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?

> 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? 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?

--Abram

On Fri, Jan 2, 2009 at 11:32 AM, Bruno Marchal <marc...@ulb.ac.be> 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
>> <halr...@alum.syracuse.edu> 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-l...@googlegroups.com
>>> [mailto:everything-l...@googlegroups.com] On Behalf Of Abram Demski
>>> Sent: Monday, December 29, 2008 12:59 AM
>>> To: everything-l...@googlegroups.com
>>> 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-dem...@googlegroups.com
>> Public archive: http://groups.google.com/group/abram-demski
>> Private address: abramdem...@gmail.com
>>
>> >
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>



-- 
Abram Demski
Public address: abram-dem...@googlegroups.com
Public archive: http://groups.google.com/group/abram-demski
Private address: abramdem...@gmail.com

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