Uncountability of the Rationals?
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From: Tony Orlow on 8 Jun 2010 11:59 On Jun 8, 10:11 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 7, 11:38 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > A quantitatively ordered set is one wherein each member is either > > greater than, less than, or equal to every other member. Yes/no? > > Trichotomy applies? If neither x<y or x>y then x=y, noyes? > > WHAT exact ordering on sets in GENERAL do you mean '<' to stand for? > > MoeBlee I was referring to sets of real (or hyperreal) quantities, specifically. Changing the subject doesn't prove anything. Tony
From: David R Tribble on 8 Jun 2010 12:51 Tony Orlow wrote: >> What is your personal definition of a sequence, again? > David R Tribble wrote: >> Fairly close to the actual definition. Essentially, it's a >> set that maps the naturals to a function (a function being >> a set itself, of course, within set theory). >> >> Here's a few example sequences: >> S1 = 0, 1, 2, 3, 4, 5, 6, ... >> S2 = 0, 2, 1, 4, 3, 6, 5, ... >> S3 = 0, 2, 4, 6, ..., 1, 3, 5, 7, ... >> S4 = 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... >> >> You'll notice that all of these particular sequences are >> *different* (which is why I gave them different names), >> even though they all contain the same values (all of the >> naturals, as it happens, in these examples). > Tony Orlow wrote: > By the axiom of extensionality, as sets they are identical. No, they are not. Each sequence is a *different* set. I'll use a more expressive notation to depict these sequences as actual sets: S1 = {<0,0>, <1,1>, <2,2>, <3,3>, <4,4>, <5,5>, <6,6>, ...} S2 = {<0,0>, <1,2>, <2,1>, <3,4>, <4,3>, <5,6>, <6,5>, ...} This notation shows more clearly the mapping between the naturals and the values of each sequence. In a purer (but harder to read) set notation, each '<a,b>' pair would be written as '{a, {b}}', or something similar. I've only bothered to show the first two sequences (the rest being slightly more complicated since they are concatenations of subsequences), but it's plainly obvious that they are all *different* sets. Different sequences, thus different sets. And to be even more instructive, here are two more different ways to write sequence S1 as a set: S1 = {<1,1>, <0,0>, <3,3>, <2,2>, <5,5>, <4,4>, <6,6>, ...} S1 = {<3,3>, <4,4>, <5,5>, <0,0>, <1,1>, <2,2>, <6,6>, ...} All three ways of writing S1 are different, but the set itself is exactly the *same* for all three. Again, and it bears repeating, by themselves *sets have no order*. They are just collections of their members. Better to think of them as special kinds of bags or sacks than as lists or sequences. Really, Tony, these concepts are very, very simple. My explanations are pretty easy to follow. I don't claim to be an expert in these matters, but these ideas are so simple I don't have to be in order to understand and explain them. > As sequences they differ. If a sequence contains the same members as > another which is monotonically increasing, then that other can be used > to calculate the bigulosity of both sets/sequences. If you say so, but what if they don't? Can you calculate the (relative?) "bigulosity" between S1 and S2? -drt
From: MoeBlee on 8 Jun 2010 13:18 > I was referring to sets of real (or hyperreal) quantities, > specifically. Changing the subject doesn't prove anything. Then you haven't given any alternative to set theoretic cardinality. We ALREADY know the standard ordering on R. You contribute nothing by harping on it. And for hyperreals, indeed those are defined by certain orderings. So, again, we don't need you to remind us of ordering. You carp and carp and carp that you don't like the notion of 'size' being based on bijection, but you've given no alternative. MoeBlee
From: Virgil on 8 Jun 2010 15:27 In article <ccfdcfaf-6f22-4601-8132-90a407737e7f(a)z8g2000yqz.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 12:52�am, David R Tribble <da...