At 19:08 -0400 29/09/2002, Wei Dai wrote:
>On Thu, Sep 26, 2002 at 12:46:29PM +0200, Bruno Marchal wrote:
>> I would say the difference between animals and humans is that humans
>> make drawings on the walls ..., and generally doesn't take their body
>> as a limitation of their memory.
>It's possible that we will never be able to access more than a
>bounded volume of space. It depends on the cosmology of our universe.
The UDA is intended to show that with comp it is the cosmology of "our
universe" which is the result of an average on our unbounded computationnal
stories, which are "living", statically, in UD* (comp platonia).
>> It is also the difference between
>> finite automata, and universal computers: those ask always for more
>> memory; making clear, imo, the contingent and local character of their
>> space and time bounds.
>My point is that our inability to compute non-recursive functions is also
>a contingent bound. It's contingent on us not discovering a non-recursive
>law of physics.
I agree. I think comp implies the existence (from our first plural point
of view) of non recursive law of physics. The amazing fact, which would
follow---empirically---from freedman NP paper or Calude "beyond turing
barrier" paper, is that such non recursive phenomenon can be exploited.
It is of course highly non trivial to show this from comp, although with
UDA we know that we must extract this or the negation of this from comp
(making comp completely empirically testable).
>> I have read and appreciate a lot of papers by Shapiro. He has edited
>> also the north-holland book "Intensionnal Mathematics" which I find
>> much interesting than its "case for Second-order Logic".
>> It is not very important because, as you can seen in Boolos 93, basically
>> the logic G and G* works also for the second order logic. Only the
>> restriction to Sigma_1 sentences should be substituted by a substitution
>> to PI^1_1 sentences. This can be use latter for showing the main argument
>> in AUDA can still work with considerable weakening of comp, but I think
>> this is pedagogically premature.
>I guess I'll have to take your word for it.
Well, just look Boolos 93 chapter 14 ... (read the definitions, until
you understand the enunciation of the theorems and take Boolos word for
the proof ...).
>BTW, you never answered my earlier question of why Arithmetical Realism
>rather than Set Theoretic Realism. Is is that you don't need more than
>Arithmetical Realism for your conclusions? What do you personally believe?
(I thought I did answer that question once (?)).
With comp, arithmetical realism is enough for the basic ontological (different
from substancial) basic level. Set theoretic realism can be used, except
that I have no idea of what it could mean. That is, I believe that each
sentence with the form ExAyEzArEtAu....P(x,y,z,r,t,u ...) is true or false
when the variable x, y, z, r .. are (positive) integers. And this
independently of my ability to know the truth value. I have just no similar
belief if the variable are allowed to represent arbitrary sets.
Set theory is like group theory, I can be platonist on the groups (= models
of group theory) and I can be platonist about the "universes-of-sets" (models
of set theory). Now, if you ask me "Is a * b = b * a" in group theory, I
will answer you by "it depends on the group you are talking about". Similarly,
if you ask me "Is the Cantor Continuum Hypothesis true or false" about sets,
I will answer that it depends on which set-universe you talk about.
If you answer me: "Come on, I am talking about the standard model of ZF
theory", I am just not sure I can know what you mean by standard. You can only
define the word "standard" in a more doubtful theory.
By a sort of miracle---akin to Church thesis---, I have a clear
(but admittedly uncommunicable) understanding of the standard model of
natural number theories. If you ask me if the prime twin conjecture is false or
true, I will just answer that I currently do not know, but that I do find
the question meaningful. I have no doubt the twin prime conjecture is
true or is false. By the same token, I have no doubt a machine will stop, or
not stop, independently of my ability to solve any stopping machine problem.