An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 13:52 David Marcus wrote: > Tony Orlow wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>>> Tony Orlow wrote: >>>>>>> David Marcus wrote: >>>>>>>> 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. >>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm >>>>>>> not interested in that nonsense, to be honest. I see it as a dead end. >>>>>>> >>>>>>> If there is a definition for "number" in general, and for "infinite", >>>>>>> then there cannot both be a smallest infinite number and not be. >>>>>> A moot point, since there is no definition for "'number' in general", as >>>>>> I just said. >>>>>> -- >>>>>> David Marcus >>>>> A very simple example is that there exists a smallest positive >>>>> non-zero integer, but there does not exist a smallest positive >>>>> non-zero real. If someone were to ask "does there exist a smallest >>>>> positive non-zero number?", the answer depends on what sort >>>>> of "numbers" you are talking about. >>>>> >>>>> Stephen >>>> Like, perhaps, the Finlayson Numbers? :) >>> If they were sensibly defined then sure you could talk about them. >>> Nothing Ross has ever said has made any sense to me, and >>> I severely doubt there is any sense to it, but I could be wrong. >>> The point is, there are different types of numbers, and statements >>> that are true of one type of number need not be true of other >>> types of numbers. >>> >>> Stephen >> Well, then, you must be of the opinion that set theory is NOT the >> foundation for all mathematics, but only some particular system of >> numbers and ideas: a theory. That's good. > > That's rather an amazing leap from what Stephen actually said. > > Do you agree that there is no logical contradiction between the fact > that there is a least positive integer and the fact that there is no > least positive real number? Yes, of course, the reals being continuous. What makes you think I would disagree with that? > > Do you agree that there is no logical contradiction between the fact > that there is a least infinite ordinal and the fact that there is no > least non-standard real number? I think that the concept of a least infinite number in any real sense violates the fact that one can always remove 1 from it and it will still be infinite. Robinson directly uses this idea. Limit ordinals directly violate it. Is it true or not? Does removal of a nonzero quantity always result in a smaller value, or not? That's the issue. So, yes, there is no deductive contradiction between the two, because they have a difference of axiomatic assumptions to begin with. That difference causes a contradiction BETWEEN the two. > > Feel free to give your reasoning, but please also answer each question > with either "Agree" or "Disagree". > agreed
From: imaginatorium on 28 Oct 2006 14:00 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: <snip> > > Here's another question - I' ve asked before, so I don't hold out much > > hope of an answer, but anyway: Suppose you could ask Abraham Robinson > > what he thought of your ideas: what do you suppose he would say? Do you > > think he might just latch onto the IFR, N^L=S (whaddeveritwas), your > > T'rrible numbers, the twilight zone, etc., or do you suppose he would > > dismiss it as total nonsense? > > I suppose that depends on whether he liked me. I don't think hed find my > ideas objectionable. He was obviously an original thinker. What do you mean by "objectionable"? How can whether you like someone affect whether you think they are making mathematical sense or not? As you say, Abraham Robinson was an original thinker, but I don't think I would be insulting him to say no more original in particular than thousands of other mathematicians all brighter than me. Do you think you are an "original thinker"? I've more or less said this before, but I think one reason I'm just light-years (in the popular sense) ahead of you on all this is that I have sat in undergraduate algebra lectures when Simon Norton asked questions. I have some concept of how much dimmer I am than a bright mathematician. You have only the proud memory of being top all the time in the village school (as I was, of course). > Do you actually think you might ever find > > a real mathematician who thought there was anything at all of merit in > > what you have to say? > > Why don't you tell me? Did Boole? Did Cantor? Did Boole or Cantor what? > Here's a suggestion: Robinson died in 1974, at a > > rather early age, but Conway is very much still alive - he's a very > > helpful person (I've seen him laboriously explaining something rather > > elementary on a geometry list I think it was), and he also created > > another non-standard collection of numbers, with what's more, > > constructions like omega/2 in it. > > Yes, the surreals. > > So send him an email, of not more > > than say 200 words, setting out your most basic ideas; ask him if he > > thinks you're wasting your time? Don't forget to mention that you are > > quite sure the set of natural numbers is not infinite. > > That's worth a try. Could be. > > Meanwhile, I've got just a bit tired of asking the same unanswered > > questions over and over again. I mean, what is a _function_ in poetry? > > > > A function in poetry? Yes, you're writing poetry, we gather. What, poetically speaking, do you see a function as being? Brian Chandler http://imaginatorium.org
