An uncountable countable set
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From: Virgil on 20 Sep 2006 17:40 In article <4511618e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Han de Bruijn wrote: > >> Mike Kelly wrote: > >> > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>> Mike Kelly wrote: > >>>> > >>>>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>>>> to schoolkids. > >>>> Look who is talking. Good luck explaining alpha_0 to schoolkids. > >>> Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > >>> this is a bad analogy, because school kids don't need to know about > >>> cardinals but they do need to know how to work with natural numbers. My > >>> point, if you really missed it, was that Tony's ideas of "infinite > >>> natural numbers" don't match up to our "naive" or "intuitive" idea of > >>> what numbers should be - how we were taught to do arithmetic in school. > >>> I for one don't understand what the hell an "infinite natural number" > >>> is. And yet supposedly the advantage of his ideas are that they're more > >>> intuitive than a standard formal treatment. > >> My point is that the pot is telling the kettle that it's black (: de pot > >> verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than > >> Tony's "infinite natural number". > > > > Your analogy is terrible, as usual. > > > > My point was that Tony's "infinite natural numbers" are not compliant > > with everyday arithmetic. Aleph_0 is part of a formalisation that leads > > to an arithmetic that works exactly as we expect it to. > > > > Oh? For what finite x is x-1=x? Does TO expect infinite arithmetic to look exactly like finite arithmetic? The more fool he!
From: Virgil on 20 Sep 2006 17:51 In article <4511667a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> If omega is the successor to the set of all finite naturals, > > > > It isn't. What does "successor to N" even mean? > > Ask von Neumann. JvN would say that the successor to any x is the union of x and {x}, so that the successor to N is the union of N and {N}. > It is the set of all naturals WRONG! It contains the set of all naturals both as a member and as a proper subset. > and any ordinal being > the set of all preceding naturals, omega is the set of all preceding > naturals. It's larger than all naturals. DO you disagree with that > simple statement? " It's larger than all naturals" is a bit ambiguous. It's larger than any natural, but not larger than the set of all naturals. > > You are working with an undefined term, with undefined > > operations, and then trying to draw conclusions > > as if you'd defined those things. > > If an infinite number is not greater than a finite number, then it's not > a number at all. That would shows that it is not an infinite number. > > > > >> as any successor is greater then all > >> those that precede it. > > > > It isn't a successor of any particular natural number. > > So given that it is NOT a successor, how do you > > think a property of "any successor" is relevant? > > Every ordinal is the set of all which came before it. Omega is AFTER the > naturals in the quantitative order. But still not the successor of any one of them. > No, but if I say I have a science of all life, it should apply equally > to broccoli and mammals, and if I have a science of animals, it applies > to mammals, but NOT broccoli. So, if I have a rule for numbers, which > aleph_0 doesn't obey, I don't consider it a number. As TO has never let us in on that "rule for numbers" why should we agree to it? And when has what TO considered been of any weight in actual mathematics?
From: Virgil on 20 Sep 2006 17:56 In article <4511685c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Han de Bruijn wrote: > >>> Tony Orlow wrote: > >>> > >>>> Han de Bruijn wrote: > >>>>> Precisely! Mathematicians get confused by the idea of a "bijection", > >>>>> which is an Equivalence Relation, which in turn is a "generalization" > >>>>> of "common equality" (yes: the one in a = b). But the funny thing is > >>>>> that EQUALITY HAS NEVER BEEN DEFINED. > > > > Idiot. Do you know the definition of an equivalence relation? Do you > > claim that bijection is not an equivalence relation? > > > > (I think that was a different crank - here's Tony...) > > > >> Consider the equally spaced staircase from (0,0) to (1,1), as the number > >> of steps increases from 1 without bound. Is it the same as the diagonal > >> line? > > > > What _exactly_ is "it" here? > > Idiot. What was the object referred to in the previous sentence? Do you > not know how to correlate pronouns to their reference? As there is no single thing referred to, but only an unbounded sequence of things with no distinct method of finding a limit to that sequence, TO has no "it". > > > If I consider the set of staircases as the > > number of steps increases without bound I get an unending set of > > staircases. The only obvious singular object is the set, which is not > > anything like a diagonal line (of course I don't think you mean this); > > otherwise there are lots of staircases. Well, now you foam at the mouth > > a bit... > > I mentioned ONE staircase, in the limit as the number of steps > approaches oo. Don't play dumb. By what standard limiting process? > I was reading the Wikipedia article on "Crank > > (person)" today. Particularly the bit about cranks' incredible > > over-rating of their own abilities. You seriously think you are so much > > cleverer than the staff of every maths department in the world that you > > alone can notice that every one of these staircases has length 2; you > > think mathematicians in general are _that_ stupid? > > I think I have a nonstandard perspective which is at least equally valid > as the standard transfinitology. Unwarranted arrogance!
From: Virgil on 20 Sep 2006 18:00 In article <45117ec0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Omega is successor to N, and the size of N, in the von Neumann system. WRONG! And stupidly wrong. Omega = N, when N regarded as an ordinal, at least in the von Neumann system.
From: Virgil on 20 Sep 2006 18:04
In article <45118ac3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > IFR does that and more. IFR can only measure ordered sets, and will give different values to the same set with a different ordering. |