An uncountable countable set
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From: David Marcus on 28 Oct 2006 12:54 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? 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? Feel free to give your reasoning, but please also answer each question with either "Agree" or "Disagree". -- David Marcus
From: stephen on 28 Oct 2006 12:50 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: >>>>> David Marcus wrote: >>>>>> Tony Orlow 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. >>>>>> Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? >>>>>> (The vertical lines denote "cardinality".) >>>>> Um, before I answer that question, I think you need to define what you >>>>> mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do >>>>> you define '-' on these "numbers"? >>>> IN and OUT are sets, not "numbers". For any two sets A and B, the >>>> difference, denoted by A - B, is defined to be the set of elements in A >>>> that are not in B. Formally, >>>> >>>> A - B := {x| x in A and x not in B} >>>> >>>> Note that the difference of two sets is again a set. For any set, the >>>> notation |A| means the cardinality of A. So, saying that |A| = 0 is >>>> equivalent to saying that A is the empty set. In particular, for any set >>>> A, we have |A - A| = 0. >>>> >> >>> Sure, in the sense of containing the same n's, they are the same set. >>> That entirely ignores the rates at which those sets are processed over >>> time, which is expressed in your floor(n/10), causing ten times as many >>> in IN as in OUT, for any given value range of the two functions defining >>> them. If you have n balls in, that took n/10 steps. If you have n balls >>> out, that took n steps. The vase accumulates more balls at every step. >>> So, the axiom of extensionality doesn't address this matter of measure >>> in the sequence, but tries to cover it up in typical set theoretic fashion. >> >>> Tony >> >> What balls and vase? Are you really incapable of answering a simple >> question about sets without bringing in total irrelevancies? >> >> Stephen > If the problem is irrelevant to your question, then your question is > irrelevant to the problem, eh? I explained the purpose of my question. If you are not going to bother to read what I write and to respond to what I write, you should not bother responding at all. Stephen
From: stephen on 28 Oct 2006 12:52 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Let me recap the discussion: Stephen suggested the following problem > (which may or may not have some relationship to any other problem that > anyone has ever considered): > Define the following sets of natural numbers. > IN = { n | -1/(2^floor(n/10)) < 0 }, > OUT = { n | -1/(2^n) < 0 }. > What is |IN\OUT|? > Stephen suggested that this problem would "not cause any fuss at all", > i.e., everyone would agree what the answer is. In reply, you wrote, "It > would still be inductively provable in my system that IN=OUT*10." We all > took this to mean that you disagreed that |IN\OUT| = 0. Now, you seem to > be saying that you agree that |IN\OUT| = 0. > Care to clear up this confusion? > -- > David Marcus Thanks for the nice summary. Hopefully it helps. Stephen
From: David Marcus on 28 Oct 2006 12:55 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> 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? > >> So, either smallest has two meanings, or infinite has tow meanings, or > >> both. Would you like to elucidate the matter by enumerating the various > >> definitions of "small" and "infinite"? A table might be nice... > > > > As many have said, "infinite" has many meanings. I'm afraid it isn't > > practical to produce a table. > > How about a list? ;) I'm sorry, but I don't have the inclination to do this. The word "infinite" is used in too many fields with too many different meanings. -- David Marcus
From: stephen on 28 Oct 2006 13:01
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 } 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 |