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From: imaginatorium on 20 Oct 2005 00:36 Tony Orlow wrote: > stevendaryl3016(a)yahoo.com said: <snip> > > Let A be any set whatsoever, finite or infinite, it doesn't matter. > > Let f be any function from A to P(A). > > Let w = { x in A | x is not an element of f(x) }. > > Let x = any set in A. > > Let u = f(x). We prove that u is not equal to w. > > > > By definition of w, we have x in w <-> x is not an element of f(x). > > So x in w <-> x is not an element of u. That means that there are > > two cases: Case 1: x in w, and x is not in u. In that case, u cannot > > equal w. Case 2: x is not in w, and x is in u. In that case, u cannot > > equal w. > > > > So what we have proved is that forall x, w is not equal to f(x). So > > w is not in the image of f. So f is not a bijection between A and P(A). > > > > There's no induction. There's no assumption that A is finite. > But there is an assumption that y is in S. If you are assuming you have the > complete set of naturals, that you have identified the last, and can therefore > identify the element that maps to the entire set, then you indeed run into a > contradiction. ... Goodness you are dim. The axiom of infinity says (in effect) "The naturals are a set". This means we talk about the set of naturals, letting all the other axioms apply to it. Why do we need to have an _axiom_ to let us talk about the naturals? Because it is an infinite set. It goes on forever, and never ends. There is no last natural. There is no end to them. The "end" is not merely "unspecified", "unidentified", "tenuous", or any such, it is *nonexistent*. (Remembering that nonexistence, like existence, is not a predicate.) Having proved that there is no last natural, we nonetheless talk about the complete, entire, total, set of all naturals. All naturals. There is no natural anywhere in any of your nonsensical "doubling" and "thinning" operations that does not already belong to the mathematical set of all naturals. That's what "all" means. Sorry, mustn't go on - it's pointless anyway, since after thousands of posts it's pretty unlikely you will ever grasp any of it. But anyway, to go back to your paragraph: Yes, we have the complete set of naturals. No, we have not "identified the last", because there isn't one. "Can therefore identify the element that maps to the entire set" makes no sense. We are considering any mapping from a set A to its power set P(A). In some mappings there is an element mapped to the complete set - e.g. (a crank favourite) 0 -> {} 1 -> {0, 1, 2, 3, ... } // the complete set of naturals 2 -> {0} 3 -> {1, 2, 3, 4, ... } // all naturals except 0 4 -> {1} .... > ... However, both the infinite set of naturals and the infinite > power set go on forever, so you never run out of naturals to map to subsets, > nor subsets to map to naturals. Right: well done!! You got something right. > ... Despite the fact that, within any range up to > S, you cannot map every subset to an element within that range, the lack of a > largest element makes it so there DOES exist an element that maps to all n > through S, but it is more than S. This is a prime example of where the value > range matters in the bijection. No it isn't. There is no "value range" in a bijection. "Value ranges" are only used in TOmatics, remember. But you are clawing your way towards grasping what was the natural assumption (that if two sets go on forever it is never possible to say there can't be a bijection between them) before Cantor pointed out that it is wrong. Here's a question for you. Obviously if the TOnats (T) are a set, within the framework of conventional set theory, the proof applies to them, and there can be no bijection T <-> P(T). But are they a set? Is it possible to have the complete, final, total, can never be extended in response to the next question, set of TOnats? Brian Chandler http://imaginatorium.org
From: Daryl McCullough on 20 Oct 2005 08:55 Tony Orlow says... >stevendaryl3016(a)yahoo.com said: >> Here it is once again: >> >> Let A be any set whatsoever, finite or infinite, it doesn't matter. >> Let f be any function from A to P(A). >> Let w = { x in A | x is not an element of f(x) }. >> Let x = any set in A. >> Let u = f(x). We prove that u is not equal to w. >> >> By definition of w, we have x in w <-> x is not an element of f(x). >> So x in w <-> x is not an element of u. That means that there are >> two cases: Case 1: x in w, and x is not in u. In that case, u cannot >> equal w. Case 2: x is not in w, and x is in u. In that case, u cannot >> equal w. >> >> So what we have proved is that forall x, w is not equal to f(x). So >> w is not in the image of f. So f is not a bijection between A and P(A). >> >> There's no induction. There's no assumption that A is finite. > >But there is an assumption that y is in S. You mean A. Yes, that's what it means to have a surjection f between two sets A and P(A) (a bijection is a kind of surjection). It means that for every w in P(A) there exists a y in A such that w = f(y). If y is not in A, then the fact that w = f(y) is irrelevant to whether there is a bijection between A and P(A). >If you are assuming you have the complete set of naturals, >that you have identified the last, and can therefore >identify the element that maps to the entire set, then >you indeed run into a contradiction. However, both the >infinite set of naturals and the infinite power set go >on forever, so you never run out of naturals to map to subsets, >nor subsets to map to naturals. Yes, if you imagine