An uncountable countable set
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From: Ross A. Finlayson on 27 Oct 2006 00:45 Virgil wrote: > > Coming from TO it damns Ross. Virgil, shut up. ZFC and IST, Nelson's Internal Set theory, and Robinson's hyperreals are quite compatible with each other, for example how ZFC is coconsistent with IST. ZF is inconsistent. There is no set of sets in ZF. Robinson's hyperreals are basically what are referred to here as Newton's notion of the reals, with fluents and fluxions, extended with infinite values. The hyperintegers are similarly a notion of the finite natural numbers having appended infinite, natural numbers. Fluxions after the first unit's aren't generally used in analysis, but they can be in chaining derivatives, composition, as Newton's are nilpotent. The sum of any number of zeros is zero. There are quite the few other even more alternative and nonstandard formulations of the real numbers than the hyperreals, so constructed in and known from the literature, re Schmieden and Laugwitz, Bishop and Cheng, myself, etcetera. There are as well useful compactifications or projective extensions of the real numbers with the as you say "point at infinity." Ross
From: Virgil on 27 Oct 2006 00:48 In article <45418573(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4540cf58(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> imaginatorium(a)despammed.com wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> As each ball n is removed, how many remain? > >>>>>>> 9n. > >>>>>>> > >>>>>>>> Can any be removed and leave an empty vase? > >>>>>>> Not sure what you are asking. > >>>>>> If, for all n e N, n>0, the number of balls remaining after n's removal > >>>>>> is 9n, does there exist any n e N which, after its removal, leaves 0? > >>>>> I don't know what you mean by "after its removal"? > >>>> Oh, I think this is clear, actually. Tony means: is there a ball (call > >>>> it ball P) such that after the removal of ball P, zero balls remain. > >>>> > >>>> The answer is "No", obviously. If there were, it would be a > >>>> contradiction (following the stated rules of the experiment for the > >>>> moment) with the fact that ball P must have a pofnat p written on it, > >>>> and the pofnat 10p (or similar) must be inserted at the moment ball P > >>>> is removed. > >>> I agree. If Tony means is there a ball P, removed at time t_P, such that > >>> the number of balls at time t_P is zero, then the answer is no. After > >>> all, I just agreed that the number of balls at the time when ball n is > >>> removed is 9n, and this is not zero for any n. > >>> > >>>> Now to you and me, this is all obvious, and no "problem" whatsoever, > >>>> because if ball P existed it would have to be the "last natural > >>>> number", and there is no last natural number. > >>>> > >>>> Tony has a strange problem with this, causing him to write mangled > >>>> versions of Om mani padme hum, and protest that this is a "Greatest > >>>> natural objection". For some reason he seems to accept that there is no > >>>> greatest natural number, yet feels that appealing to this fact in an > >>>> argument is somehow unfair. > >>> The vase problem violates Tony's mental picture of a vase filling with > >>> water. If we are steadily adding more water than is draining out, how > >>> can all the water go poof at noon? Mental pictures are very useful, but > >>> sometimes you have to modify your mental picture to match the > >>> mathematics. Of course, when doing physics, we modify our mathematics to > >>> match the experiment, but the vase problem originates in mathematics > >>> land, so you should modify your mental picture to match the mathematics. > >>> > >> I disagree. When you formulate a theory, whether scientific or > >> mathematical, the goal should be to draw conclusions in line with > >> observations. In science, it's no problem to disprove a theory, if there > >> is a verifiable situation which it predicts incorrectly. When it comes > >> to math, there is no such test, but the whole of mathematics should be > >> consistent, and where one theory contradicts another, that's an > >> indication that one or the other is less than correct. > > > > That depends. > > > > If the apparently contradictory results follow from different axiom > > systems, they may both be quite valid. > > > > They cannot be mutually consistent, and a larger mathematical system > including both cannot be internally consistent. Why not? Since each is based on assumptions which are not required to be true, different assumptions may lead to different conclusions without any problems. > The universe is > consistent, and math creates and describes it. I'll take a side of > logic, and hold the contradictions. But despite his high moral tone above, TO always takes the side opposed to logic and includes his own contradictions.
