An uncountable countable set
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From: Ross A. Finlayson on 18 Oct 2006 01:19 Virgil wrote: > In article <4535944f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > In article <26453$4534c7d5$82a1e228$20375(a)news1.tudelft.nl>, > > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > >> Virgil wrote: > > >> > > >>> In article <45341a3a$1(a)news2.lightlink.com>, > > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > >>> > > >>>> Well, I think that, while the empty set may easily be taken to represent > > >>>> 0, 1 is not the set containing 0. That doesn't seem, even at first > > >>>> glance, like a very accurate model of what 1 is. > > >>> If TO is not happy with the set representing 1 containing a single item > > >>> does TO want the set representing 1 to contain more or less that single > > >>> item? > > >> That single item is the EMPTY set, pasted between curly braces. > > > > > > HdB is missing my point here. > > > > > > If TO accepts {} as representing 0 but does not like {{}} as > > > representing 1, what does TO suggest replace {{}} as representing1? > > > > 1 represents the finite unit. 1 is arbitrary in that respect. 1 is 0+1. > > It marks the end of an interval containing infinitely many reals, not a > > container for a container full of nothing. > > Naw, that's 1.0, not 1, that you are talking about. Your 1.0 is only > good for measuring, but no good for counting. They're the same number. I'll go perhaps a bit farther than might seem reasonable, and suggest that Virgil stopped talking to me because he sees reason in the null axiom theory, not just because of embarrassment over all of his gaffes. Virgil, ya self-described defender of the orthodoxy, gadfly, addict, and basically irritant: there is no universe in ZF. Orthodoxy doesn't need your help, far from it, for it to progress. Your abuse of Tony and others, usenet, is disrespect to everybody else. There is a universe, and where functions between physical objects are physical objects, the universe is infinite, and, infinite sets are equivalent, Bob. At some point, each one of those balls was in the vase, when? For any time you note where a ball has been removed, there is that difference in time between that time and noon, and, as that difference goes to zero, the count of balls in the vase diverges. Look at it another way. You're fooling yourself if you think it's so cut and dried there's not an unobvious answer. Remember when we were talking about implicit composition of functions to map between a closed set and the reals? You have that for any ball n, you can designate a time when it removed. Contrarily, for any time t, there is a later time where more balls are in the vase than there are at time t. It's the inductive impasse. Ross
From: Ross A. Finlayson on 18 Oct 2006 01:44 Virgil: > > > Given an arbitrary set of individually identifiable balls, S, and an > empty vase capable of containing then all. > > Assume on some day that for each ball in S there is a time before noon > at which that ball is placed in the vase and a later time, also before > noon at which that ball is removed from the vase. > > Regardless of the arrangement of the times of insertion or removal of > individual balls, so long as they all satisfy those assumptions, the > vase will be empty at noon of that day. Also, for each later time t less than noon, there are more balls in the vase than at time t, because for each difference there are ten balls added for each removed. So, how does the ball remove itself? Consider the ball and vase as a related rates tank problem. You pour in a dram of this syrupy liquid, and it goes to the bottom of the tank, where there's a hole in the bucket, Elvira, and the liquid drains out. That would seem similar to your problem here with the balls and vase, no? It even offers a mechanism, where you need to expand on some of the explanation there of the mechanics of the balls and vase. Your non-explanation is not a constraint, it's handwaving. So, the tank obviously fills. Ross
From: Han de Bruijn on 18 Oct 2006 03:11 David Marcus wrote: > Han de Bruijn wrote: > >>No matter how much textbooks about set theory I read, it all remains >>abacedabra for me. > > I just looked in two books on set theory on my shelf and both explicitly > state that the set in the axiom of infinity need not be the natural > numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set > Theory, An Introduction to Independence Proofs" by Kenneth Kunen. Hmm, I have that book by Halmos myself ... But why are the finite ordinals not equivalent with the naturals (I mean in mainstream mathematics)? > Which math textbooks have you read? Have you worked the problems in the > ones you've read? Taken any math courses at a university? I'm a (theoretical) physicist by education. Han de Bruijn
From: imaginatorium on 18 Oct 2006 03:15 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>> <snip> > > > > <snipitty-snip> > > > >>> Do you accept the above statements, or do you still claim that there is > >>> /no/ valid proof that ball 15 is not in the vase at t=0? > >>> > >> 15 is a specific finite number for which we can state its times of entry > >> and exit. > > > > Agreed, > > > >> At its time of exit, balls 16 through 150 reside in the vase. > > > > Agreed. > > > >> For every finite n in N, upon its removal, 9n balls remain. > > > > "upon its removal" = "at the time of ball n's removal"; Agreed. > > > >> For every n > >> e N, there is a finite nonzero number of balls in the vase. > > > > "For every n e N, there is a finite non-zero number of balls in the > > vase at t = -1/n". Agreed. > > > >> Every > >> iteration in the sequence is indexed with an n in N. > > > > "Balls are only added or removed at a time t = -1/n for some natual n." > > Agreed. > > > >> Therefore, nowhere > >> in the sequence... > > > > ..., i.e, at no time t such that t = -1/n for some natural n, ... > > > >> is there anything other than a finite nonzero number of > >> balls in the vase. > > > > Agreed. > > > >> Now, where, specifically, in the fallacy in that argument? > >> > > > > Well, what do you state is the conclusion of this argument? > > You have agreed with everything so far. At every point before noon balls > remain. You claim nothing changes at noon. Is there something between > noon and "before noon", when **those balls** disappeared? Tony, which balls does "those balls" refer to here? You mention *any* time before noon, and we can work out what balls are in the vase, and we can also work out exactly when those balls [refers to the balls earlier in this sentence] will all have been removed, and we know "in advance" that that time of removal will be before noon. So at *no* time before noon will there be any balls in the vase with any chance of lingering after noon. Hmm. So your "those balls" must have been introduced into the vase in this mysterious zone "between before noon and noon". But, see, in mathematics, it's quite clear there is no such zone - here's a proof. Let B = { t : t is a time, and t is before noon } // the set of all times before noon Let N = {noon} // the singleton set of noon I suggest that if there is a time _between_ before noon and noon, it must be a member of the following set: Let Z = { t : t is a time, t is after b for all b in B, t is before n for all n in N } Do you agree? Would you like to prove that Z is the empty set, just as a little exercise? Brian Chandler http://imaginatorium.org
