Uncountability of the Rationals?
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From: Rob Johnson on 3 Jun 2010 08:14 [article below posted to the newsgroups listed in the header] In article <f1b4a519-f904-43ca-8370-4cbb1b75e998(a)q8g2000vbm.googlegroups.com>, "christian.bau" <christian.bau(a)cbau.wanadoo.co.uk> wrote: >Cantor's argument breaks down for rational numbers: Assume we have a >list of all rational numbers. Now we create a rational number that is >not on the list by taking the first decimal of the first number on the >list and adding 1, the second decimal of the second number on the list >and adding one, then the third decimal of the third number adding one >and so on. The resulting number is not in the list. So where does the >argument break down? Very simple: The resulting number is not >rational. In article <d95e1d01-9992-4e8d-9dfb-1eb215dae964(a)z33g2000vbb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: >Thanks for (most of) the responses. I get what I missed, but probably >have to play with it a little to be fully convinced intuitively that >the diagonal of a list of all rationals must produce an irrational >real. That would be a good response. I think that you are missing the point. In the Cantor diagonal proof, the number constructed from the diagonal is easily a real number. However, when we try the same argument with the rationals, the proof breaks down because we cannot show that the number constructed from the diagonal is a rational. We don't need to prove it's irrational. It's up to the proof writer to show that the constructed number is rational; if they can't, the proof fails. It is provable that the diagonal element in the argument with the rationals is irrational. However, the only relevance of this fact to the discussion above is to assure us that no one can show that the diagonal element is rational. christian.bau gives an outline of an enumeration of the rationals: In article <f1b4a519-f904-43ca-8370-4cbb1b75e998(a)q8g2000vbm.googlegroups.com>, "christian.bau" <christian.bau(a)cbau.wanadoo.co.uk> wrote: > Start with >zero. Then all fractions where numerator + denominator = 2, followed >by the negative ones, then those where numerator + denominator = 3, >the negatives, and so on. There is nothing "artificial" about this, >just a straightforward way to achieve what you want. And every >fraction will eventually turn up. Since the rationals are countable, if all of them are in a list, the diagonal element from that list must be irrational. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: Tony Orlow on 3 Jun 2010 08:34 On Jun 3, 4:07 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > Transfer Principle wrote: > > On Jun 2, 12:27 pm, Brian Chandler <imaginator...(a)despammed.com> > > wrote: > > > > Looking at some of Herc's recent threads about Cantor's diagonal proof > > > > of the uncountability of the reals, a question occurs to me which I > > > > actually don't know how a subscriber to transfinite set theory would > > > > answer. I'm curious. > > > There are no "subscribers" to set theory (unless this has something > > > curious to do with using a "full-price" newsreader) > > > This is a thinly veiled reference to me, > > I'd have called it a "weak joke", rather than a "thinly-veiled > reference", but never mind. FWIW, you might notice I'm posting this > from Google, so you and I share the same low esteem rating, it seems. I used to use a real news reader at Cornell, but now I have Roadrunner, and they conveniently discontinued newsgroup support right before I joined. Hrummmmph!! Not my fault. > > > > only those who > > > understand it (as distinct from those who don't, of course). > > > I agree with TO here, except that instead of "subscriber," > > I prefer to use a word like "adherent" ... > > > The word "adherent" often describes those who have a > > particular religion. ... Adherents of ZFC believe that its axioms > > are true, just as adherents of a religion believe that its > > unprovable claims are true. > > Hmm. But non-adherents of religion (and possible adherents, too, with > respect to all but the one they adhere to*) can generally say just > where they disagree with the axioms of the religion. "Non-adherents" > to ZFC typically flunk out when challenged to exactly what they > disagree with. Actually, Brian, that statement is pure bunk, no offense intended. Most adherents of a given religion simply believe that theirs is the right belief because their parents or some authority figure told them that some prophet or book says so, and rarely do they really understand the tenets of their own religion, much less those in competition. Can you say, for instance, why you disagree with Islam or Tibetan Buddhism? If not, do you then accept what they believe? 