A possible structure isomorphic to reality

2010-12-04 Thread Bruno Marchal

On 03 Dec 2010, at 18:56, Brian Tenneson wrote:

I'm going to try to concentrate on each issue, one per post. Let me
say again that your feedback is absolutely invaluable to my work.

In an earlier post you say something that implies the following:
Suppose M1, M2, and M3 are mathematical structures
Let denote the elementarily embedded relation
Let [=] denote elementarily equivalence

(A) If M1 M3 and M2 M3 then M1 [=] M2.

Consequently, one of my theorems must be wrong since all structures
elementarily embedded within U implies all structures are elementarily
equivalent, which is false.

Firstly, is (A) implied by your statement quoted here?
[quote]
If they are all elementary embeddable within it [my structure U], then
they are all
elementary equivalent, given that the truth of first order formula
are
preserved. All mathematical theories would have the same theorems.
So
eventually there has to be something wrong in your theorem. [unquote]

I don't think so. I think (A) is false.

Secondly, I believe I can prove (A) is false, thus restoring
plausibility of my theorem on page 12. I still need to prove theorem
on page 12 if possible, of course fixing the flaw which might be a
flaw in the way it was stated. Perhaps I can rescue the main theorem
even if I weaken the theorem on page 12.

Hmm...

However, if (A) is not implied by your remarks then I wouldn't need to
try to prove it false as my proof would be a moot point.
If (A) is implied by your remarks then I will show my proof that (A)
is false.

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Re: A possible structure isomorphic to reality

2010-12-04 Thread Brian Tenneson

On Dec 4, 2:52 am, Bruno Marchal marc...@ulb.ac.be wrote:

 I just said that if M1  M2, then M1 [=] M2. This means that M2 needs  
 higher order logical formula to be distinguished from M1.
 Elementary embeddings () are a too much strong notion of model  
 theory. It is used in context where we want use non standard notions,  
 like in Robinson analysis.

Doesn't the archemedian property show that R is not elementarily
equivalent to R*?  I mean the following 1st order formula true in only
one of R and R*:
for all X there is a Y such that (Y is a natural number and XY)

This is true in R but not in R*.  This would appear to me to be an
example of why R is not [=] to R*.

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Re: A possible structure isomorphic to reality

2010-12-03 Thread Brian Tenneson

I'm going to try to concentrate on each issue, one per post.  Let me
say again that your feedback is absolutely invaluable to my work.

In an earlier post you say something that implies the following:
Suppose M1, M2, and M3 are mathematical structures
Let  denote the elementarily embedded relation
Let [=] denote elementarily equivalence

(A) If M1  M3 and M2  M3 then M1 [=] M2.

Consequently, one of my theorems must be wrong since all structures
elementarily embedded within U implies all structures are elementarily
equivalent, which is false.

Firstly, is (A) implied by your statement quoted here?
[quote]
If they are all elementary embeddable within it [my structure U], then
they are all
elementary equivalent, given that the truth of first order formula
are
preserved. All mathematical theories would have the same theorems.
So
eventually there has to be something wrong in your theorem. [unquote]

Secondly, I believe I can prove (A) is false, thus restoring
plausibility of my theorem on page 12.  I still need to prove theorem
on page 12 if possible, of course fixing the flaw which might be a
flaw in the way it was stated.  Perhaps I can rescue the main theorem
even if I weaken the theorem on page 12.

However, if (A) is not implied by your remarks then I wouldn't need to
try to prove it false as my proof would be a moot point.
If (A) is implied by your remarks then I will show my proof that (A)
is false.

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Re: A possible structure isomorphic to reality

2010-10-17 Thread Bruno Marchal



On 16 Oct 2010, at 23:45, Brian Tenneson wrote:



If they are all elementary embeddable within it, then they are all
elementary equivalent, given that the truth of first order formula  
are

preserved.


How would all structures be elementarily equivalent?


If M1 is an elementarily substructure of M2, you cannot distinguish M1  
and M2 elementarily, that is by a first order closed formula  
(sentence). See Mendelson page 98, for example. Or Chang and Kiesler.  
Only some higher order formula can distinguish them, by definition of  
*elementary* embedding. So if you can embed elementarily all  
structures in one structure, they will all verify the same first order  
sentences and be elementarily equivalent (although not equivalent in  
general). Think about the standard model of PA and its elementary  
embedding in some non standard model.





