Bruno, although I did not see in my list-post my comment to Marc's report about the German conference (sent before your and Saibal's posts) I may continue it (maybe copying the missing text below); Saibal's : "uncompoutable numbers, non countable sets etc. don't exist in first order logic,..." is interesting: it may mean that the wholeness-view (like Robert Rosen's 'complexity' and my wholistiv views as well) do not "compute" with 1st order logic - what may not be fatal IMO. I use "model" a bit opposite to Skolem's text (sorry, I could not read your URL, my new comnputer does not (yet) have ppt installed) - but as I decyphered Skolem's long text, it is a math-construct based on 'a theory'. A different "model". My 'model' (and R. Rosen's) is an extract of the totality, a limited cut from the interconnectedness by topical, functional ideational etc. boundaries and THEORIES are based on that (usefully, if not extended beyond the margins of the model - limitedly observed to formulate them.) If this is beyond "1st order logic" that is not the fault of such model-view - with uncomputable (impredicative) "numbers" and unlimited variables - rather shows a limitation of the domain called 1st order logic. (I put numbers in quotation, I used the word to apply it according to the here ongoing talks.) Rosen (a mathematician) also called it "Turing un-emulable". Your explanation about the ZF uncountability and the uncomputability is intgeresting, I could not yet digest its meaning as how it may be pertinent to my thinking.
John M ----- Original Message ----- From: "Bruno Marchal" <[EMAIL PROTECTED]> To: <email@example.com> Sent: Friday, November 03, 2006 9:08 AM Subject: Re: Zuse Symposium: Is the universe a computer? Berlin Nov 6-7 In "conscience et mécanisme" I use Lowenheim Skolem theorem to explain why the first person of PA "see" uncountable things despite the fact that from the 0 person pov and the 3 person pov there is only countably many things (for PA). I explain it through a comics. See the drawings the page "deux-272, 273, 275" in the volume deux (section: "Des lois mécanistes de l'esprit). It explains how a machine can eventually infer the existence of other machine/individual). Here: http://iridia.ulb.ac.be/~marchal/bxlthesis/Volume2CC/2%20%203.pdf Note also that the word "model" (in http://www.earlham.edu/~peters/courses/logsys/low-skol.htm ) refers to a technical notion which is the opposite of a theory. A model is a mathematical reality or structure capable of satisfying (making true) the theorem of a theory. Like a concrete group (like the real R with multiplication) satisfy the formal axioms of some abstract group theory. (Physicists uses model and theory interchangeably, and this makes sometimes interdisciplinary discussion difficult). ZF can prove the existence of non countable sets, and still be satisfied by a countable model. This means that all sets in the model are countable so there is a bijection between each infinite set living in the model and the set N of natural numbers. What is happening? just that the bijection itself does not live in the model, so that the inhabitants of the model cannot "see" the bijection, and this shows that uncountability is not absolute. It just means that from where I am I cannot enumerate the set. But, contrariwise, "uncomputability" is absolute for those "enough rich" theories. Here I am close to a possible answer of a question by Stathis (why comp?), and the answer is that with comp you have robust (absolute, independent of machine, language, etc.) notion of everything. Comp has a "Church thesis". few notion of math have such facility. Tegmark's whole math, for example, is highly ambiguous. Thanks to Saibal for Peter Suber web page on Skolem (interesting). Bruno Le 03-nov.-06, à 13:50, Bruno Marchal a écrit : > > It is not a question of existence but of definability. > For example you can define and prove (by Cantor diagonalization) the > existence of uncountable sets in ZF which is a first order theory of > sets. > Now "uncountability" is not an absolute notion (that is the > Lowenheim-Skolem lesson). > Careful: uncomputability is absolute. > > Bruno > > > Le 03-nov.-06, à 13:43, Saibal Mitra a écrit : > >> uncompoutable numbers, non countable sets etc. don't exist in first >> order logic, see here: >> >> http://www.earlham.edu/~peters/courses/logsys/low-skol.htm >> ================= Copy of my "lost?" note to Marc (Bov.3 - 6:59AM): Marc, I do not argue with 'your half' of the 'answer' you gave to the conference announcement of Jürgen Schm , I just ask for the 'other part': what should we call "a computer"? 'Anything' doing Comp? (meaning: whatever is doing it)? Will the conference be limited to that technically embryonic gadget - maybe even on a binary bases - we use with that limited software-input in 2006? a Turing machine? John M ========================================= --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to firstname.lastname@example.org 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 -~----------~----~----~----~------~----~------~--~---