Uncountability of the Rationals?
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From: Tony Orlow on 3 Jun 2010 14:36 On Jun 3, 1:10Â pm, kunzmilan <kunzmi...(a)atlas.cz> wrote: > 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 rationals have the same cardinality as the naturals, though there are surely more rationals per unit interval of the real line than there are naturals. > 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. One might think there were something like aleph_0^2 rationals, but that's not standard theory. > 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. That's not a bad alternative argument. Of course, they still have the same cardinality, but the axioms never say anything about cardinality to begin with. Nice comment. You get five stars! > kunzmilan- Hide quoted text - > > - Show quoted text - Regards, Tony
From: Virgil on 3 Jun 2010 15:55 In article <1bb4e64e-9dd5-45a2-8c42-9ffa430f836a(a)c7g2000vbc.googlegroups.com>, Tony Orlow <tony(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. How does TO define 'sequence' without reference to something like the SET of naturals as indices?
From: K_h on 3 Jun 2010 17:53 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:1bb4e64e-9dd5-45a2-8c42-9ffa430f836a(a)c7g2000vbc.googlegroups.com... 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: > 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 Tony, first, the von-Neuman ordinals do not form a set, rather they form a proper class. Second, cardinal size is only one measure of size. There are other measures, for example ordinal size. The ordinal w2 has an ordinal size greater than w but they both have the same cardinal size. The same is true of w+1 and w: w<w+1 ordinally but |w|=|w+1| cardinally. To be clear on your issue about Cantor's diagonal argument, there is nothing wrong with Cantor's argument. If there is a list containing all rationals then its `anti-diagonal' numbers will not be rational. The proof is easy: use one of the common `zig-zag' bijections between the naturals and the rationals so that every ratio p/q appears in the list once and only once with p and q containing no common factors except 1. (Note negative rationals are in the list and this is not a problem: there will be two anti-diagonals because of opposite signs). Create an `anti-diagonal' number for the list and then note that this number is nowhere in the list and so it must be different from each p/q. Therefore, it cannot be written as the ratio of two integers p/q. _
From: porky_pig_jr on 3 Jun 2010 18:51 On Jun 3, 1:10Â pm, kunzmilan <kunzmi...(a)atlas.cz> wrote: > 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 So we had Herc claiming that reals are countable. now we have kunzmilan claiming the rationals are uncountable. The plot thickens.
From: Jesse F. Hughes on 3 Jun 2010 23:40
kunzmilan <kunzmilan(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) |