All mathematical theories would have the same theorems. So
eventually there has to be something wrong in your theorem. My friend
found the error. Your theorem page 12 is wrong, and the error is page
13 in he last bullet paragraph, when you do the negation induction
step for your lemma. When you say: Since it is not the case that A  
l=
ψ(Fi (b)), it is not the case that Aj l= ψ(Fi (b)(j)) by  
induction.

This will be true for *some* j, not for an arbitrary j. If you negate
for all j Aj l= ψ(Fi (b)(j)) , it means that there is a j such that
it is not the case that Aj l= ψ(Fi (b)(j)). After that your j is no
more arbitrary.





It took me a while to understand what you're saying but indeed I see
the error of my proof at this point.


OK.



I'm going to try to prove it in a different way but my hope is quite
limited.


I suggest you find a far less strong notion than elementary embedding.  
It is hard for me not to think about simulation, or recursive  
injection, or something related to computations. Of course in that  
case the 'universal structure' is the universal computer or  
dovetailer. This follows from Church's thesis, which is a rather  
unique statement of universality in mathematics.




Although I have nothing to show for my efforts, I do feel like I
learned a bit along the way.  Thanks for your feedback.


You are welcome. Sorry for not having seen the mistake at once. You  
almost get me. I have to revise a bit of model theory.


Bruno



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Re: A possible structure isomorphic to reality

2010-10-16 Thread Brian Tenneson


 If they are all elementary embeddable within it, then they are all  
 elementary equivalent, given that the truth of first order formula are  
 preserved.

How would all structures be elementarily equivalent?

 All mathematical theories would have the same theorems. So  
 eventually there has to be something wrong in your theorem. My friend  
 found the error. Your theorem page 12 is wrong, and the error is page  
 13 in he last bullet paragraph, when you do the negation induction  
 step for your lemma. When you say: Since it is not the case that A l=  
 ψ(Fi (b)), it is not the case that Aj l= ψ(Fi (b)(j)) by induction.  
 This will be true for *some* j, not for an arbitrary j. If you negate  
 for all j Aj l= ψ(Fi (b)(j)) , it means that there is a j such that  
 it is not the case that Aj l= ψ(Fi (b)(j)). After that your j is no  
 more arbitrary.




It took me a while to understand what you're saying but indeed I see
the error of my proof at this point.
I'm going to try to prove it in a different way but my hope is quite
limited.

Although I have nothing to show for my efforts, I do feel like I
learned a bit along the way.  Thanks for your feedback.

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Re: A possible structure isomorphic to reality

2010-10-10 Thread Bruno Marchal

On 09 Oct 2010, at 17:02, Brian Tenneson wrote:

I am starting a new thread which begins with some quotes by myself and
to continue the conversation with Bruno.

I figure this is especially of interest because of the references to
Tegmark's works.
From a logician's standpoint, it may be of interest that I show that
there is a structure U such that all structures, regardless of
symbol set, can be elementarily embedded within it.
From a physicist's point of view, at least one who might subscribe
to Tegmark's 4-level hierarchy of parallel universes, a structure
with this property might be of interest under the hypothesis that
reality is a mathematical structure. If we suppose that reality is
something which is all encompassing, then the structure with the
aforementioned property could be said to be all encompassing.
Now that I have this structure in hand, I can try to go further by
looking at the structure from a model-theoretic point of view. This
task to further the investigation will be undertaken soon.
Here is a link

http://www.alphaomegadimension.info/media/A_Mathematical_Structure_Isomorphic_to_Reality_ver_5-12_anon.pdf

Any feedback is encouraged, critical or otherwise.

[quote]
Let us call universe, the ultimate reality.
Then I agree with this: if the universe is a mathematical object,
then
NF is the best tool to attempt a description of that universal
object.
The universe, when being a mathematical object, has to belong to
itself, so we need a theory à-la Quine, instead of the usual
Zermelo-
Franekek or Von Neuman Bernays Gödel. In that sense it improves
the
raw description Tegmark makes of level 4.
[end quote]

Belong in the context of the paper is elementary embedding. Since
every structure is elementarily embeddable within itself, there is no
violation of any kind of foundation axiom and no anti-foundedness
assumption is required. Also, the universal set is barely used; what's
more important in my paper is the stratified comprehension theorem.
The universal set is invoked in any mention of power set such as for
relations.

