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
>
>
> --Abram
>
> On Fri, Jan 2, 2009 at 11:32 AM, Bruno Marchal <marchal.domain.name.hidden>
> wrote:
>>
>>
>> On 02 Jan 2009, at 16:01, Abram Demski wrote:
>>
>>>
>>> Hal,
>>>
>>> I went back and reviewed some of your old postings. My
>>> interpretation
>>> of your system was closer to the mark than I'd suspected!
>>>
>>> I think enumeration via inconsistency can be equivalent to
>>> enumeration
>>> by incompleteness... depending on exactly how things are defined.
>>> Enumeration by inconsistency seems more intuitive to me:
>>> inconsistency
>>> can be readily detected (derive P&~P), whereas incompleteness
>>> cannot.
>>
>>
>> I don't think so. You cannot derive that you cannot derive P&~P, but
>> you can derive that your are incomplete, assuming you will not derive
>> P&~P. Indeed you can derive that: IF you cannot derive P&~P, THEN you
>> cannot derive that you cannot derive P&~P (Godel incompleteness).
>> This
>> gives extension by "self-consistency" bets.
>>
>> But I think I see what you mean. In artificial and pragmatic
>> intelligent procedure, with non monotonic logic, you could have local
>> inconsistencies, and build from that (with revision procedure). In
>> the
>> realm of the ideal machine which derives the ideal correct physics,
>> it
>> is better to extend by consistencies, I think.
>>
>> 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.
>>
>> You can extend this in the transfinite, because you can describe in
>> arithmetic transfinite ordinal sequences like
>>
>> PA, PA+conPA, PA+conPA+con(PA +conPA), ... PA + con(PA+conPA+con(PA
>> +conPA)...), ...
>>
>> Bruno
>>
>>
>>
>>
>>
>>
>>>
>>>
>>> --Abram
>>>
>>> On Mon, Dec 29, 2008 at 6:47 PM, Hal Ruhl
>>> <HalRuhl.domain.name.hidden> wrote:
>>>>
>>>> Hi Abram:
>>>>
>>>> My sentence structure could have been better. The Nothing(s)
>>>> encompass no
>>>> distinction but need to respond to the stability question. So they
>>>> have an
>>>> unavoidable necessity to encompass this distinction. At some point
>>>> they
>>>> spontaneously change nature and become Somethings. The particular
>>>> Something
>>>> may also be incomplete for the same or some other set of
>>>> unavoidable
>>>> questions. This is what keeps the particular incompleteness trace
>>>> going.
>>>>
>>>> In this regard also see my next lines in that post:
>>>>
>>>> "The N(k) are thus unstable with respect to their "empty"
>>>> condition. They
>>>> each must at some point spontaneously "seek" to encompass this
>>>> stability
>>>> distinction. They become evolving S(i) [call them eS(i)]."
>>>>
>>>> I have used this Nothing to Something transformation trigger for
>>>> many years
>>>> in other posts and did not notice that this time the wording was
>>>> not as
>>>> clear as it could have been.
>>>>
>>>> However, this lack of clarity seems to have been useful given your
>>>> discussion of inconsistency driven traces. I had not considered
>>>> this
>>>> before.
>>>>
>>>> Yours
>>>>
>>>> Hal
>>>>
>>>> -----Original Message-----
>>>> From: everything-list.domain.name.hidden
>>>> [mailto:everything-list.domain.name.hidden] On Behalf Of Abram Demski
>>>> Sent: Monday, December 29, 2008 12:59 AM
>>>> To: everything-list.domain.name.hidden
>>>> Subject: Re: Revisions to my approach. Is it a UD?
>>>>
>>>>
>>>> Hal,
>>>>
>>>> I do not understand why the Nothings are fundamentally
>>>> incomplete. I
>>>> interpreted this as inconsistency, partly due to the following
>>>> line:
>>>>
>>>> "5) At least one divisor type - the Nothings or N(k)- encompass no
>>>> distinction but must encompass this one. This is a type of
>>>> incompleteness."
>>>>
>>>> If they encompass no distinctions yet encompass one, they are
>>>> apparently inconsistent. So what do you mean when you instead
>>>> assert
>>>> them to be incomplete?
>>>>
>>>> --Abram
>>>>
>>>>
>>>>
>>>>>
>>>>
>>>
>>>
>>>
>>> --
>>> Abram Demski
>>> Public address: abram-demski.domain.name.hidden
>>> Public archive: http://groups.google.com/group/abram-demski
>>> Private address: abramdemski.domain.name.hidden
>>>
>>>>
>>
>> http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
>>
>>>
>>
>
>
>
> --
> Abram Demski
> Public address: abram-demski.domain.name.hidden
> Public archive: http://groups.google.com/group/abram-demski
> Private address: abramdemski.domain.name.hidden
>
> >
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Jan 03 2009 - 04:25:03 PST