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
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
Any feedback is encouraged, critical or otherwise.
Let us call universe, the ultimate reality.
Then I agree with this: if the universe is a mathematical object,
NF is the best tool to attempt a description of that universal
The universe, when being a mathematical object, has to belong to
itself, so we need a theory à-la Quine, instead of the usual
Franekek or Von Neuman Bernays Gödel. In that sense it improves
raw description Tegmark makes of level 4.
"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
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.
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
"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
What is really 'baggage-free' in first order theories is the relation
of consequences, between premise and conclusion. But all premise are
Such universal machine cannot know in which computational history
would belong, still less in which mathematical structure she
but below its level of substitution, she belongs to an infinity of
universal history (number relations, combinators relation, Horn
relations) 'competing' in term of a measure of credibility.
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
What about incompleteness?
You say "she can know", but how do you define "knowing" for a
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.
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.
The problem is that I am not really a set theoretical realist. So I am
not even sure those question are meaningful.
So with mechanism the physical is not something mathematical among
mathematical, it is a very special structure which sums on all
mathematical structures is a way specified by computer science and
logic of self-references. It is based on distinction of different
internal sel-referential views.
A major shortcoming of the paper appears to be the lack of explanation
for the physical.
And of consciousness. I'm afraid too.
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.
You can save the theory by supposing some strong anti-comp axioms. but
most usual weakening of comp, like with supermachine, or machines with
oracles, will not do. All this follows from what is argued in the
SANE04 paper (or my thesis, alas in french).
My work refute Tegmark approach (even if much older!). Level 1 and 2
appears to concerned geographical question, and the laws of physics
are given by what is universally observable by universal machine,
blurring level 3 and 4. I should probably write a paper, but this
follows easily from the Universal Dovetailer Argument, once you get
the non triviality of Church Thesis.
Also, I am not convinced by your argument that from the premise
exists a reality completely independent of us human" it follows
reality is a mathematical structure". You beg the question by
identifying a baggage free description with a mathematical
A physicalist argues in general that baggage-free description is
him provides: particles, waves, fields, and that mathematics is an
approximate language conveying human ideas on those things. Your
seems to me just a platonist act of faith.
The human baggage of concepts like particle, waves, and fields are in
that these concepts are defined based on observations made by humans,
with tools or whatever. Therefore, these concepts are not baggage-
free and not consistent with reality being independent of humans,
which is an assumption I make.
You still beg the question. (First to be sure I do agree with you on
the conclusion). A physicalist will just tell you that, if observation
needs tool, their work consists from abstracting from those tools to
get the real, perhaps still unknown, thing. And that thing will be
conceive as substantial, primitively material or physical. They will
say that anything mathematical is a product of the human mind, and
that it simplifies the physical reality, and that we should never
Now, what I show, is that if we assume that we are digitalisable
machine, then physical reality is given by a very special mathematical
'structure' (which by the way are not elementary embaddable in first
order theories, already (the model of the modal logic G (and S4Grz)
are not definable by first order relations, nor are the arithmetical
quantum logic. This is of course coherent with the fact that your
theory need an anti-computationalist foundation.
You make a good point in that it is
possible that there could be other baggage-free objects out there and
thus the conclusion that reality is a mathematical structure would
have to be weakened to just the conclusion that a complete description
of reality (ie a Toe) is --something-- which is baggage-free.
A mathematical structure, being baggage free, is quite a tempting
choice for being a description of reality independent of humans due to
the generality of what a mathematical structure could be (eg, every
group, ring, field, algebra, etc. are mathematical structures).
Another weakened alternative is that there are many different complete
descriptions of reality and this mathematical structure concept is one
of the many.
Yet another way to weaken the conclusion is to consistently state that
the mathematical structure isn't literally reality but is isomorphic
to reality, ie, sharing intrinsic properties.
You miss the importance of the consciousness problem, concerning
Indeed, the nature of the structure U in the paper is not
investigated. The arguments in the paper are regarding a complete
description of reality without specifying what that complete
description is. The conclusion that reality is isomorphic to a
mathematical structure does not depend on which complete description
of reality one is referencing.
In essence, what that complete description is is bypassed.
And what do you mean by 'complete description' of reality? Sometimes I
call that a realm. By incompleteness there is no complete theory of
the arithmetical realm, but (assuming mechanism) the mathematical
structure (N, +, x) is already reality-complete, even if non
Mechanism makes it possible to reduce the mind-body problem to a
reduction of physics to number self-reference theory. The splitting
such theory into the deductive and the inductive part of those
makes it possible to derive a notion of both quanta and qualia.
