An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 28 Oct 2006 08:59 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! 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? > 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? 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. 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. 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.
From: Tony Orlow on 28 Oct 2006 09:03 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?
From: Tony Orlow on 28 Oct 2006 09:07 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Mike Kelly wrote: >>>>>>>>> Now correct me if I'm wrong, but I think you agreed that every >>>>>>>>> "specific" ball has been removed before noon. And indeed the problem >>>>>>>>> statement doesn't mention any "non-specific" balls, so it seems that >>>>>>>>> the vase must be empty. However, you believe that in order to "reach >>>>>>>>> noon" one must have iterations where "non specific" balls without >>>>>>>>> natural numbers are inserted into the vase and thus, if the problem >>>>>>>>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is >>>>>>>>> this a fair summary of your position? >>>>>>>>> >>>>>>>>> If so, I'd like to make clear that I have no idea in the world why you >>>>>>>>> hold such a notion. It seems utterly illogical to me and it baffles me >>>>>>>>> why you hold to it so doggedly. So, I'd like to try and understand why >>>>>>>>> you think that it is the case. If you can explain it cogently, maybe >>>>>>>>> I'll be convinced that you make sense. And maybe if you can't explain, >>>>>>>>> you'll admit that you might be wrong? >>>>>>>>> >>>>>>>>> Let's start simply so there is less room for mutual incomprehension. >>>>>>>>> Let's imagine a new experiment. In this experiment, we have the same >>>>>>>>> infinite vase and the same infinite set of balls with natural numbers >>>>>>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note >>>>>>>>> that time is a real-valued variable that can have any real value. At >>>>>>>>> time -1/n we insert ball n into the vase. >>>>>>>>> >>>>>>>>> My question : what do you think is in the vase at noon? >>>>>>>> A countable infinity of balls. >>>>>>> So, "noon exists" in this case, even though nothing happens at noon. >>>>>> Not really, but there is a big difference between this and the original >>>>>> experiment. If noon did exist here as the time of any event (insertion), >>>>>> then you would have an UNcountably infinite set of balls. Presumably, >>>>>> given only naturals, such that nothing is inserted at noon, by noon all >>>>>> naturals have been inserted, for the countable infinity. Then insertions >>>>>> stop, and the vase has what it has. The issue with the original problem >>>>>> is that, if it empties, it has to have done it before noon, because >>>>>> nothing happens at noon. You conclude there is a change of state when >>>>>> nothing happens. I conclude there is not. >>>>> So, noon doesn't exist in this case either? >>>> Nothing happens at noon, and as long as there is no claim that anything >>>> happens at noon, then there is no problem. Before noon there was an >>>> unboundedly large but finite number of balls. At noon, it is the same. >>> So, noon does exist in this case? >> Since the existence of noon does not require any further events, it's a >> moot point. As I think about it, no, noon does not exist in this problem >> either, as the time of any event, since nothing is removed at noon. It >> is also not required for any conclusion, except perhaps that there are >> uncountably many balls, rather than only countably many. But, there are >> only countably many balls, so, no, noon is not part of the problem here. >> As we approach noon, the limit is 0. We don't reach noon. > > To recap, we add ball n at time -1/n. We don't remove any balls. With > this setup, you conclude that noon does not exist. Is this correct? > I conclude that nothing occurs at noon in the vase, and there are countably, that is, potentially but not actually, infinitely many balls in the vase. No n in N completes N.
From: Tony Orlow on 28 Oct 2006 09:18 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
From: Tony Orlow on 28 Oct 2006 09:20
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. |