An uncountable countable set
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From: MoeBlee on 20 Sep 2006 16:13 Tony Orlow wrote: > > No, you need to DEFINE a division operation on infinite sets or > > infinite cardinals if you are to speak of "divding" as a kind of > > arithmetic. It is the fact that there is NOT a unique result from such > > a proposed "divsion operation" that blocks arguments that set theory is > > inconsistent due to erratic divsion results. > That is simply not true for any well-defined infinite value. We've > already established that the average value in the infinite set of reals > in [0,1] is 1/2. Similiarly, we can apply such methods to the naturals, > giving that the evens are half the set. The fact that transfinite > "arithmetic" does not get a compatible result is evidence that it's not > compatible with mathematics. Damn it. Give a rigouous mathematical definition already or shut up. MoeBlee
From: MoeBlee on 20 Sep 2006 16:16 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Given that all evens are natural, but only every other natural is even, > >> is is logically deducible that the naturals are twice as numerous in any > >> range as the evens. > > > > Define 'twice as numerous' for infinite sets. > > IFR does that and more, in terms of formulaic relations between sets > over a given range. Where the mapping from the naturals to another is > f(x)=x/2, the other has twice as many elements over any range that's a > multiple of 1, and asymptotically, over the entire real range. All of that being UNDEFINED VERBIAGE. MoeBlee
From: Virgil on 20 Sep 2006 16:30 In article <3a6c6$4510f00a$82a1e228$27505(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David R Tribble wrote: > > > mueckenh wrote: > > > >>>Nothing has changed. There is no complete set of natural numbers. Any > >>>set that can be established is a finite set. Hence, the probability to > >>>select a number divisible by 3 is 1/3 or very very close to 1/3. > > > > Virgil wrote: > > > >>That presumes that the allegedly finite set of naturals that can be > >>constructed is nearly uniform with respect to divisibility by 3 at > >>least, and probably by other numbers as well. What is the justification > >>for this assumption? > > Wolfgang says litteraly: "_or_ very very close to 1/3". Which requires "_nearly uniform_ with respect to divisibility by 3".
From: Virgil on 20 Sep 2006 16:31 In article <1158742382.376116.13260(a)b28g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > The limit of a sequence need not have a property that each of its > > elements has. > > The limit of the sequence 1,1,1,1,1,1,1,1, ... ,1, ... is 1. > > Han de Bruijn But "need not" does not require "can not"!
From: MoeBlee on 20 Sep 2006 16:36
Tony Orlow wrote: > Thank you. We're talking about the von Neumann ordinals because the > alephs are the cardinals based on them, being the sizes of the limit > ordinals, which are sets. A cardinal is an ordinal that is not bijectable with a member of itself. > are supposed to be sets/sizes/numbers which have some relation to the > finite naturals or reals, which really boils down to one of order. It boils down to the standard ordering of ordinals, which is the membership relation. It doesn't entail that that is the same as the standard ordering of reals. > It's supposed to be the smallest infinite value, somehow less than which > there is no other infinite value, because removing anything finite from > it doesn't change it any. 'because' there is dubious. > So, it's what comes right after (with some > sort of a pause) all the finite ordinals. But not right after any particular natural. So 'right after' is not a term we use in this regard, lest it lead ignoramuses like you to think that there is something going on that is not, even though you'll insist on thinking it anyway. > The distinction between cardinals and ordinals is artificial. Oh for god's sake. It's just a distinction between an object having a certain property and not having that property. An ordinal is bijectable with a member of itself or it is not bijectible with a member of itself. That is the disctinction between an ordinal that is not a cardinal and an ordinal that is a cardinal. It's as "artifical" as the distinction between a number that is divisible by two and a number that is not divisible by two. MoeBlee MoeBlee |