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/ > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
