Hi Abram and Bruno:
My goal some time ago was to find an origin to a dynamic in the Everything.
It seemed that many on the list were pointing to such a dynamic - the UD for
example.
I came up with the Nothing to Something incompleteness dynamic initiator
maybe 10 or more years ago.
Since then I have been trying to make the resulting model as simple as I
could.
I have looked at Abram's idea of adding inconsistency derived traces in the
dynamic:
I have in recent changes stopped using "information" to avoid the
complications this term seemed to bring with it. This lead to a compact
model with just two definitions, one assumption, and the stability trigger
question resulting in the dynamic. To maintain this simplicity I note that
when a Nothing in a particular All containing just one copy of the Nothing
converts to a Something this also converts the particular All into a
Something. The All is inconsistent by reason of its absolute completeness.
The absence of its Nothing which was consistent but incomplete is not likely
to make the Something the All became consistent Something. So this
Something may be a source of inconsistency driven traces.
As far as learning how to communicate this model in a more mathematical
language [logic, set theory, etc.] to aid understanding by others, I have
consumed what little time I had available over the years just getting to the
current state of the model. It has been said that it takes 10,000 hours of
practice in some endeavor to become an expert in it. Since I understand
less than half the mathematical logic based comments in this tread regarding
my model I am far from expert in such a language.
My engineering career gives me some formal exposure and practical
understanding of it, and I have studied small additional pieces of it in the
course of developing this model. However, the current realities of life
have made adding new time intensive endeavors such as becoming sufficiently
fluent in such a communication method an "overcome by events" effort. I
might find maybe an hour a week for my total participation on the list. This
seems extremely insufficient. Thus I suspect that despite my real interest
in developing an alternative means of communication for my ideas in this
area, my primary reliance for communicating the model will unfortunately
have to remain using as small a set of words as I can muster.
Hal
-----Original Message-----
From: everything-list.domain.name.hidden
[mailto:everything-list.domain.name.hidden] On Behalf Of Bruno Marchal
Sent: Saturday, January 03, 2009 3:25 AM
To: everything-list.domain.name.hidden
Subject: Re: Revisions to my approach. Is it a UD?
On 03 Jan 2009, at 02:04, Abram Demski wrote:
>
> Bruno,
>
> Interesting point, but if we are starting at nothing rather than PA,
> we don't have provability logic so we can't do that! How can we tell
> if an *arbitrary* set of axioms is incomplete?
"nothing" is ambiguous and depends on the theory or its intended
domain. Incompleteness means usually arithmetically incomplete.
The theory with no axioms at all? Not even logical axioms? Well, you
can obtain anything from that.
The theory with nothing ontological? You will need a complex
epistemology, using reflexion and comprehension axioms, that is a bit
of set theory, to proceed.
Nothing physical? You will need at least the numbers, or a physics:
the quantum emptiness is known to be a very rich and complex entity.
It needs quantum mechanics, and thus classical or intuitionistic
logic, + Hilbert spaces or von Neumann algebra.
I would say that "nothing" means nothing in absence of some logic, at
least.
No axioms, but a semantic. Right, the empty theory is satisfied by all
structure (none can contradict absent axioms). But here you will have
a metatheory which presupposes ... every mathematical structure. The
metatheory will be naïve set theory, at least.
I suspect since some time that Hal Ruhl is searching for a generative
set theory, but unfortunately he seems unable to study at least one
conventional language to make his work understandable by those who
could be interested.
>
>
>> This can be related with the so-called autonomous progressions
>> studied
>> in the literature, like: PA, PA+conPA, PA+conPA+con(PA +conPA), etc.
>> The "etc" here bears on the constructive ordinals. "conPA" is for "PA
>> does not derive P&~P.
>
> I have been wondering recently, if we follow the "..." to its end, do
> we arrive at an infinite set of axioms that contains all of
> arithmetical truth, or is it gappy?
The "..." is (necessarily) ambiguous. If it is constructive, it will
define a constructive ordinal. In that case the theory obtained is
axiomatizable but still incomplete. If the "..." is not constructive,
and go through all constructive ordinals at least, then Turing showed
we can get a complete (with respect to arithmetical truth) theory,
but, as can be expected from incompleteness, the theory obtained will
not be axiomatizable.
> In other words, is the hole that
> Godel pointed out flexible enough to fill in any hole eventually if we
> keep adding con(x), or are there "non-godelian" holes?
I am not sure what you mean by a hole filling any hole, nor what you
mean by a non-godelian hole. The point is that the hole provided by
Godel's incompleteness phenomena cannot be fill completely in any
effective way. Even ZF + strong infinity axioms cannot prove all
arithmetical truth. All effective theory are arithmetically incomplete.
Bruno
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Received on Sat Jan 03 2009 - 11:39:06 PST