(a)tribble.com> wrote: > > MoeBlee wrote: > > >> (1) For only occasional sets do we have a "usual" ordering. In > > >> general, there is no definition provided (especially by you [Tony]) for > > >> "the > > >> usual ordering of a set". So your analysis in that respect doesn't > > >> even get off the ground. > > > > David R Tribble wrote: > > >> Indeed. Here are a few ways to write the set of naturals, > > >> all of them being exactly the same set: > > >> �{ 0, 1, 2, 3, 4, 5, 6, ... } > > >> �{ 0, 2, 1, 4, 3, 6, 5, ... } > > >> �{ 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } > > >> �{ 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } > > >> �{ ..., 5, 4, 3, 2, 1, 0 } > > >> �{ 0, ..., 4, 3, 2, 1 } > > > > >> Tony must realize that a set is simply a collection, > > >> specifically an *unordered* collection. Just because all > > >> of its members are natural numbers (or reals, or rationals, > > >> or whatever) does not automatically imbue it with an > > >> ordering. > > > > Tony Orlow wrote: > > > Why are all of these infinite sets you cite some kind of sequence? > > > > They're not. Not a single one. > > > > And (I repeat), they are all exactly the same set. > > > > Again, you come across as though you truly don't get it, > > intentionally, or sarcastically, or otherwise. > > > > -drt- Hide quoted text - > > > > - Show quoted text - > > Wow. Those aren't sequences, with or without internal limit > "ordinals"? Look again. What is your personal definition of a > sequence, again? For most people, a sequence is a function whose domain is the set of naturals (or. at least, some set order-isomorphic to the naturals). None of the sets displayed are even functions, thus also not sequences.
From: Virgil on 8 Jun 2010 15:37
In article <abd15b73-5071-49f3-9a78-b5107bf1dc37(a)d8g2000yqf.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 1:08�am, Virgil <Vir...(a)home.esc> wrote: > > In article > > <d6c03228-ad6c-4d32-833a-b742f3b09...(a)y4g2000yqy.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > A quantitatively ordered set is one wherein each member is either > > > greater than, less than, or equal to every other member. Yes/no? > > > Trichotomy applies? If neither x<y or x>y then x=y, noyes? > > > > Then how, if at all, do your "quantitatively ordered sets" differ from > > the more standard "ordered sets" or "totally ordered sets" or "linearly > > ordered sets"? > > In that greater quantities occur after smaller quantities. > Monotonically increasing. You've heard of it. But without any of the order required by being an ordered set (having an order relation on it), how do you distinguish greater from lesser? > > > > > Unless there is some difference, I see no reason for creating a new name > > for what is already satisfactorily and multiply named. > > > > I think it is up to others to speak for themselves, as it is up to you > > > to do so. If it is not clear to you, then please ask a clarifying > > > question. > > > > My questions: > > > > � �What do you mean by "infinite-case induction"? > > > > � �Is there some more standard term for it, as there is for your > > � �"quantitatively ordered sets" or it it an invention of your own. > > > > � �Can you give an example of it? > > > > I've done this about a million times. Did you read my post to Transfer > explaining the T-riffics? > > > > > > > > > Anything that TO is at all able to comprehend must be superficial, and > > > > injection and bijection of all but trivially small sets are not > > > > dependent on any particular mapping formula. > > > > > Dear, they are. > > > > Not at all. For all but finite sets there are for every instance of a > > surjection mapping formula from a given set an infinity of alternates > > each of which is just as satisfactory. > > > > > > To you, perhaps. To mathematics in general. In establishing the existence of surjections, it is totally immaterial WHICH of the possible surjections one shows exists. > > > Yes, there is no explicitly defined infinite set which is not also a > > > sequence > > > > The rationals standardly ordered are not a sequence. > > The reals standardly ordered are not a sequence. > > The complexes, as standardly constructed, are not even ordered, much > > less a sequence. > > When you have to snip the end of my sentence in order to make my > statement sound stupid the you're stretching and making yourself look > > > > I do not know that what you call IFR makes any sense at all. > > Then you haven't paid any attention to it. I have given it more than it deserves, but still found o sense in it. > > > > > > You know infinite-case induction is currently workable > > > > No I don't. I am much to picky about what I "know" to know that. > > > > > and how N=S^L works with IFR. > > > > No I don't. > > > > > Virgil, you know it all. You know you're > > > invited as a guest when I accept the Fields Medal... > > > > If I really know it all then what I know of your IFR makes it nonsense. > > Besides, re your invitation, I cannot possibly live that long. > > I'll bring your urn. My "urn" won't last that long either. |