Le 20-juin-06, à 06:00, Tom Caylor a écrit (replying to Norman Samish) :


>
>
> Norman Samish wrote:
>> Gentlemen:
>>
>> I've endured this thread long enough!  Let's get back to something I 
>> can understand!
>>
>> "Why?" you'll ask.
>>
>> I'll reply, "Because your audience is shrinking!  I've plotted the 
>> Audience vs. Topic, and find that, in 12.63 months, there is a 91% 
>> probability that, if the topic doesn't become understandable to one 
>> with an IQ of 120, your audience will be zero, and the only expositor 
>> will be Bruno.  Not that there's anything wrong with that, but we 
>> must acknowledge that Bruno speaks a language that very few of us can 
>> understand.  Bruno, and probably Russell and a few others, are 
>> clearly Homo Superior, while the rest of us are mere Homo Sapiens."
>>
>> You will then say, "Our discourse is meant for Homo Superior.  If you 
>> can't stand the heat, get out of the kitchen."
>>
>> I'll reply, "Damn!  I was hoping to learn something!"
>>
>> Norman Samish
>>
>
> Norman:
>
> Even though the other current topic "Calculus of personal identity" has
> the word "calculus" in it, I think it's an understandable and
> interesting thread.  And you can also start a new thread.  For me, I'm
> hoping to learn something, too, as long as Bruno lasts, and feels like
> he's benefiting.  This current topic I think is just starting to really
> get good, in my view.  Or it may evolve to the next level and be less
> mathematical and more philosophical.  Or maybe someone smarter than I
> am will pipe up and make it even more interesting.  Who knows what will
> happen, but it's up to whoever wants to participate to make it happen.
> My thoughts.


I appreciate. I will solve the four diagonalization questions later (I 
recall them below(*).
I intend also to make clearer the motivation. I was just trying to help 
george with Smullyan's FU "heart of the matter" but Norman is right at 
least on this: I should find a way to explain the general idea without 
being technical, and then, for those interested, I can explain the 
technics.
It would be better for everyone, though, to ask questions once there is 
a unclear point. With all my respect for Norman, he reminded me those 
students who wait for the exams for asking questions!
I do think such clarifications could help to clarify nuances between 
theoretical computer science notions (number, program, function, 
computation, ...) which are important in many other thread. For example 
I said recently that Kolmogorov complexity is a well defined notion, 
that it is a non constructive (nor computable) notion, and that it does 
not depend on the chosen universal machine except for a constant term. 
But all that presupposes importantly Church thesis. So it is not a luwe 
to dig on it a little more.

Aaaaaarghh...  Sorry for that teacher's tone.... it is June. Exam Month 
here ... :-(

Bruno

(*) So the question was: what does diagonalization prove on those 
following list of functions:

0) r1 r2 r3 r4 r5 r6 r7 r8 ...
This is an arbitrary list of functions from N to N  (not necessarily 
computable one);

1) h0 h1 h2 h3 h4 h5 h6 ...
This is a list of total computable functions from N to N that we can 
generate mechanically (I mean we can generate their codes). It means 
that we can generate the codes of each hi, written in some language, 
and that, for some reason, we are sure that each hi is total 
computable. Examples: Caylor last funny enumeration; all the 
(transfinite collection of) sequences of growing functions we have 
defined in this thread (since "Smullyan Smullyan ...");

2) f0 f1 f2 f3 f4 f5 f6 f7 ...
This is a list of *all* total computable functions;

3) F0 F1 F2 F3 F4 F5 F6 F7 F8 ...
This is the list of all programmable things in some "universal 
language" like fortran.  CT asserts fortran is universal so that the 
total computable function fi will be dispersed *among* those Fi things, 
so that a universal machine can really compute all the fi, among other 
things.


Now the same diagonalization argument proves 4 different propositions 
according to which list we are talking about. Which one?

Before answering, I let people muse a little about what are those 4 
different consequences,  given that the only way to really grasp those 
propositions consists in rediscovering them by oneself,  ... or at 
least in searching enough so as to be curious listening to the 
solutions.

Bruno

http://iridia.ulb.ac.be/~marchal/


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