It sounds like this number self-reference theory would be connected to
a mathematical structure and as such would be elementarily embedded
within the U of the paper.
But models of G are not first oder theoretical, even for just the
propositional part. You can represent them IN term of first-order
theoretically describable objects (like set), but the Kripke structure
of G needs second order axiom to be handled directly. Going through
*some* theory of sets, in my opinion, introduced baggages.
Once again, I am reminded that the nature
of U isn't investigated which is why the paper is in that sense
unfinished. I have some plans as for what to investigate but mostly
it will turn into a study of U from a logic standpoint. However, if I
can include this reduction you're mentioned, which I have no knowledge
of, then that would add much substance to the paper.
See my Sane04 paper:
Mechanism makes the inference of consistency (a part of
a key ingredient in the making of the physical realities, which
to be first person plural sharable computations.
These computations certainly sound like they involve some mathematical
structure such as arithmetic, ie, the structure whose domain is a set
of numbers, whose function symbols would include successor, and
relation symbol which would reference comparison (eg "less than").
Yes. With mechanism, elementary (first order) arithmetic, even without
the induction axioms, is already the 'theory of everything'.
I can explain how to derive the whole of physics from it, including
both quanta and qualia.
To be sure *any* first order logic Sigma_1 complete theory will do.
The Pi_i comes from inside, and already belongs to the mind of the
numbers or finite thing. I could use the hereditarily finite sets.
Even equational theory without logic (like a Diophantine equation, or
the SK combinators) can do the job.
If we are machine, it is just absolutely undecidable to know if there
is more at the basic existence level. From inside, by reflexive
incompleteness, there is *much* more. The intuitive reason is that
machine cannot distinguish realities from dreams (computations).
Mechanism intertwined completely the level 3 and 4, in an highly
mathematically structured way. This answers a criticism by Deutch
that kind of everything theory, because mechanism makes physics the
mathematical non trivial border of the universal (lobian) person.
makes mechanism testable. And indeed indeterminacy, non locality
non)-clonability are 'easily' derivable. The person, alias any
universal system, becomes Löbian when it can prove its own Sigma_1
completeness: it proves p -> Bp for all p Sigma_1, so that Bp ->
B(BB->p)->Bp, etc. Such machine are aware of their incompleteness.
see a reference on incompleteness in your reference, but none in
I'm trying to understand your notation. Is p a statement
yes, p is arbitrary (sigma_1) arithmetical sentence, like Ex(x+s(s(0))
and B a
predicate something along the lines of "believes"?
It is Gödel provability predicate (Beweisbar). Bp, for example B('Ex(x
+s(s(0)) = s(s(s(s(s(0))))))') is an arithmetical sentence, with 'Ex(x
+s(s(0)) = s(s(s(s(s(0))))))' the Gödel number of the string "Ex(x
+s(s(0)) = s(s(s(s(s(0))))))".
If that machine
can prove p->Bp I take that to mean that it can prove that if a
statement is true then it believes that statement.
You would have to quantify on "p". That is interesting, but I use only
that: for all Sigma-1 p the machine prove p -> Bp. It is the difficult
part in Hilbert Bernays proof of Gödel's second incompleteness
theorem. See Boolos 1993 for a readable proof. Now, Bp is itself
Sigma_1, so that such machine verify that for all arithmetical p, Bp -
> BBp, and this makes such machine Löbian (we get Löb's formula: B(Bp
-> p) -> Bp). The main axiom of G (and G*).
Note that you need induction for this. The believer is a richer theory
than the theory of everything!
In other words,
prove that its own set of beliefs is closed under consequence. Am I
That would be B(p -> q) & Bp . -> Bq, or equivalently B(p -> q) ->
(Bp -> Bq).
We have that too. No need of induction here.
Your paper is a nice little paper. I am not a NF expert, so I
judge the originality, but I took pleasure reading it, and you have
probably reawaken my taste for NF. Nice presentation of first
Thanks. I am very indebted to you for giving so much feedback.
In terms of originality, I am far from an expert and it will be
interesting to see what your friend knowledgeable in NF says.
opinion, there is nothing deep about my paper in terms of the tools I
use to get the result. However, I do show the existence of a set
consisting of all structures which may be of interest outside physics
and the structure U might also be interesting to logicians.
have yet to investigate U, even basic questions like uniqueness. The
big hurtle I was shooting for was to answer the question posed by
Tegmark: which structure is isomorphic to reality?