generating subsets of the naturals one at a time, then you are only going to generate countably many subsets. The point is that for any rule for generating subsets, there exists a subset that will *never* be generated by that rule. So no rule can generate all possible subsets of the naturals. (Cantor's theorem is not limited to sets and functions that are defined by any particular rule, but for concreteness, let's restrict ourselves to those.) Notice that this is not the case with *finite* subsets. If S is any finite subset of naturals, then we can associate it with the natural number n = sum of S_i 2^i where S_i = 1 if i is an element of S, and S_i = 0 otherwise. This is a rule for associating every finite set of naturals with a natural, and there are no finite subsets that are left out. Therefore, there is a bijection between the set N of naturals and the set FS(N) the finite subsets of N. If you are limiting yourself to describable mathematical objects, then Cantor's theorem has the following form: If you give me any rule for mapping sets of naturals to naturals, then I will give you a rule for creating a set of naturals that is not in the range of your mapping. Even though you can never complete the process of generating an infinite set, you *can* complete the process of giving a rule for one. For example, the rule "x is in the set if and only if x is even" defines the infinite set of even natural numbers. The rule "x --> 2*x" is a rule for mapping the set N of natural numbers to the set E of even numbers. -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 20 Oct 2005 10:36 albstorz(a)gmx.de said: > > William Hughes wrote: > > > > > > Coincidently natural numbers and cardinalities are undistinguishable in > > > finity. > > > > They are very similar, but they are not quite "undistinguishable". > > A natural number is a set, a cardinality is an equivalence class. > > You make me hopefull. Some experts make "=E4=E4=E4h", "h=F6mm" and > "=FC=FC=FCh" if I said "A natural number is a set." One sees a > correspondence between natural numbers and von Neumann sets after all. > You are free to say "A natural number is a set." without "=E4=E4=E4h-", > "h=F6mm-" and "=FC=FC=FCh-" comments. Be lucky. You are right. > And natural numbers don't behave in any other way than sets. So, if > there is an infinite set there is an infinite number. If there is no > infinite number there is no infinite set. And vic versa. > > > > Regards > > AS > > And, of course, there are infinite sets and infinite numbers, natural and real. :D -- Smiles, Tony
From: Virgil on 20 Oct 2005 14:00 In article <MPG.1dc07f6746a0fede98a4f8(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > albstorz(a)gmx.de said: > > > > William Hughes wrote: > > > > > > > > > Coincidently natural numbers and cardinalities are undistinguishable in > > > > finity. > > > > > > They are very similar, but they are not quite "undistinguishable". > > > A natural number is a set, a cardinality is an equivalence class. > > > > You make me hopefull. Some experts make "=E4=E4=E4h", "h=F6mm" and > > "=FC=FC=FCh" if I said "A natural number is a set." One sees a > > correspondence between natural numbers and von Neumann sets after all. > > You are free to say "A natural number is a set." without "=E4=E4=E4h-", > > "h=F6mm-" and "=FC=FC=FCh-" comments. Be lucky. You are right. > > And natural numbers don't behave in any other way than sets. So, if > > there is an infinite set there is an infinite number. If there is no > > infinite number there is no infinite set. And vic versa. > > > > > > > > Regards > > > > AS > > > > > And, of course, there are infinite sets and infinite numbers, natural and Only in the weird world of TOmatics are any standard reals or standard naturals anything but finite. Any "numbers" which are not finite are not in the set of reals nor any subset of the set of reals.
From: Tony Orlow on 20 Oct 2005 14:08
William Hughes said: > > albstorz(a)gmx.de wrote: > > William Hughes wrote: > > > > > > > > > Coincidently natural numbers and cardinalities are undistinguishable = > in > > > > finity. > > > > > > They are very similar, but they are not quite "undistinguishable". > > > A natural number is a set, a cardinality is an equivalence class. > > > > You make me hopefull. Some experts make "=E4=E4=E4h", "h=F6mm" and > > "=FC=FC=FCh" if I said "A natural number is a set." One sees a > > correspondence between natural numbers and von Neumann sets after all. > > You are free to say "A natural number is a set." without "=E4=E4=E4h-", > > "h=F6mm-" and "=FC=FC=FCh-" comments. Be lucky. You are right. > > And natural numbers don't behave in any other way than sets. So, if > > there is an infinite set there is an infinite number. If there is no > > infinite number there is no infinite set. And vic versa. > > > > You have made exactly this mistake before. Yes every number > is a set. No, not every set is a number. For example > {peach, apple, plum, fiddle} is a set but not a number. Huh! I coulda swore that was 4! > Just because you have a set does not mean you have a number. So, not every set has a size? > So yes, there is an infinite set. But this does not mean > that this set is a number. Indeed, no infinite set is a number. It's not aleph_0? What is that thing anyway? > > - William Hughes > > > P=2ES Actually it is not true that natural numbers must be sets, but > they can be. As you insist on using a model in which the natural > numbers are sets, I am playing along to be polite. That's very nice of you, William. > > -- Smiles, Tony |