Le 29-nov.-07, à 17:22, Torgny Tholerus a écrit : > > Quentin Anciaux skrev: >> Hi, >> >> Le Wednesday 28 November 2007 09:56:17 Torgny Tholerus, vous avez >> écrit : >> >>> >>> You only need models of cellular automata. If you have a model and >>> rules for that model, then one event will follow after another event, >>> according to the rules. And after that event will follow another >>> more >>> event, and so on unlimited. The events will follow after eachother >>> even >>> if you will not have any implementation of this model. Any physics >>> is >>> not needed. You don't need any geometric properties. >>> >>> >> Sure, but you can't be ultrafinitist and saying things like "And >> after that >> event will follow another more event, and so on unlimited". >> > > > There is a difference between "unlimited" and "infinite". "Unlimited" > just says that it has no limit, but everything is still finite. If you > add something to a finite set, then the new set will always be finite. > It is not possible to create an infinite set.
Come on! Now you talk like a finitist, who accepts the idea of "potential infinity" (like Kronecker, Brouwer and the intuitionnist) and who rejects only the so called actual infinities, like ordinal and cardinal "numbers" (or sets). At the ontic level, (or ontological, I mean the minimum we have to bet on at the third person pov), comp is mainly finitist. Judson Webb put comp (he calls it mechanism) in Finitism. But that is no more ultrafinitism. With finitism: every object of the "universe" is finite, but the universe itself is infinite (potentially or actually). With ultrafinitism, every object is finite AND the universe itself is finite too. Jesse wrote: How would the set "omega-1" be defined? It doesn't make sense unless you believe in a "last finite ordinal", which of course a non-ultrafinitist will not believe in. Actually this makes sense in what is called non standard model of arithmetic. In Peano Arithmetic you cannot define what is a finite number (the notion of finiteness is typically a second order notion). So it is consistent to add some "infinite numbers" in the model of PA. But you can prove in PA that any number, except zero, has a predecessor, so a non standard model of arithmetic can have infinite objects, but then also its predecessor, and all their successive predecessors. A non standard model will have the following order on it: 0 1 2 ... ... infinity-1 infinity infinity+1 .... (and possibly other so called Z-chain). Such order are NOT ordinals, note. All this is not important now, but I say this in passing. > My instinct would be to say that a "well-defined" criterion is one > that, given any mathematical object, will give you a clear answer as > to whether the object fits the criterion or not. And obviously this > one doesn't, because it's impossible to decide where R fits it or not! > But I'm not sure if this is the right answer, since my notion of > "well-defined criteria" is just supposed to be an alternate way of > conceptualizing the notion of a set, and I don't actually know why > "the set of all sets that are not members of themselves" is not > considered to be a valid set in ZFC set theory. Frege and Cantor did indeed define or identify sets with their defining properties. This leads to the Russell's contradiction. (I think Frege has abandoned his work in despair after that). One solution (among many other one) to save Cantor's work from that paradox consists in formalizing set theory, which means using "belongness" as an undefined symbol obeying some axioms. Just two examples of an axiom of ZF (or its brother ZFC = ZF + axiom of choice): is the extensionality axiom: AxAyAz ((x b z <-> y b z) -> x = y) "b" is for "belongs". It says that two sets are equal if they have the same elements. AxEy(z included-in x -> z b y) with "z included-in x" is a macro for Ar(r b z -> r b x). This is the power set axiom, saying that the set of all subsets of some set is also a set). Paradoxes a-la Russell are evacuated by restricting Jesse's "well-defined criteria" by 1) first order formula (in the set language, that is with "b" as unique relational symbols (+ equality) ... like the axioms just above. 2) but such first order formula have to be applied only to an already defined set. For example, you can defined the set of x such that x is in y and has such property P(x). With P defined by a set formula, and y an already defined set. Also, ZFC has the foundation axiom which forbids a set to belong to itself. In particular the informal collection of all sets which does not belongs to themselves is the universe itself, which cannot be a set (its power set would be bigger than the universe!). But there are version of ZF without foundation, or even with diverse versions of the negation of the foundation axiom, like Stephen Paul King appreciates so much. Actually, I will not use set theory at all, but I will mention often ZF and its many brothers (like ZFC, ZF+<some big cardinal>, ...) as example of very rich lobian machine. ZF can prove the complete (propositional) theology of a lesser rich (in provability power) machine like PA. Bruno PS BTW, on the FOR list Charles gives this interesting reference on Everett. It looks his son has made a movie on his father: http://www.newscientist.com/channel/opinion/mg19626311.800-interview- parallel-lives-can-never-touch.html 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 -~----------~----~----~----~------~----~------~--~---
