An uncountable countable set
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From: Dik T. Winter on 31 Oct 2006 19:14 In article <b29fk2drh5cg1120isnglpruj5mo6j6739(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: .... > I don't ignore it. I considered it as understood but even if you > include it above such that you have "there is a positive least > integer" and "there is no positive least real" you're still stuck with > the implication that there is a distinction between integers and > reals. Yes, there is. Not all reals are created equal, some are superior and also integer. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 31 Oct 2006 19:52 Randy Poe wrote: > Tony Orlow wrote: > > Randy Poe wrote: > > > Tony Orlow wrote: > > >> Randy Poe wrote: > > >>> Do you deny me the ability to create a set of variables > > >>> t_n, n = 1, 2, ...? Why do vases have to come into it? > > >> I thought we were trying to formulate the problem. > > > > > > No, we (some of us) are trying to formulate a completely > > > different problem, with balls and vases (possibly even > > > times) explicitly removed so that other aspects can be examined. > > > > > > Yet you keep trying to put balls and vases back in, after being told > > > that they are not present in the new problem. > > > > I am pointing out that your formulation doesn't match the original > > problem. > > It's a new problem. > > Are you capable of contemplating a second problem, > throwing away balls and vases and starting from scratch, > asking different questions about a different problem? That's a good question. I was thinking of asking Tony the same thing. If Tony can only discuss one problem, it does limit the discussion. Best we know that up front. -- David Marcus
From: David Marcus on 31 Oct 2006 20:04 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> I am beginning to realize just how much trouble the axiom of > >>>>>> extensionality is causing here. That is what you're using, here, no? The > >>>>>> sets are "equal" because they contain the same elements. That gives no > >>>>>> measure of how the sets compare at any given point in their production. > >>>>>> Sets as sets are considered static and complete. However, when talking > >>>>>> about processes of adding and removing elements, the sets are not > >>>>>> static, but changing with each event. When speaking about what is in the > >>>>>> set at time t, use a function for that sum on t, assume t is continuous, > >>>>>> and check the limit as t->0. Then you won't run into silly paradoxes and > >>>>>> unicorns. > >>>>> There is a lot of stuff in there. Let's go one step at a time. I believe > >>>>> that one thing you are saying is this: > >>>>> > >>>>> |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not the > >>>>> correct translation of the balls and vase problem into Mathematics. > >>>>> > >>>>> Do you agree with this statement? > >>>> Yes. > >>> OK. Since you don't like the |IN\OUT| translation, let's see if we can > >>> take what you wrote, translate it into Mathematics, and get a > >>> translation that you like. > >>> > >>> You say, "When speaking about what is in the set at time t, use a > >>> function for that sum on t, assume t is continuous, and check the limit > >>> as t->0." > >>> > >>> Taking this one step at a time, first we have "use a function for that > >>> sum on t". How about we use the function V defined as follows? > >>> > >>> For n = 1,2,..., let > >>> > >>> A_n = -1/floor((n+9)/10), > >>> R_n = -1/n. > >>> > >>> For n = 1,2,..., define a function B_n by > >>> > >>> B_n(t) = 1 if A_n <= t < R_n, > >>> 0 if t < A_n or t >= R_n. > >>> > >>> Let V(t) = sum_n B_n(t). > >>> > >>> Next you say, "assume t is continuous". Not sure what you mean. Maybe > >>> you mean assume the function is continuous? However, it seems that > >>> either the function we defined (e.g., V) is continuous or it isn't, > >>> i.e., it should be something we deduce, not assume. Let's skip this for > >>> now. I don't think we actually need it. > >>> > >>> Finally, you write, "check the limit as t->0". I would interpret this as > >>> saying that we should evaluate the limit of V(t) as t approaches zero > >>> from the left, i.e., > >>> > >>> lim_{t -> 0-} V(t). > >>> > >>> Do you agree that you are saying that the number of balls in the vase at > >>> noon is lim_{t -> 0-} V(t)? > >>> > >> Find limits of formulas on numbers, not limits of sets. > > > > I have no clue what you mean. There are no "limits of sets" in what I > > wrote. > > > >> Here's what I said to Stephen: > >> > >> out(n) is the number of balls removed upon completion of iteration n, > >> and is equal to n. > >> > >> in(n) is the number of balls inserted upon completion of iteration n, > >> and is equal to 10n. > >> > >> contains(n) is the number of balls in the vase upon completion of > >> iteration n, and is equal to in(n)-out(n)=9n. > >> > >> n(t) is the number of iterations completed at time t, equal to floor(-1/t). > >> > >> contains(t) is the number of balls in the vase at time t, and is equal > >> to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). > >> > >> Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. > > > > You seem to be agreeing with what I wrote, i.e., that you say that the > > number of balls in the vase at noon is lim_{t -> 0-} V(t). Care to > > confirm this? > > No that's a bad formulation. I gave you the correct formulation, which > states the number of balls in the vase as a function of t. Let's try some numbers. t = -1, 9*floor(-1/t) = 9, V(t) = 9. t = -1/2, 9*floor(-1/t) = 18, V(t) = 18. Looks to me like V(t) = 9*floor(-1/t) for t < 0. So, lim_{t->0-) 9*floor(-1/t) = lim_{t->0-} V(t). So, it does seem that what I said you are saying is what you are saying. -- David Marcus
From: David Marcus on 31 Oct 2006 20:07 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> So, please do as I asked and consider this as a completely separate > >>> math problem. There are no balls, vases, time, or noon. Here is the > >>> problem again (with slightly changed notation): > >>> > >>> For j = 1,2,..., let > >>> > >>> a_j = -1/floor((j+9)/10), > >>> b_j = -1/j. > >>> > >>> For j = 1,2,..., define a function f_j: R -> R by > >>> > >>> f_j(x) = 1 if a_j <= x < b_j, > >>> 0 if x < a_j or x >= b_j. > >>> > >>> Let g(x) = sum_j f_j(x). What is g(0)? > >>> > >>> Do you say that g(0) = 0? > >> No, I say that lim(x->0: g(x))=oo. > > > > I assume you mean lim_{x->0-} g(x) = oo. However, that isn't the > > question I asked. The question I asked is "What is g(0)?". Please answer > > the question that I asked. > > oo Please confirm. With these definitions: For j = 1,2,..., let a_j = -1/floor((j+9)/10), b_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 1 if a_j <= x < b_j, 0 if x < a_j or x >= b_j. Let g(x) = sum_j f_j(x). You say that g(0) = oo. Is that correct? -- David Marcus
From: David Marcus on 31 Oct 2006 20:09
MoeBlee wrote: > Tony Orlow wrote: > > When I say there are problems with the axiom of extensionality, I refer > > to the application of the fact that two sets, when viewed statically, > > contain the same elements. Yes, that means that, without regard to time > > or order or anything else, the sets are, theoretically, the same. I > > don't dispute that. But, I do dispute the application of that fact to > > the exclusion of specifically stated time constraints and their > > resulting definition of iterations. I am not saying that the axiom > > itself is wrong, as a definition. It just doesn't capture the essence of > > what's going on. Static sets are not sequences of arithmetical events. > > Do you disagree? > > Okay, fair enough. But then you need to find axioms and definitions > that capture your notion of dynamic sets. Or, just use functions, as people have been doing for hundreds of years. -- David Marcus |