From: David Marcus on 27 Oct 2006 00:50 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Uh, if Robinson's thesis is built upon transfinite set theory, then that > >> is evidence right there that it's inconsistent, since you have a > >> smallest infinity, omega, but Robinson has no smallest infinity. > > > > We JUST agreed that 'smallest infinity' means two different things when > > referring to ordinals and when referring to certain kinds of other > > orderings! It is AMAZING to me that even though I took special care to > > make sure this was clear, and then you agreeed, you NOW come back to > > conflate the two ANYWAY! > > Ahem. I said that Robinson's analysis seems to have nothing to do with > transfinitology. They appear to be unrelated. However, they cme to two > very different conclusions regarding a basic question: is there a > smallest infinite number? It seems clear to me there is not, for the > very same reason that Robinson uses: if there is an infinite number, you > can subtract 1 and get a different, smaller infinite number. It's the > same logic y'all use to argue that there's no largest finite. It's > correct. The Twilight Zone between finite and infinite CANNOT really be > pinpointed that way. > > So, that's a verrry basic discrepancy. There is clearly a contradiction > between the two theories. They can't both be right about that, can they? > Is there, and at the same time is not, a smallest infinite number? > > Tach me more about mathematical logic and consistency. We've been trying, but you don't seem to want to learn. Your question "Is there a smallest infinite number?" lacks context. You need to state what "numbers" you are considering. Lots of things can be constructed/defined that people refer to as "numbers". However, these "numbers" differ in many details. If you assume that all subjects that use the word "number" are talking about the same thing, then it is hardly surprising that you would become confused. The two theories can both be right about the "numbers" that they are talking about, since they are talking about different things. To avoid confusion, the simplest solution is to be specific. We could say that there is a smallest infinite ordinal, but there is no smallest infinite non-standard real number. When phrased this way, no contradiction is apparent. The apparent contradiction is due to your using "infinite number" to mean both "infinite ordinal" and "infinite non-standard real number". A similar problem would arise if you used the word "cat" to refer to both domestic cats and lions and were to say that "Cats make good pets". -- David Marcus
From: Ross A. Finlayson on 27 Oct 2006 00:50 David Marcus wrote: > > When mathematicians talk, they know which words have technical meanings > and which don't. Ross simply uses the words without knowing the > technical meanings. So, it gives the appearance of mathematics, but > there is no actual communication of mathematical ideas. > > I once tried to teach some mathematics to some friends who weren't > mathematicians via a weekly lunch seminar. We took a (fairly advanced) > math book, and started reading it. When I read a math book, I can > immediately categorize each sentence as definition, theorem, proof, or > remark, even if the sentence isn't labeled as such. I was a bit > surprised to discover that my friends weren't picking up on this at all. > As such, they couldn't even begin to follow the book, since they > couldn't tell what the purpose of each sentence was in the logical flow. > They weren't even aware of the convention that a word in italics means > the sentence is a definition of the word. > > -- > David Marcus I dispute that. Let's see, the nearest mathematics book is a Dover reprint of Meyer's _Introduction to Mathematical Fluid Dynamics_. So, explain fluid mechanics. Perhaps you feel more secure in your grasp of the subject talking about set theory? So do I. Ross
From: imaginatorium on 27 Oct 2006 01:04
Virgil wrote: > In article <45417528$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: <snip> > > For what it's worth, and I know this doesn't add a lot of credibility to > > Ross in your eyes, coming from me, but I think Ross has a genuine > > intuition that isn't far off with respect to what's controversial in > > modern math. Sure, he gets repetitive and I don't agree with everything > > he says, but his cryptic "Well order the reals", which I actually > > haven't seen too much of lately, is a direct reference to his EF > > (Equivalence Function, yes?) between the naturals and the reals in > > [0,1). The reals viewed as discrete infinitesimals map to the > > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > > answer your question, I think Ross makes some sense. But, of course, > > coming from me, that probably doesn't mean much. :) > > Coming from TO it damns Ross. Even by your standards, Virgil, this is egregiously silly. TO skips the basic exposition in Robinson's book, but finds a sentence he likes. So this "damns" Robinson's non-standard analysis, does it? Brian Chandler http://imaginatorium.org |