An uncountable countable set
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From: David Marcus on 9 Oct 2006 12:49 Han de Bruijn wrote: > David Marcus wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>David Marcus wrote: > >> > >>>Consider this situation: At time 5 one ball is added to a vase. At time > >>>6, the ball is removed. > >>> > >>>Is the following a valid translation into mathematics? > >>> > >>>Let the value 1 denote that that the ball is in the vase and the value 0 > >>>that the ball is not in the vase. Let A(t) be the location of the ball > >>>at time t. Let > >>> > >>>A(t) = { 1 if 5 < t < 6; 0 if t < 5 or t > 6 }. > >> > >>No. > > > > Then what is the translation into Mathematics? > > The mathematical model for the number of balls in the vase is this. > Let t_k = - 1/2^k for (k = 0,1,2, ... in N). Note that t < 0 . > Then the number of balls is B(k) = 9 + 9.ln(-1/t_k)/ln(2) = 9.(k+1) > for t_k < t < t_(k+1) : a staircase. I meant what is the translation of the situation I stated above, i.e., "At time 5 one ball is added to a vase. At time 6, the ball is removed."? -- David Marcus
From: David Marcus on 9 Oct 2006 12:56 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <1160310643.181133.6720(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > AXIOM OF INFINITY Vla There exists at least one set Z with the > > > following properties: > > > (i) O eps Z > > > (ii) if x eps Z, also {x} eps Z. > > > > > > There are several verbal formulations dispersed over the literature > > > without any "all". > > > > Perhaps. Does this mean that there are some x in Z such that {x} not in > > Z? Or, what do you mean? > > He means that the axioms can be stated without using universal > quantifiers or existential quantifiers, as simple logical implications > between propositions, I believe. WM, am I close? If you see "if x is in Z, then {x} is in Z" in a math book (without any other explicit restriction on what x is), then the convention is that you are to interpret it as meaning "for all x, if x is in Z, then {x} is in Z". Otherwise, you have a letter whose meaning you do not know. -- David Marcus
From: Virgil on 9 Oct 2006 13:26 In article <6b14e$452a1745$82a1e228$13843(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <e3ab0$4529ffd9$82a1e228$22026(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote: > >> > >>>In article <1160332251.241188.301420(a)i3g2000cwc.googlegroups.com>, > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>>But ah, a picture says more than a thousand words: > >>>> > >>>>http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > >>> > >>>But the function one should be graphing is not continuous at any of the > >>>time of entry or exit of any ball, nor at noon (except it is > >>>right-continuous at noon). > >> > >>Correct. I have "continuized" the function. The true (discrete) function > >>should be like a staircase with a stair at each red ball. But a discrete > >>version likewise explodes (i.e. not implodes) at noon. > > > > Actually not precisely "at" noon. > > > > Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition. > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t. > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point. > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > Wrong. Because you cannot jump over noon. But HdB easily jumps over moon. > > The vase has become so infinitely heavy at that time that a black hole > is being formed. And clocks begin to tick slower and slower, sooo slow > that it's forever impossible to reach noon. Look what modern GR can do! But such GR need not apply in a mathworld, paricularly since mathematical ping pong balls are massless. > > Han de Bruijn
From: Virgil on 9 Oct 2006 13:38 In article <452a65be(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4529ac7e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > I'll take your lack of response as assent. If I took lack of TO's response's as assent, he would have long since and repeatedly conceded the error of his ways. > > >> > >>>> Between every removal of 1 and the successive > >>>> removal of 1 is an insertion of 10. So, each term '+10' must be coupled > >>>> with a term '-1', which together make a '+9' per iteration. Where you > >>>> violate the condition of the experiment that specifies this coupling of > >>>> events into an iteration, you create your "discontinuity" at noon, and > >>>> the monster under your bed. > >>> Let A_n(t) be equal to > >>> 0 at all times, t, when the nth ball is out of the vase, > >>> 1 at all times, t, when the nth ball is in the vase, and > >>> undefined at all times, t, when the nth ball is in transition. > >>> > >>> Note that noon is not a time of transition for any ball, though it is a > >>> cluster point of such times. > >> Does time obey the law of trichotomy? Can something occur, not before, > >> not after, and not at the same time as another event, given that the > >> events happen instantaneously? > > > > In general relativity , two events cannot be said to occur at "the same > > time"" unless they are also in the same place. > > ???? Where did anyone mention relativistic spacetime? I asked about time > as a dimension. Can we consider it to be linear and continuous? Is there > a fourth option besides the three of trichotomy? In GR time is not separable from the other dimensions and there is another option besides trichotomy, as "before" versus "after" can depend on the observer. > > >>> let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > >>> at any non-transition time t.Let A_n(t) be equal to > >>> 0 at all times, t, when the nth ball is out of the vase, > >>> 1 at all times, t, when the nth ball is in the vase, and > >>> undefined at all times, t, when the nth ball is in transition. > >> What is "in transition". Can't we consider the addition or removal