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. By demanding to know with which axioms a non-transfinitologist disagrees, mathematicians obscure the question. After some years of this discussion have arisen some obvious culprits: the von Neumann ordinals as some complete set, and the simplistic assumption of equivalence of "size" based on bijection alone. If bijection determines equal cardinality that's fine, but to equate cardinality with set size simply does not work for infinite sets to the satisfaction of most people's intuitions. To claim that they are being illogical by objecting to the theory without identifying an axiom at fault is disingenuous, since the axioms are clearly not the problem. > > * I'm reminded of the review of Martin Gardner's seminal anti-woo > bookl published around 1950. The second edition is generally preferred > because it includes samples of the hate mail he received after the > first edition, from people who generally loved all chapters but > exactly one of the book. ** > > ** Sorry, I chickened out, because I just couldn't choose between "all > but exactly one chapter" and "all but exactly one chapters" ... The physics isn't wrong. The bomb is. The axioms aren't wrong (though some could be better stated). The von Neumann limit ordinals are simply bunk. Everything Jesus said was true and wise, but the Trinity and the Immaculate Conception are Christianity's von Neumann limit ordinals, thrown in later for Constantine's political goal of state religion incorporating pagan beliefs and christian. > > > I have nothing against ZFC or religion per se -- just that > > adhering to either is somewhat similar. > > Ah, now I'm stuck. What's brown and sticky? > > Brian Chandler Peace, Tony
From: Ludovicus on 3 Jun 2010 08:47 On Jun 2, 2:56 pm, Tony Orlow <t...(a)lightlink.com> wrote: >.. in any such list of rationals, it is > always true that we can concoct another rational which is not on the > list. Irrational numbers need not even be considered. So, are the > rationals not to be considered uncountable, by Cantor's own logic? > Tony That is false. There is not rational that is not in the list. Given any rational n/d, its place N in Cantor's list is: Be S =n + d If S is even, N = (S-1)(S-2) + d If S is odd , N = (S-1)(S-2) + n Ludovicus
From: kunzmilan on 3 Jun 2010 13:10 On 2 Ävn, 20:56, Tony Orlow <t...(a)lightlink.com> wrote: > Hi All - > > Looking at some of Herc's recent threads about Cantor's diagonal proof > of the uncountability of the reals, a question occurs to me which I > actually don't know how a subscriber to transfinite set theory would > answer. I'm curious. > > In Cantor's list of reals, for every digit added, the list doubles in > length. 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. Irrational numbers need not even be considered. So, are the > rationals not to be considered uncountable, by Cantor's own logic? > > BTW, I understand that the rationals are considered countable because > of a rather artificial bijection with N, 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? > > Thanks and Smiles, > > Tony The uncountability of the reals is simply based on the fact, that there are more rational numbers than there are the natural numbers. The rational numbers are defined as the natural number i divided by the natural number j. Thus, rational numbers can be greater than 1, too. When we form the matrix R of rational numbers, all rationals lesser than 1 are bellow its diagonal, all rationals greater than 1 are ower its diagonal. In the first row are all natural numbers defined as n/1. All rationals greater than 1, as 3/2, 4/3, are in the triangle over the diagonal. All rationals lesser than 1, as 2/3, 3/4, are in the triangle under the diagonal. Since in both triangles are more different elements than in the first row of the matrix R, there are more rational numbers than in the set of the natural numbers and the rational numbers are uncoutable. kunzmilan
From: Rick Decker on 3 Jun 2010 13:55
On 6/3/10 1:10 PM, kunzmilan wrote: > On 2 čvn, 20:56, Tony Orlow<t...(a)lightlink.com> wrote: >> Hi All - <snip> >> >> Tony > The uncountability of the reals is simply based on the fact, that > there are more rational numbers than there are the natural numbers. > The rational numbers are defined as the natural number i divided by > the natural number j. Thus, rational numbers can be greater than 1, > too. > When we form the matrix R of rational numbers, all rationals lesser > than 1 are bellow its diagonal, all rationals greater than 1 are ower > its diagonal. In the first row are all natural numbers defined as n/1. > All rationals greater than 1, as 3/2, 4/3, are in the triangle over > the diagonal. > All rationals lesser than 1, as 2/3, 3/4, are in the triangle under > the diagonal. > Since in both triangles are more different elements than in the first > row of the matrix R, there are more rational numbers than in the set > of the natural numbers and the rational numbers are uncoutable. No. While there are "more" rationals than natural numbers, in the sense that the naturals can be viewed as a proper subset of the rationals, that does not prove that the two sets are equinumerous. In fact, there is a rather trivial bijection between the two sets. Regards, Rick |