Uncountability of the Rationals?
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From: Tony Orlow on 4 Jun 2010 16:40 On Jun 4, 4:02 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > > In Cantor's list of reals, for every digit added, the list doubles in > > length. > > I assume you mean a list of finite-length binary fractions > representing reals in [0,1). > > > In order to follow his logic, we need not consider any real > > numbers which are not rational. In any such list of rationals, it is > > always true that we can concoct another rational which is not on the > > list. > > Oh come on, Tony. You know this isn't right. > > A list of finite-length rationals is either a finite list itself, > being a list of reals of n bits or less and therefore containing > no more than 2^n reals in total; or it's an infinite list of > finte-length reals (i.e., all of them are binary fraction that > end with a trailing list of repeating 000... digits). > > In the first case, it's not Cantor's list of reals, so it's moot. > > In the second case, the real that's not in the list is a > sequence of binary digits that differs by at least one digit > from every other (rational) real in the list. This diagonal > number, by the way, is most likely not rational (unless it > just so happens to end in a repeating binary digit sequence). > > > Irrational numbers need not even be considered. So, are the > > rationals not to be considered uncountable, by Cantor's own logic? > > Well, first you have to have the correct description of Cantor's > diagonal argument. You don't. > > > BTW, I understand that the rationals are considered countable because > > of a rather artificial bijection with N, > > Or any other bijection with N. There are an infinite number of > them to choose from. I personally like the one I independently > re-discovered a few years ago (as a response to one of your posts, > in fact, if I recall correctly), which is at: > > http://david.tribble.com/text/sequence1.html > > > but if Cantor's argument > > really has anything to do with the reals, as opposed to the powerset > > (which is what it's really all about), then why is it entirely > > unnecessary to even consider irrational reals in his construction? > > Again, you have to have the right description of the list. > Once you do, you can see that your conclusion is not warranted. > > -drt Hi David I expected you to excoriate me, but you kind of complimented me. Thanks. I'm glad I made you think. I've already been corrected several times in this thread on my original post, and freely accept it. It was more of a question than a challenge, and my "hypothesis" stood corrected, until I told it to sit down and shut up. I'm printing out your web page and will peruse it with anticipation. :) Take care, Tony
From: David R Tribble on 4 Jun 2010 16:55 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, ... . > 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
From: Michael Stemper on 4 Jun 2010 17:10 In article <83ef0ace-2110-4d92-86d2-078f686290e8(a)e21g2000vbl.googlegroups.com>, Tony Orlow <tony(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". 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. 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+ 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.) 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. -- Michael F. Stemper #include <Standard_Disclaimer> This sentence no verb.
From: Virgil on 4 Jun 2010 17:24 In article <51fafd21-e48d-461b-82cc-c2a93ed9a728(a)d8g2000yqf.googlegroups.com>, Tony Orlow <tony(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. > It is simply > consistent in the sense of not contradicting them. Meaning that it can be deduced therefrom. > 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. > > > > > 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". > > Peace, Virgil. > > Tony
From: Virgil on 4 Jun 2010 17:28
In article <2ff234ce-4fc9-49fb-b5ba-027951c1e6bc(a)c33g2000yqm.googlegroups.com>, Tony Orlow <tony(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? |