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
> 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. 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.
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.
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.
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. 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.
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 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.
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. 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.
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").
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 and B a
predicate something along the lines of "believes"? 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. In other words,
prove that its own set of beliefs is closed under consequence. Am I
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. In my
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. Again, I
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?
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.
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?
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.
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