Bruno, here is my "out of order and off topic" remark.
----------------------------
We are here in theoretical theorizing by theory-laden theoretic ways.
It is ALL the product of  a mental exercise. Even a Loebian kick in
the ass can be a theoretical halucination.
You wrote:
"... -  ...
But does 'M" exist? ,,,  -  ..."
(Never mind in what context. )

"exist" is a hard word. Contemplating in a generalized way, I would say:
"Everything (not in Hal's sense) exists what we THINK of, if not
otherwise: in our ideas.
Does 'K' or 'S' have a better than mental existential veracity? We can
think of a symbol that it does or does not exist, it does not change
that it DOES indeed exist in our mental domain.
Do you have a better 'domain' (e.g. a physical existence)? I doubt.
In our 1st person  world  it would not make sense.
-----------------------

Excuse my rambling and please, consider it 'entertainment' rather than
discussion-post.

John M

On Wed, Feb 6, 2008 at 10:40 AM, Bruno Marchal <[EMAIL PROTECTED]> wrote:
> Le 05-déc.-07, à 23:08, Mirek Dobsicek a écrit :
>
>
>  "But thanks to that crashing, *Church thesis remains consistent*. I
>  would just say "An existence of a universal language is not ruled out".
>
>
>
>  I am ok with you. Consistent (in math) means basically "not rule out".
>  "Formally consistent" means "not formally ruled out", or "not
>  refutable".
>
>  That is:
>
>  "Consistent(p") is the same as "~ Provable(~ p)"    " ~" = negation
>
>  like "Provable(p)" is equivalent with "~ Consistent( ~ p)"
>
>
>
>  Some thoughts:
>  Thanks to Godel "completeness" theorem for the first order theory
>  (1930) you can also read consistent(p) by there is a world satisfying p
>  (a world "where" p is true).
>
>  This relates a syntactical notion (the non existence of a chain of
>  formula derived from the axioms by the use of the inference rules and
>  ending with f) with a semantical: the existence of a mathematical
>  structure satisfying the formula.
>
>  At least in the frame of many formal classical theories, it is related
>  to the recurrent modal duality:
>
>
>  Permitted p <====> ~ Obligatory ~p
>  Obligatory p <====> ~ Permitted ~p
>
>  Somewhere p <====> ~ Everywhere ~p
>  Everywhere p <====> ~ Somewhere ~p
>
>  Sometimes p <====> ~ Always ~p
>  Always p <====> ~ Sometimes ~p
>
>  Like the usual first order quantifiers: (Ax = for all x; Ex = it exists
>  a x)
>
>  Ex F(x) <====> ~ Ax ~ F(x)
>  Ax F(x) <====> ~ Ex ~F(x)
>    (all cats are ferocious  <====> it does not exist a non ferocious cat)
>
>  And with formal provability we have also:
>
>  Consistent p <====> ~ provable ~p
>  Provable p <====> ~ consistent ~p
>
>
>  But yes, it is by allowing the machine to crash, and actually by
>  allowing it to crash in a *necessarily* not always predictible way,
>  which makes it possible to be universal.
>
>  In a nutshell: Universality ==> insecurity ====> kicking back reality
>
>  and then
>  (knowledge of your universality) ==> (knowledge of your relative
>  insecurity) ====> (knowledge of a kicking back reality) ===>
>  anticipating an independent "reality"
>
>  (knowledge of your universality)  = lobianity (this I intend to explain
>  later)
>
>
>  Mirek asked also in trhe same post:
>
>
>  <<And my last question, consider the profound function
>  f such that f(n) = 1 if there is a sequence of n consecutive fives in
>  the decimal expansion of PI, and f(n) = 0 otherwise
>  Is this an example of a partial computable function?>>
>
>   Yes.
>
>   <<Or is this function
>  as such already considered as un-computable function?>>
>
>