From: imaginatorium on 28 Oct 2006 14:03 Lester Zick wrote: > On 27 Oct 2006 11:38:10 -0700, imaginatorium(a)despammed.com wrote: > > > > >David Marcus wrote: > >> Lester Zick wrote: > >> > On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >> > >A very simple example is that there exists a smallest positive > >> > >non-zero integer, but there does not exist a smallest positive > >> > >non-zero real. > >> > > >> > So non zero integers are not real? > >> > >> That's a pretty impressive leap of illogic. > > > >Gosh, you obviously haven't seen Lester when he's in full swing. (Have > >_you_ searched sci.math for "Zick transcendental"?) > > Hell, Brian, on some of my better days I can even prove the pope's > catholic. Lester, I'm glad to see age is not wearying you, but isn't this the second time you've replied to the same message? Brian Chandler http://imaginatorium.org
From: Randy Poe on 28 Oct 2006 14:39 Tony Orlow wrote: > Virgil wrote: > > In article <4542201a(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> cbrown(a)cbrownsystems.com wrote: > > > >>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > >>> time t, we mean a real number t" > >> That doesn't mean t has to be able to assume ALL real numbers. The times > >> in [-1,0) are all real numbers. > > > > By what mechanism does TO propose to stop time? > > By the mechanism of unfinishablility. But that's why I asked you a question about variables labelling times yesterday, when noon clearly occurred. I can define a list of times t_n = noon yesterday - 1/n seconds, for all n=1, 2, 3, ... Clearly this list of times has no end. But didn't noon happen? How does my list affect the existence of noon yesterday? It's unfinishable. Why don't your time-stopping rules work retroactively? What's different about this set of times t_n and the set of times in the balls-and-vase problem? Why does assigning these labels to times yesterday not affect anything, but if I assign those labels today, it stops time? - Randy
From: Tony Orlow on 28 Oct 2006 14:40
stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> 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. >>>> OOoooOOoooohhhh a predicate! >>> This is a non answer. >>> > >> That's because it followed a non question. :) > > How is "formulaic relationship?" a non question? I do not know > what you mean by that phrase, so I asked a question about. > Presumably you do know what it means, but your refusal to > answer suggests otherwise. > > >>>> If an element satifies >>>>> the predicate, it is in the set. If it does not, it is not in >>>>> the set. >>>>> >>>> Ever heard of algebra or formulas? Ever seen a mapping between two sets >>>> of numbers? >>> This is a lame insult and irrelevant comment. It says nothing >>> about what a "forumulaic relationship" between sets is. >>> > >> What is there to say? You know what a formula is. > > Yes, but I do not know what a "formulaic relationship" is. > >>>>> 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 >>>> Do 1+n, 2*n and sin(n*pi) look like formulas to you? They do to me. >>>> Maybe they're just the names of your cats? >>> Sure they are formulas. But I am interested in your phrase >>> "formulaic relationship", the explanation of which you seem to be avoiding. >>> > >> It's the mapping between set using a quantitative formula. Observe... > >>>> A can be expressed 1+n>=1, or n>=0, and is the set mapped from the >>>> naturals neN (starting from 1) by the formula f(n)=n-1. The inverse of >>>> n-1 is n+1, indicating that over all values, this set has one more >>>> element than N, namely, 0. >>> I said that n was restricted to non-negative integers, so this >>> set equals N. >>> > >> Ooops, missed that. Sorry. n is restricted to nonnegative integers, but >> f(n) isn't. What you mean is that, in this case, f(n) is restricted to >> nonnegative integers, which means n>=2, and f(n)>=1. So, yes, the set is >> size N, from 1 through N. > >>>> B can be simplified by subtracting n from both sides, without any worry >>>> of changing the inequality, so we get n>=0, neN. That's the same set, >>>> again, mapped from the naturals by f(n)=n-1. >>> Also N. > >> Yes, by the same reasoning. > >>>> C is simply the set of all integers, which we can consider twice the >>>> size of N. There's really nothing to formulate about that. >>> Once again N. >>> > >> Sure. > >>> So all three sets are N. So in fact, there is only one set. >>> A, B, and C are all the same set. A, B, C, IN and OUT are all >>> the same set, namely N. You still have not answered what >>> a "formulaic relationship" is. >>> >>> Stephen > >> Take the set of evens. It's mapped from the naturals by f(x)=2x. Right. >> Many feel that there are half as many evens as naturals, and this is >> reflected in the inverse of the mapping formula, g(x)=x/2. Over the >> range of N, we have N/2 as many evens as naturals. Over the range of N, >> we have sqrt(N) as many squares as naturals, and log2(N) as many powers >> of 2 in N. That's IFR, using formulaic relationships between infinite >> sets. Byt he way, it works for finite sets, too. :) > > What does that have to do with the sets IN and OUT? IN and OUT are > the same set. You claimed I was losing the "formulaic relationship" > between the sets. So I still do not know what you meant by that > statement. Once again > IN = { n | -1/(2^(floor(n/10))) < 0 } > OUT = { n | -1/(2^n) < 0 } > I mean the formula relating the number In to the number OUT for any n. That is given by out(in) = in/10. > Given that for every positive integer -1/(2^(floor(n/10))) < 0 > and -1/(2^n) < 0, both sets are in fact the same set, namely N. > > Do you agree, or not? Or is it the case that the > "formulaic between the sets is lost." > ? > > Stephen The formulaic relationship is lost in that statement. When you state the relationship given any n, then the answer is obvious. |