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From: John Mikes <jamikes.domain.name.hidden>

Date: Wed, 6 Feb 2008 17:56:52 -0500

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 <marchal.domain.name.hidden> 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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Received on Wed Feb 06 2008 - 17:57:03 PST

Date: Wed, 6 Feb 2008 17:56:52 -0500

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 <marchal.domain.name.hidden> wrote:

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To post to this group, send email to everything-list.domain.name.hidden

To unsubscribe from this group, send email to everything-list-unsubscribe.domain.name.hidden

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Received on Wed Feb 06 2008 - 17:57:03 PST

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