Which reality? Assuming mechanism there are only numbers' dream, and
some can cohere to give the 'illusion' of physical reality to
population of numbers or machines.
Eventually I think the big whole is not nameable by its internal
creature. I do like the idea that there is no universal set (in ZF,
VNBG, ...). From outside we might say that arithmetical truth is a
name for that big whole, in the 'theology' of the simple Löbian
arithmetic (Robinson arithmetic + induction). By Tarski theorem, such
a notion of truth is not describable by the machine itself, and this
makes it possible for arithmetical truth to play the role of the ONE
in Plotinus in the theology of such a simple machine/theory. Mechanism
leads to a "simple" reinterpretation of the basic of Plato-Plotinus,
and shows that mechanism is incoherent with Aristotelianism (with a
basic physical universe with creature in there).
So what it amounts to is that the nature of reality might be equally
impenetrable as the nature of U. I do hope that analyzing U will give
insights into the nature of reality.
Nice goal. You might be interested by what I think to be consequences
of the assumption that "I" am Turing emulable. It gives a bigger role
to computer emulation than to elementary embeddings though. You can
read my papers. The UDA is done in 8th steps, and I think most people
have not much trouble to understand the 7th first steps. There are
still (pedagogical?) problem in the 8th step. If you agree that
mechanism entails that consciousness on some computation does not
depend on the presence or absence of a *inactive* piece of material,
for that computation, then the conclusion follows, physics is given by
a relative credibility measure on the relative computations. This
entails more than indeterminacy and non-locality, that entails also
non-clonability. If moreover you accept the classical (by Plato)
theory of knowledge (in the Theaetetus), and its arithmetical
interpretation, then you get the complete axiomatization of
propositional physics. This is done in the second part of sane04. but
is also the object of my Plotinus paper, and all the details of the
morphism and representational embeddings are given in my long thesis
(also in french, alas).
You might consult the archive of this list, or my url, but assuming
mechanism, and even strong weakening of mechanism, entails that you
cannot make the physical, nor the mental, a purely mathematical
except in a necessarily informal way (mentioning the logician
of standard model of arithmetic, for example). Both the mental and
physical, or the coupling first person/its third person possible
computations, emerges from the purely (first order) arithmetical
relation existing among numbers, or combinators, or lisp programs.
it only "emerges" from inside, and that inside, including the first
person", can never completely self-reflect itself in it completely
(which justifies a tree of transfinite progression and
diversification, some very deep one like most probably ours).
But does mechanism imply that physical and mental --can not-- be
isomorphic to a mathematical structure, even if it implies they are
not literally mathematical structures?
It is an open problem, with the mechanist assumption. My feeling is
that consciousness is not a mathematical structure. It is more like a
mathematical 'phenomenon'. A tendency of universal number to infer
coherent universal extension to their dreams. To be short.
You might improve your theory by addressing concrete problems, like
why physics has this shape? why qualia? why quanta? are there
constant in physics? why superposition?, why complex numbers? why
dream, why pleasure, why symmetry, why irreversibility (if any?),
These things certain would improve the theory and this is an analysis
of the structure U which I haven't done of course, only existence.
For such things as physics, dreams, pleasure, symmetry, etc., it would
help to prove that they are isomorphic to mathematical structures and
perhaps that can be done without elaborating on the nature of those
structures. Of course, knowing the full properties of said structures
would make the theory more complete.
As it stands, the only things I can see to investigate are logic-
related and I would love to figure out a way to investigate it with
focus not on the logic properties of U but also being on concrete
issues like those you mentioned.
Good luck. Your paper is very clear, and interesting (despite it
misses somehow both mind and matter!). I hope it is original enough to
be published. Don't hesitate to submit to some logical journal,
perhaps the bulletin of the society for symbolic logic:
Of course it would even be better if you succeed in taking into
account the mechanist constraints, or at least make clear that you
need some non-comp axioms. Even in that case you will have to define
your notion of 'intersection of structures' in a way which take into
account the infinitely many first order theories describing your
BTW doing catgeory theory in NF would be cute too. That might give a
uniform theory for large and small categories. Do you know the n-
categories? They define quickly amazing structure in knot theory. Such
shortcuts are promising for deriving highly no trivial happenings,
like gravity, from simple combinatorial truth.
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