Uncountability of the Rationals?
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From: Tony Orlow on 5 Jun 2010 01:50 On Jun 4, 4:24 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 3:20 pm, David R Tribble <da...(a)tribble.com> wrote: > > > Tony Orlow wrote: > > > One might think there were something like aleph_0^2 rationals, but > > > that's not standard theory. > > > Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0. > > Orlow can't be bothered to learn such basics. > > MoeBlee Piffle to you both. I already stated that very fact very early in this thread. Don't start crying "quantifier dyslexia". You know better. Tony
From: Tony Orlow on 5 Jun 2010 02:01 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? > > > If set > > theory would like to hide the recursive nature of infinite sets in > > order to draw precarious conclusions about sequences as if they "just > > exist", > > You often seem to have problems with this "just exists" concept. > > If we say that the set E of all even naturals exists, we mean that > E = {0, 2, 4, ...} is there, in abstract mental idea space, in its > entirety, all at once. A set (or any other mathematical entity) > simply exists, all at once, fully formed. > > Such entities are not processes that have to "execute" in order > to be "finished". Irrational numbers, for instance, are not incomplete > sequences of digits continuously growing by some mysterious > mathematical digit-appending daemon. > > -drt It's not so much that such a sequence does not exist, but that the existence of each but the first is dependent on the existence of at least one other. The size of such a structure simply cannot be measured, except relatively with respect to some other countably infinite sequence, such as N. Peace, Tony
From: Tony Orlow on 5 Jun 2010 02:17 On Jun 4, 5:10 pm, mstem...(a)walkabout.empros.com (Michael Stemper) wrote: > In article <83ef0ace-2110-4d92-86d2-078f68629...(a)e21g2000vbl.googlegroups..com>, Tony Orlow <t...(a)lightlink.com> writes: > > > > > > >On Jun 4, 1:12=A0pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > >> On Jun 4, 12:20=A0pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> > On Jun 3, 11:40=A0pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> > > There are not more rational numbers than naturals -- that is, |N|=|Q|. > > >> > > Even Tony knows that. > >> > 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. > > >> You have been given an explicit injection from Q to N. > > >> If you think it is wrong, explain why. > > >> Otherwise, explain this idiocy in which you believe that there are > >> more rationals than > >> integers. > >I have stated clearly that I understand there is a bijection between > >the countable infinities of N and Q. Are you daft? Why correct > >something which isn't wrong? Sure, they have the same cardinality. So, > >what? > > >You cannot disagree with the fact that the rationals are a dense set > >whereas the naturals are sparse, with a countably infinite number of > >rationals lying quantitatively between any two given naturals, can > >you? Do you not see that in some sense there appear to be more > >rationals than naturals? Are you incapable of considering any other > >system of set measurement besides what was drilled into you in class? > > Y'know, if you want, you're perfectly free to propose a different > way to compare two sets. Nobody's ever said that such comparisons > need to be based on the notion of cardinality. > > For starters, you might want to investigate the concept of "Lebesgue > measure". As I understand it (and I haven't actually reached this in > my formal studies yet), the Lebesgue measure of some uncountable > proper subsets of R is less than that of R. This might be close to > what you're seeking. On the other hand (again, as I understand it), > the Lebesgue measure of any countable subset of R is zero, so this > still gives N and Q being of the "same size". Yes, I've looked into Lebesgue measure, and the notion that any countable (finite or infinite) set of contiguous points has a spatial measure of zero is consistent with my theory, insofar as it refers to standard measure. However, when assuming a unit infinitesimal into account, as the measure of a point, the Lebesgue measures of these sets might be extended so that they are distinguishable on that scale. > > However, it might be a starting point that you could use as a springboard > for developing a method of comparing the "sizes" of sets in some new > way. I assure you that, if you come up with a well-defined way to do > this that doesn't give ambiguous (as opposed to unintuitive) results, > the real mathematicians here would be interested. One can only hope. We'll see how it goes. :) > > Something to use as a test case for any comparison method that you come > up with is to compare following sets: > > 1. The positive rationals, R+ That's a little tricky to quantify with respect to N if one wants to keep all quantities unique regardless of expression, but I have a few ganglia working on it. > 2. The expression of all positive rationals as decimal fractions > 3. The expression of all positive rationals as octal fractions > > (The last two sets are sets of strings.) Yes, that's where I apply N=S^L. Good reminder!! Of course, there's still that problem with the rationals in general... :( > > Be ready to show how your method of comparison treats each of these sets, > and answer which is "biggest", "smallest", and show how your method arrives > at those answers. Indeed. Thanks for the advice. I'm sure some questions will remain after I'm "done". You guys figgered out that continuum thingy yet? Mine don't gots dat problem... ;) > > -- > Michael F. Stemper > #include <Standard_Disclaimer> > This sentence no verb.- Hide quoted text - > > - Show quoted text - Thanks, TOny
From: Tony Orlow on 5 Jun 2010 02:27 On Jun 4, 5:24 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <51fafd21-e48d-461b-82cc-c2a93ed9a...(a)d8g2000yqf.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 3, 3:55 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...(a)c7g2000vbc.googlegroups.com>, > > > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > In the same vein, most objectors to transfinitology have a hard time > > > > identifying where exactly they disagree with the construction of the > > > > theory, but have at least a few examples of conclusions drawn from it > > > > which are completely incorrect when approached through other avenues. > > > > This is partly because mathematicians misrepresent the theory, > > > > claiming that every conclusion they draw "follows logicallly from the > > > > axioms." This claim is simply not true, as cardinality is not > > > > mentioned in the axioms, much less anything about omega or the alephs, > > > > except for a declaration that something homomorphic to the naturals > > > > exists, which is ultimately not a set, but a sequence. > > > > The issue is whether cardinality CAN BE defined within an axiom system, > > > not whether it is explicitly mentioned in some axiom(s) of that system. > > > > And it can be. > > > And yet, it does not logically follow from the axioms. > > If you mean that one can avoid deducing the properties of cardinality, > note that one can, by totally ignoring the system, refuse to deduce > anything from it at all. Yeah, man, like, wow. Why deduce anything when you just know? Here, I grew this flower for you, in my hair. Doesn't it smell nice? I'm sure it will convince you..... ;) > > > It is simply > > consistent in the sense of not contradicting them. > > Meaning that it can be deduced therefrom. No, meaning that there does not arise any contradiction. That is to say, no set of assumptions within the model can be used to derive the opposite of any subset of the axioms. The axioms do not imply the model. The model simply fails to directly contradict the axioms. I'm sure you understand this, deep down inside your soul, man. Dig it. ;) > > > Therefore, other > > measures of sets can also exist, consistent with the axioms, but > > inconsistent with cardinality. > > Incompatible measures perhaps, but not inconsistent with cardinality > unless inconsistent with itself. > I'm sure I won't be the only one to disagree with this statement so I'll reserve comment for now. > > > > > How does TO define 'sequence' without reference to something like the > > > SET of naturals as indices? > > > 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? Love, Tony
From: Tony Orlow on 5 Jun 2010 02:28
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 |