An uncountable countable set
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From: stephen on 28 Oct 2006 11:19 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. > 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. >> 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. > 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. > 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. > 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. 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
From: stephen on 28 Oct 2006 11:20 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
From: Tony Orlow on 28 Oct 2006 12:01 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. :) >> 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. >>> 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. :)
From: Tony Orlow on 28 Oct 2006 12:02 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?
From: imaginatorium on 28 Oct 2006 12:09
Tony Orlow wrote: > Virgil wrote: > > In article <454286e8(a)news2.lightlink.com>, > > 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? :) > > > > Any set of numbers whose properties are known. Are the properties of > > "Finlayson Numbers" known to anyone except Ross himself? > > Uh, yeah, I think I understand what his numbers are. Perhaps you've seen > our recent exchange on the matter? They are discrete infinitesimals such > that the sequence of them within the unit interval maps to the naturals > or integers on the real line. Is that about right, Ross? Do they form a field? Brian Chandler http://imaginatorium.org |