But your mathematical structures are sets, this is why you work in a
set theory (NF). They are models of first order theory, and the
function and relation symbols are interpreted in term of sets and
subsets. OK?

Once this
ultimate structure is found, I think the means to finding it (eg,
NFU) are largely irrelevant in the same vain as the Dedekind cut
construction of the reals is largely irrelevant when actually dealing
with real analysis at least in the sense that Dedekind cuts are rarely
mentioned when you do calculus.

But you have to know if you are using real as cut or real as Cauchy
limit if you do constructive analysis.
I know only the partial computable functions as something really
machine or language independent. Theories are never universal, except
*perhaps* with respect to injection (inclusion, embedding). All first
order recursively enumerable set theory only scratch the arithmetical
truth.

What is really 'baggage-free' in first order theories is the relation
of consequences, between premise and conclusion. But all premise are
already baggages.

[quote]
Such universal machine cannot know in which computational history
she
would belong, still less in which mathematical structure she
belongs,
but below its level of substitution, she belongs to an infinity of
universal history (number relations, combinators relation, Horn
clause
relations) 'competing' in term of a measure of credibility.
[end quote]

Well if the paper is accurate, she can know that as herself, being a
mathematical structure, she is elementarily embeddable within U as
argued in the paper. Elementary embedding is not literally belonging
as in is an element of, so I'm not sure if this directly contradicts
the hypotheses you are using.

Are you not confusing mathematical object with first order theories
describing those mathematical objects, or with the models of those
theories.

What about incompleteness?
You say she can know, but how do you define knowing for a
mathematical structure.

Let me drop my pen. How do you predict what will happen in your
theory, and how do you relate this to possible reality. mechanism
shows that such question are not trivial, and that physics (as
conceived today) just miss the second part of the question, by
assuming a mind-brain identity thesis which is not correct. A lot of
discussion have been done on just this.

A possible structure isomorphic to reality

2010-10-09 Thread Brian Tenneson

I am starting a new thread which begins with some quotes by myself and
to continue the conversation with Bruno.

It would be nice to say something like the universal set V is what is
isomorphic to reality. However, the argument presented entails that a
baggage-free complete description of reality (ie, a TOE) is a
mathematical structure instead of a mathematical set. Once this
ultimate structure is found, I think the means to finding it (eg,
NFU) are largely irrelevant in the same vain as the Dedekind cut
construction of the reals is largely irrelevant when actually dealing
with real analysis at least in the sense that Dedekind cuts are rarely
mentioned when you do calculus.

However, this statement of yours is not inconsistent with my paper. I
would presume that one could say that she is in a sort of intersection
of all structures containing (ie elementarily embeddable within)
herself, which is the smallest structure she is embeddable within. I
know that intersection is vague at this point regarding math
structures. For example, what is the intersection of a lattice
structure and the complex number field? It would have something to do
with intersecting the universes, functions, and relations involved.

[quote]
So with mechanism the physical is not something mathematical among
the
mathematical, it is a very special structure which sums on all
mathematical structures is a way specified by computer science and
the
logic of self-references. It is based on distinction of different
internal sel-referential views.
[end quote]

A major shortcoming of the paper appears to be the lack of explanation
for the physical. Then again, this is a description of the level 4
universe, and not lower levels so one would view this as a piece of
the puzzle that is meant to complete the picture painted by Tegmark in
his works. In truth, it is a house of cards and if the level 4
universe does not fit, then everything in the paper falls apart as
then the underlying hypotheses would be false.

But that remains to be seen.

[quote]
Also, I am not convinced by your argument that from the premise
there
exists a reality completely independent of us human it follows
that
reality is a mathematical structure. You beg the question by
identifying a baggage free description with a mathematical
structure.
A physicalist argues in general that baggage-free description is
what
him provides:

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

Re: A possible structure isomorphic to reality

A possible structure isomorphic to reality

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