Uncountability of the Rationals?
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From: Virgil on 5 Jun 2010 12:56 In article <a459d982-9855-4f03-b105-77c333ef5726(a)j4g2000yqh.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 4, 4:55�pm, David R Tribble <da...(a)tribble.com> wrote: > > Tony Orlow wrote: > > > Sequences are sets with order. Sets in general have no order. > > > > I get what you're trying to say, but to be pedantic, sequences are > > not sets at all. Consider the sequence S = 1, 1, 1, 1, ... . > > Yes, granted. Monotonically increasing sequences are sets with order > of magnitude correlated positively with order of occurrence. Better? > That's the way the simplest infinite "sets" are defined, no? The issue here being how SEQUENCES are defined, TO, as usual, goes off on a tangent.
From: Virgil on 5 Jun 2010 12:59 In article <db5cbe4b-a8b8-4b6a-ae47-05fa05dd6587(a)i28g2000yqa.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Yes, that's where I apply N=S^L. Which is a wrong now as when first dropped on an unsuspecting world.
From: Virgil on 5 Jun 2010 13:04 In article <796c4157-f47d-4aba-b056-8ddc38d465a4(a)c10g2000yqi.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > It is a set wherein every element is either before or after (not > > > immediately) every other element. That's one possible definition, > > > though you undoubtedly have some objection. > > > > Both the rationals and the reals, with their usual orders, satisfy YOUR > > definition of sequences, and while the rationals, with a suitable but > > different ordering may be a sequence, there is no ordering on the reals > > which is known to make them into a sequence, at least for any generally > > accepted definition of "sequence". > > > > Surely you remember the T-Riffics? Does Tony Orlow really want to maintain that ANY part of his idiotic "T-Riffics" was ->generally accepted<- ?
From: Virgil on 5 Jun 2010 13:05 In article <426447ef-9569-4bea-b017-c4f89c9cda43(a)g7g2000yqj.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 4, 5:28�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <2ff234ce-4fc9-49fb-b5ba-027951c1e...(a)c33g2000yqm.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > > > > > > > On Jun 3, 11:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > kunzmilan <kunzmi...(a)atlas.cz> writes: > > > > > The uncountability of the reals is simply based on the fact, that > > > > > there are more rational numbers than there are the natural numbers. > > > > > > This is, of course, utterly butt-wrong. > > > > > > There are not more rational numbers than naturals -- that is, |N|=|Q|. > > > > > > Even Tony knows that. > > > > > > -- > > > > Jesse F. Hughes > > > > > > One is not superior merely because one sees the world as odious. > > > > -- Chateaubriand (1768-1848) > > > > > Hi Jesse - > > > > > Just because I concede that both sets are countably infinite and > > > therefore of the same cardinality, nevertheless the sparse proper > > > subset of the rationals called the naturals should not be equated in > > > size with its dense proper superset. > > > > > Tony > > > > Physical objects can have diverse measures of size, such as mass, volumn > > and surface area, so why are you so violently opposed to having a > > variety of "size" measures for non-physical objects?- Hide quoted text - > > > > - Show quoted text - > > Do you really think I'm violent? (sniffle) > > Tony More like "violet".
From: Virgil on 5 Jun 2010 13:15
In article <5d346c04-9c57-4d64-9beb-af3434ded0e7(a)d37g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > > > Look, "Tonico", you can cop whatever attitude you want with me, but in > > > a conversation in this very newsgroup not long ago it became obvious > > > that seasoned mathematicians don't even agree on what a sequence is, > > > some considering it isomorphic to the naturals, and others to the > > > entire class of von Neumann ordinals. Besides, this is unrelated to > > > the topic, and merely finger exercise for your pallid digits and > > > tangled neurons. > > > > By one fairly standard definition, a sequence (or, more properly, an > > infinite sequence) is a surjection from the set of naturals to any > > non-null set. Any other definition has to be fairly much equivalent.- Hide > > quoted text - > > > > Yeah. That's "fairly standard", at least in the sense of being fair to > middling. I mean, "fairly standard"? Isn't mathematics all about > precision? Or, perhpaps there is some kind of double standard? Since To seems to think that my definition is less than perfectly precise, I challenge TO to find a more precise definition expressed only in words. Mine: an infinite sequence is a surjection from the set of naturals to a set (note the "non-null" is not actually needed) Tony's: It is a set wherein every element is either before or after (not immediately) every other element. I'm sure Tony can improve on his present definition, but I doubt he can improve on mine. |