From: imaginatorium on 18 Oct 2006 03:30
Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> imaginatorium(a)despammed.com wrote: > >>> Tony Orlow wrote: > >>>> imaginatorium(a)despammed.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> imaginatorium(a)despammed.com wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > > > > <snip> > > > >>> Suppose I define the following function, referring to sliver-1, which > >>> is the area between y=-2/x and y=-1/x for x<0. ("sleight" stands for > >>> 'sliver height', not 'sleight of hand'...) > >>> sleight(x) = -2/x +1/x for x<0; 0 elsewhere > >> Uh huh. For x<0 as opposed to x>=0. No declared point of discontinuity > >> there.... > > > > OK. Let's see if it's possible to understand what, if anything, you > > mean by "function". Why don't you answer any of my questions? Somewhere buried here is a key to your near-total miscomprehension of mathematics. If you cooperated, we might be able to diagnose the problem. > > Do you agree that the graph of y=-1/x for x < 0 is one lobe of a > > hyperbola? > > > > Do you agree that the graph of y=-2/x for x < 0 is one lobe of another > > hyperbola? > > > > Do you agree that in the unbounded x-y plane, these two lobes define a > > "sliver", a boomerang-shaped area, extending indefinitely 'left' and > > also extending indefinitely 'upward' (using these directional terms in > > the sense of looking at a conventional graph)? > > > > Do you agree that for any simply-connected area (think that's the right > > term) within the x-y plane we could consider the function that maps x > > to the vertical measure* of the area at the particular x value? > > > > ( * a term I've made up. If you don't understand ask; if anyone knows a > > proper word, please tell me) > > > > By way of a different example, consider the circle radius 1, centre (0, > > 0), and find its 'height()' function. For any value of x outside the > > range (-1, +1), the vertical measure is zero, because, obviously, the > > circle only extends horizontally from -1 to +1. Within that range, the > > vertical measure is equal to the height of an ellipse centred on the > > origin, of width 2 and height 4, so (if I calculate correctly) the full > > function is given by: > > > > height(x) = 2 * sqrt(1-x^2) for -1 < x < 1 > > height(x) = 0 otherwise > > > > Please tell me: is this a function? Is it a continuous function? If so, > > does it have a "declared discontinuity"? > > > > You might like to do the same for the function height() of a rectangle > > diagonal from (0,0) to (3, 57). > > > > If you somehow claim that there _is_ no function representing the > > height of the sliver at a particular value of x, you really need to > > give us your definition of "function". > > > > If you agree there is such a function, why not try to write it down? > > > > You may or may not agree that this function is discontinuous - in any > > event, please explain whether my description above of the hyperbola > > lobes and the "sliver" has already included a "declared discontinuity". > > If not, does that mean there might be different ways of writing the > > same function, possibly some including a "declared discontinuity", > > others not. > Your examples of the circle and rectangle are good. Neither has a height > outside of its x range. The height of the circle is 0 at x=-1 and x=1, > because the circle actually exists there. To ask about its height at x=9 > is like asking how the air quality was on the 85th floor of the World > Trade Center yesterday. Similarly, it makes little sense to ask what > happens at noon. There is no vase at noon. Do you understand what "function" means in mathematics? The above paragraph suggests that you have simply no idea at all. If so all further "discussion" is bound to be fruitless. > Additionally, when it comes to 1/x and functions of that ilk, the > discontinuity at x=0 is not declared, but surmised on the basis that > oo<>-oo. When the number circle is considered, and oo as the inverse of > 0 considered to be both positive and negative, your discontinuity > disappears, and your sliver in the upper left and lower right can be > said to be connected at a point above and below, and a point to the > right and left, at oo. Just a conceptual picture to amuse you. <big snip> Look, apart from the general hopelessness of trying to discuss mathematics with someone who doesn't know what it is, there is a pervading problem of logical inconsistency. The following seems to be fairly typical. Claim 1: > >>>> You throw all the naturals in a bag and pretend you have some > >>>> specific number of them, TO claims I pretend to have a "specific number" of naturals. > >>> No I do not. It is only _you_ who talks of "specific infinities" and > >>> various other nonsense in which you pretend "Big'un" (or whatever it is > >>> at the moment) is part of finite arithmetic. I deny that I have a "specific number" of naturals. > >> Oh. What was Aleph_0 again? TO repeats his misconception that Aleph_0 is a "specific number" of naturals. > > You have, I'm sure been told dozens, if not hundreds of times - Aleph_0 > > is the name for the cardinality one might explain to children as "you > > can count, and reach any of them, but the counting never stops". I explain that Aleph_0 is not a "specific number" of naturals. > That's not a specific infinite number then. TO agrees. So have you retracted your claim 1 above? Any chance you could say "Sorry, my misconception"? > > Aleph_0 is very definitely not "part of finite arithmetic". > > It's not part of quantitative expression either. Care to define what "quantitative expression" means? (Don't bother. Try reading a book. Try to understand what 'function' - or 'mapping' - means.) Brian Chandler http://imaginatorium.org |