>  It could be uncomputable on some value, that is, everywhere the
>  function has value 1, you can in principle compute it (just search the
>  sequence: if it exists you will find it because PI is constructive). If
>  the value is zero, it could be that you will be able to know it, but it
>  could be that you will never know it ...
>
>  * * *
>
>  Something else:
>
>  Mirek, Brent, Barry, Tom (and all those inclined to do a bit of math):
>  don't read what is following unless you don't want to find the crashing
>  combinators by yourself.
>
>  I give the solution for the crashing combinators: it is enough to ...
>  mock a mockingbird.
>
>  Raymond Smullyan calls "mocking bird"  a combinator M such that Mx = xx.
>  It is a sort of diagonalisor or duplicator. Now if you apply M on
>  itself, M, that is if you evaluate MM, this matches the left of
>  equation Mx = xx, so MM gives MM gives MM gives MM gives MM ...
>  (crashing!).
>
>  But does M exists? If you recall well,  we know only the existence of K
>  and S, and their descendants: like KK, KS, S(KS), SK(KS)(S(KK)), ...
>
>  (Recall we don't write any left parenthesis, but something like
>  SK(KS)(S(KK)) really abbreviate the result of applying (SK) to (KS)
>  i.e. ((SK)(KS)) on (S(KK)), i.e.
>  (((SK)(KS))(S(KK))). each combinator can be thought as a function of
>  one variable (itself varying on the combinators).
>
>  We search a combinator playing the role of M (defined by its behavior
>  Mx = xx).
>
>  We have only K, S, and their combinations. And we have the two axioms
>  giving the behavior of K and S.
>
>  Kxy = x   K axiom
>
>  and
>
>  Sxyz = xz(yz)   S axiom
>
>  Explanation. You can see K as a projector sending (xy) on x, for any y.
>  (imo it is the *subjective* entity per excellence, in particular K
>  discards or eliminate informations like projection does. Church will
>  not allow K or any eliminators in its main systems).
>  Functionally K is Lx Ly . x The variable y is abstracted in some
>  irrelevant way.
>
>  We want Mx = xx.
>  But xx does not match either x or xz(yz), so that we could use the
>  axioms above directly.
>  But imagine we dispose of the subroutine combinators I such that Ix =
>  x. The identity combinators. Then Mx = xx = Ix(Ix), and this does match
>  xz(yz), so that Ix(Ix) is really SIIx (in Sxyz = xz(yz), so SIIx =
>  Ix(Ix) = xx. So SII can play the role of M, it behaves like M. We could
>  define M by SII.
>  Let us verify MM = SII(SII) does crash the system:
>
>  SII(SII) = I(SII)(I(SII)) = SII(SII) = I(SII)(I(SII)) = SII(SII) =
>  I(SII)(I(SII)) = SII(SII) = I(SII)(I(SII)) = SII(SII) = ... (crashing).
>
>  Now we have to still find an identity combinator I such that Ix = x.
>
>  Now x does match the right of the first axiom Kxy = x. Except that K on
>  x wait for a second argument. So let us give to it a second argument
>  such that we get something matching the second (S) axiom:
>
>  x = Kx(Kx) = SKKx
>
>  So SKK does the job. So we can take I = SKK.
>  So M = SII = S(SKK)(SKK)
>
>  and a crashing expression, sometimes called INFINITY is given by
>
>  MM = SII(SII) = S(SKK)(SKK)(S(SKK)(SKK))
>
>  So, a solution was
>
>  S(SKK)(SKK)(S(SKK)(SKK))
>
>  Remark:
>  Note that an existential quantification "ExP(x)" is a sort of
>  projection too. Eventually, the lobian machine observation-act-decision
>  is just that: projection by elimination of worlds (elimination of
>  accessibility of possibilities, a bit like when you get married, of get
>  a job, etc ....).
>
>
>  Bruno
>  http://iridia.ulb.ac.be/~marchal/
>
>  >
>

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