An uncountable countable set
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From: Virgil on 9 Oct 2006 16:56 In article <452ab540(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452aa5c9(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> If you increment a natural n times, you have added n to it. If successor > >> is increment, and there are an infinite !number! of such increments, you > >> have added this infinite number to your starting value. Adding an > >> infinite number to a finite yields an infinite. Therefore, the infinite > >> set includes infinite values. > > > > Without a limit ordinals in between or as the larger, one never can have > > two ordinals separated by infinitely many successors of the smaller. > > Proof, please? Why? TO never proves any of his claims.
From: David Marcus on 9 Oct 2006 17:02 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David R Tribble wrote: > >>>>>>> Virgil wrote: > >>>>>>>>> Except for the first 10 balls, each insertion follow a removal and with > >>>>>>>>> no exceptions each removal follows an insertion. > >>>>>>> Tony Orlow wrote: > >>>>>>>>> Which is why you have to have -9 balls at some point, so you can add 10, > >>>>>>>>> remove 1, and have an empty vase. > >>>>>>> David R Tribble wrote: > >>>>>>>>> "At some point". Is that at the last moment before noon, when the > >>>>>>>>> last 10 balls are added to the vase? > >>>>>>>>> > >>>>>>> Tony Orlow wrote: > >>>>>>>> Yes, at the end of the previous iteration. If the vase is to become > >>>>>>>> empty, it must be according to the rules of the gedanken. > >>>>>>> The rules don't mention a last moment. > >>>>>>> > >>>>>> The conclusion you come to is that the vase empties. As balls are > >>>>>> removed one at a time, that implies there is a last ball removed, does > >>>>>> it not? > >>>>> Please state the problem in English ("vase", "balls", "time", "remove") > >>>>> and also state your translation of the problem into Mathematics (sets, > >>>>> functions, numbers). > >>>> Given an unfillable vase and an infinite set of balls, we are to insert > >>>> 10 balls in the vase, remove 1, and repeat indefinitely. In order to > >>>> have a definite conclusion to this experiment in infinity, we will > >>>> perform the first iteration at a minute before noon, the next at a half > >>>> minute before noon, etc, so that iteration n (starting at 0) occurs at > >>>> noon-1/2^n) minutes, and the infinite sequence is done at noon. The > >>>> question is, what will we find in the vase at noon? > >>> OK. That is the English version. Now, what is the translation into > >>> Mathematics? > >> Can you only eat a crumb at a time? I gave you the infinite series > >> interpretation of the problem in that paragraph, right after you > >> snipped. Perhaps you should comment after each entire paragraph, or > >> after reading the entire post. I'm not much into answering the same > >> question multiple times per person. > > > > I snipped it because it wasn't a statement of the problem, as far as I > > could see, but rather various conclusions that one might draw. > > I drew those conclusions from the statement of the problem, with and > without the labels. I'm sorry, but I can't separate your statement of the problem from your conclusions. Please give just the statement. -- David Marcus
From: Tony Orlow on 9 Oct 2006 17:02 Virgil wrote: > In article <452ab325(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: > >>> I snipped it because it wasn't a statement of the problem, as far as I >>> could see, but rather various conclusions that one might draw. >> I drew those conclusions from the statement of the problem, with and >> without the labels. > > But without the labels, it is a different problem. Define "different". Why do the labels matter. Do they take precedence over order in set theory, where order matters not? Well, order is specifically stated in the gedanken, so don't violate it. >> While >>> these may be correct, until the problem is stated mathematically, it is >>> impossible to tell. Please give the statement of the problem in >>> mathematics without also giving any conclusions or derived properties. >> Yeah, I compared and contrasted the two approaches. Please reread. > > As TO was, as usual, wrong, please rewrite. As Vigil was, as usual, clueless, please rethink. >>> When discussing mathematics, communication works best one step at a >>> time. >>> >> So, yes, you can only take a spoonful at a time. > > TO can't even take that. I have taken each of your micro-points and turned it into nothing, Virgil.
From: David Marcus on 9 Oct 2006 17:05 Ross A. Finlayson wrote: > David Marcus wrote: > > Ross A. Finlayson wrote: > > > David Marcus wrote: > > > > I thought you said there was a contradiction in ZF. In the context of > > > > ZF, the Burali-Forti argument shows that there is no set of all > > > > ordinals, but does not lead to a contradiction. So, do you still say > > > > there is a contradiction in ZF? If so, what is it? > > > > > > {For any x: x is a set} = emptyset <=/=> {For any x: x is a set} = U > > > (V, L) > > > > > > That says, for any x, that's the empty set, and, for any x, that's the > > > universal set, it seems sufficient to show the universe non-empty. > > > > > > There is no set of ordinals nor cardinals in ZF. Yet, because there's > > > the axiom of infinity, those infinite ordinals/cardinals as there ever > > > would be are claimed to exist, basically where they're all hereditarily > > > finite, those ordinals of the cumulative hierarchy. > > > > Sorry, but I don't follow. Are you saying this is a contradiction within > > ZF? By "within" I mean that ZF proves this contradiction. > You seem like you know what you're talking about, which is good. > > Basically, yes, I say the existence of the (universal) quantifier, > where the word universal is in parentheses because there's the mutually > implicit existential quantifier, that the existence of the universal > quantifier in ZF lead to illustration of a contradiction derivable from > ZF. OK. But the Burali-Forti argument does not (as far as I know) lead to a contradictionn derivable from ZF. So, what is the contradiction that you can derive from ZF? > I have some other arguments along those lines as well, of similar tack, > where I advocate axiom-free natural deduction, as a return of sorts to > a more "naive" set theory with post-Cantorian acknowledgement. > > That is to say, there are thousands of pages more of my opinion about > these matters readily available. -- David Marcus
From: David Marcus on 9 Oct 2006 17:07
Tony Orlow wrote: > David Marcus wrote: > > Ross A. Finlayson wrote: > >> David Marcus wrote: > >>> I thought you said there was a contradiction in ZF. In the context of > >>> ZF, the Burali-Forti argument shows that there is no set of all > >>> ordinals, but does not lead to a contradiction. So, do you still say > >>> there is a contradiction in ZF? If so, what is it? > >> {For any x: x is a set} = emptyset <=/=> {For any x: x is a set} = U > >> (V, L) > >> > >> That says, for any x, that's the empty set, and, for any x, that's the > >> universal set, it seems sufficient to show the universe non-empty. > >> > >> There is no set of ordinals nor cardinals in ZF. Yet, because there's > >> the axiom of infinity, those infinite ordinals/cardinals as there ever > >> would be are claimed to exist, basically where they're all hereditarily > >> finite, those ordinals of the cumulative hierarchy. > > > > Sorry, but I don't follow. Are you saying this is a contradiction within > > ZF? By "within" I mean that ZF proves this contradiction. > > > > Hi David - > > This part of Ross I don't quite follow, but perhaps we should discuss > it. What is your background, if you don't mind my asking? Is that relevant? Or, are you just curious? -- David Marcus |