of a > >> ball to be instantaneous? > > > > Would TO's "occurring instantaneously" mean that it does not take place > > at any point in the time stream? I don't think so. The function I > > described is undefined at all such points in time at which there is a > > transition in position from before that moment to after that moment. > > It means it takes place at a particular point in time, and that there is > not a measurable "period" of transition. If you say it's "undefined" at > that point, then define it. All you have to do is choose whether the > event occurs at that point, or before, or after it. That is, in > transition from f(before t)=x to f(after t)=y, either f(t)=x or f(t)=y. > Either x<f(t)<=y or x<=f(t)<y. TO can "define" it at those points if he wishes, but my function retains those points of non-definition, as there is no need to define any value at those points in order to make my point. > > > > >>> Note that noon is not a time of transition for any ball, though it is a > >>> cluster point of such times. > >>> > >>> let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > >>> at any non-transition time t. > >>> > >>> B(t) is clearly defined and finite at every non-transition point, as > >>> being, essentially, a finite sum at every such non-transition point. > >> Correct, a linearly increasing finite value. > > > > Not increasing between points of discontinuity nor over any interval of > > time containing times past noon. > > Linearly increasing with each iteration, with a condensation point at > noon where iterations occur infinitely fast. Actually AT noon, nothing happens, it has all finished happening. > > > > >>> Further, A_n(noon) = 0 for every n, so B(noon) = 0. > >>> Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > >>> > >>> > >>> B(t) is clearly defined and finite at every non-transition point, as > >>> being, essentially, a finite sum at every such non-transition point. > >>> > >>> Further, A_n(noon) = 0 for every n, so B(noon) = 0. > >>> Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > >> I understand your logic, but the basis is incorrect. As WM > >> understandably complains, using the "completed set of naturals" as a > >> measure of anything simply does not work. > > > > It works in ZFC and NBG. That it does not work for TO or for "Mueckenh" > > (though for different reasons) is their problem, not ours. > > > > Not for different reasons at all. The difference is whether this problem > causes one to reject infinite values entirely, or seek a better > representation of them. > > > > >> You're focused on the Twilight > >> Zone. > > > > Even twilight beats the stygian darkness in which TO is operating. > > Says he in the Cave of Treasures. My view of mathematics is that, as it is, it is a cave of treasures, but brightly lit from within. That others find that cave dark is their loss.
From: Virgil on 9 Oct 2006 13:43
In article <452a6847(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4529afa4(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45296779(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <452946ad(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>>> Less than any finite distance. Silly! > >>>>>>> And what is the smallest finite distance? > >>>>>>> > >>>>> Note question not answered!! But the correct answer of zero would have > >>>>> blown TO's argument to blazes, so one can see why he would not care to > >>>>> answer it. > >>>> I was away a couple days, but I answered this, not paying attention to > >>>> that apparent contradiction, since I don't consider 0 really a finite > >>>> number at all. It's a point with no measure, as every number is measured > >>>> relative to that point. > >>> If zero is not a number, how does TO keep the positive numbers separated > >>> from negatives? > >> It's not a finite number. It's the origin. A finite number is a finite > >> distance from the origin. The origin is no distance from itself. > > > > Then TO's set of real numbers is two sets separated by a non-number? > > > > That does not match anyone else's set of reals. So TO casts himself > > again into outer darkness re res mathematical. > > It's a 1-D continuum with an origin, a metric space. But where are the values of that metric if zero is not one of them? > > >>>>>> When you claim that there are ordinals greater than any finite > >>>>>> ordinal, > >>>>>> are you obligated to name the largest finite ordinal? > >>>>> When you claim there is a LUB to the reals strictly between 0 and 1, > >>>>> are you required to name the largest real strictly between 0 and 1? > >>>> No. That's my point. Why should I name the smallest object which is not > >>>> infinitesimal? > >>> That is not at all what I asked. So TO is doing his STRAW MAN fallacy > >>> thing again. > >> You have no clue what the line of discussion was at this point, do you? > > > > I have no idea what TO is talking about, and am reasonably sure he > > doesn't either. > > Then don't make yourself look silly defending questions and comments > that are irrelevant. > > >>>>> A "LUB" of the naturals does not have to be a natural any more than the > >>>>> LUB of the the reals strictly between 0 and 1 has to be a real > >>>>> strictly > >>>>> between 0 and 1. > >>>> If it's a discrete set, then I disagree. > >>> The set {(n-1)/n: n in N} is a discrete set with a LUB which is not a > >>> member of the set. In fact every strictly increasing sequence having a > >>> LUB has a LUB which is not a member of the sequence. > >> That is not "the reals strictly between 0 and 1" but a subset thereof. > > > > So there is still no element within either set which is its LUB. > > If the Finlayson reals are used, indeed the LUB is the maximal member of > the set of reals in [0,1). Ross, is that correct? TO appealing to Ross is the blind asking for a lead from the blind. > > Tony |