An uncountable countable set
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From: stephen on 27 Oct 2006 13:38 Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: >> Tony Orlow wrote: >>> stephen(a)nomail.com wrote: >>>> What are you talking about? I defined two sets. There are no >>>> balls or vases. There are simply the two sets >>>> >>>> IN = { n | -1/(2^floor(n/10)) < 0 } >>>> OUT = { n | -1/(2^n) < 0 } >>> For each n e N, IN(n)=10*OUT(n). >> >> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT >> (n)". So, you seem to be answering a question he didn't ask. Given >> Stephen's definitions of IN and OUT, is IN = OUT? >> > Yes, all elements are the same n, which are finite n. There is a simple > bijection. But, as in all infinite bijections, the formulaic > relationship between the sets is lost. What "formulaic relationship"? There are two sets. The members of each set are identified by a predicate. If an element satifies the predicate, it is in the set. If it does not, it is not in the set. I could define "different" sets with different predicates. For example, A = { n | 1+n > 0 } B = { n | 2*n >= n } C = { n | sin(n*pi)=0 } Are these sets "formulaically related"? Assuming that n is restricted to non-negative integers, does A differ from B, C, IN, or OUT? Stephen
From: MoeBlee on 27 Oct 2006 13:45 Tony Orlow wrote: > Ahem. I said that Robinson's analysis seems to have nothing to do with > transfinitology. You agreed it is not an ordering by cardinality. > 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? YOU ARE NOT LISTENING. They TWO DIFFERENT ORDERINGS. And the use of the word 'infinite' is A DIFFERENT SENSE in the two different contexts. So there is NO CONTRADICTION. The reasons you don't get any of this are (1) you don't read the material except to look for quotes and OUT OF CONTEXT passages that you think you can use to bolster your own nonsense. You don't learn the mathematical logic and set theory that are the basis for the material and even pretty much ignore the mathemtatical logic and set theory that the author himself summarizes in the book. (You need to start by learning how to work in the predicate calculus), and (2) you don't listen when someone tries to warn you about the confusions you are making due to your not understanding the basis and context. > > No, it is NOT a contradiction with set theory and there being a > > smallest infinite ordinal and smallest infinite cardinal that there are > > also non-standard orderings (which are NOT cardinality or ordinal > > ordering, as even YOU recognized) that have what are CALLED 'infinite > > elements' but with no least one. > > In transfinitology? Why would anyone argue against me saying there is no > smallest infinite? Sorry, you're not meshing. Because, AGAIN, for the TENTH TIME, 'smallest infinite' regarding ordinals and cardinals is a DIFFERENT SUBJECT from 'smallest infinite' regarding elements in certain non-standard models. > Eat me. Do you maintain that the two theories are compatible with each > other? Is there, and also not, a smallest infinity. They're not in conflict, becuase 'smallest infinite' means something DIFFERENT in the different contexts. How many times will I say that while you STILL refuse to hear it? > Geeze, calm down. I am not conflating them as if they were the same > thing. I am clearly stating that they are obviously mutually > incompatible, No, that IS conflating them by saying there is an incompatiblity. I would agree that the terminology is unfortunate, since there are two different senses of 'infinite' in play. But were we to fully formalize, we would have predicate symbols, not English language words, so we wouldn't have this problem. But anyone who knows anything about the subject can understand, even without such formalization, that 'infinite' is a WORD that is being used in DIFFERENT senses and there is no incompatiblitiy in the mathematics if one just recognizes the DIFFERENT senses that are being used. > You need to relax. I worked on the first chapter for a bit, and got most > of the way through, but started to get bogged down, so I skipped ahead > to see what the next chapter held, and it did. So, sue me. At least I > didn't run to the Cliff Notes... You got "bogged down" because that chapter is a graduate level SUMMARY of material that most readers of that book will have already digested before even picking up that book. You need to learn the basics of mathematical logic and set theory to even be able to very much benefit from such a summary of material as in that book that is even more advanced in mathematical logic than the very basic mathematical logic you still have not learned. > Well, at least I'm not being such an obnoxious jerk. Or, maybe you think > I am. Yes, you are being a real obnoxious jerk. > Read what Robinson says, and think about it. If you're really so > interested in "alternative" theories, I'd think you'd go to the source. You're being an obnoxious jerk AGAIN. Of course I want to read that book, along with Martin Davis's book and others on non-standard analysis, and Edward Nelson and internal set theory, and all kinds of things. But unlike you, I lack the arrogance and foolishness to think that I can skip the basics. Therefore, given that I am not a Zeno machine of mathematical learning, I have to put certain reading on hold while I get caught up on certain basics and even on certain advanced material that is related. That is not a reflection of insincerity in my endeavor to know about alternatives, but rather of my sincerity in that endeavor. MoeBlee
From: Lester Zick on 27 Oct 2006 13:53 On 26 Oct 2006 21:27:18 -0700, imaginatorium(a)despammed.com wrote: >MoeBlee wrote: >> Lester Zick wrote >> >> even more sheer brilliance! > ><...> > >> I won't quote more that opening and close, as one can become >> overwhelmed by so much wisdom from just one man in just one day. > >Ah, I see you've noticed. Meanwhile, "Have you tried searching the >archive for Zick + transcendental?" > >http://groups.google.com/group/sci.math/search?group=sci.math&q=zick+transcendental&qt_g=1&searchnow=Search+this+group Ah, Brian, ever the amanuensis. ~v~~
From: Lester Zick on 27 Oct 2006 13:56 On 27 Oct 2006 01:11:02 -0700, imaginatorium(a)despammed.com wrote: > >David Marcus wrote: >> imaginatorium(a)despammed.com wrote: >> > >> > 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? >> >> Virgil said "Ross", not "Robinson", I believe. > >Yes, of course. But Virgil's implication is that "TO says person P is >right about something" implies P is wrong. This may, contingently, be >true about Ross, but the argument could equally be applied to Robinson, >in which case the conclusion is obviously not true. Well technically, Brian, you're being reasonable for a change. ~v~~
From: MoeBlee on 27 Oct 2006 14:01
Lester Zick wrote: > Ah, Brian, ever the amanuensis. Zick, ever the nuisance. MoeBlee |