An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: MoeBlee on 17 Oct 2006 12:47 Tony Orlow wrote: > David Marcus wrote: > > MoeBlee wrote: > >> Tony Orlow wrote: > >>> No, set theory confuses the issue with its concentration on omega. > >> Oh boy, here we go again with "Set theory confuses...". Please just way > >> which axioms of set theory you reject and which ones you use instead. > > > > Good suggestion. But, what do you think the chance is that Tony will > > actually do that? > > > > I have done it lots of times. I've talked about infinite case induction, > the Inverse Function Rule, N=S^L, and in this case specifically, that > you cannot rearrange events which are stated to be in a specific order. > That last one is just a general no-no. We have been over the rearranging > of infinite series, which was claimed to prove their case, and its > invalidity. MoeBlee can pretend we haven't discussed all that. I don't care. I don't pretend that you have not tossed about all kinds of home grown verbiage. But none of your word concoctions qualify as axiomatizations, though YOU seem to pretend they do. MoeBlee
From: Lester Zick on 17 Oct 2006 13:11 On Mon, 16 Oct 2006 14:06:37 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <7542c$4533389f$82a1e228$8559(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> Virgil wrote: >> >> > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> > >> >> I find the idea absurd that natural numbers can be built >> >>by putting curly braces around the empty set. >> > >> > It appears as if much of useful mathematics is ultimately based on what >> > HdB finds absurd. >> >> The natural numbers can be defined without employing set theory. >> >> Han de Bruijn > >How? Infinitesimal subdivision. ~v~~
From: Lester Zick on 17 Oct 2006 13:18 On 16 Oct 2006 12:42:47 -0700, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: >> On 16 Oct 2006 08:07:56 -0700, imaginatorium(a)despammed.com wrote: >> >> > >> >Han de Bruijn wrote: >> >> Virgil wrote: >> >> >> >> > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, >> >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >> > >> >> >> I find the idea absurd that natural numbers can be built >> >> >>by putting curly braces around the empty set. >> >> > >> >> > It appears as if much of useful mathematics is ultimately based on what >> >> > HdB finds absurd. >> >> >> >> The natural numbers can be defined without employing set theory. >> > >> >I should think they could be. Though I fancy the natural numbers will >> >never be properly defined by anyone who is incapable of understanding >> >the set theoretic definition. >> >> Well, Brian, if by "incapable of understanding . . ." you mean >> "unwilling to agree with set theoretic assumptions regarding >> definition of the naturals" I would certainly have to disagree. > >Well, I don't. By "incapable of understanding the set theoretic >definition" I refer to anyone who lacks the capability for abstract >thought required to understand the basics of set theory. I hesitate to >say simply "anyone too stupid", though it is true it would be shorter. So you find that anyone simply "unwilling to agree with set theoretic assumptions regarding definition of the naturals" is not too stupid necessarily to understand the basics of set theory?I agree but I'm not too sure what the drift of your comment is. I should have thought it was that anyone too lazy or stupid to appreciate the virtues of set theoretic assumptions regarding definition of the naturals would be unable to define the naturals properly. Clearly that is not the case. ~v~~
From: imaginatorium on 17 Oct 2006 13:37 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". 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. > > <snip> > > On the contrary, the process *at noon* is completely well-defined. > > Then how come no one can say what happens "at noon", which doesn't > happen "before noon"? You're stretching to the point of breaking. No, I'm slightly lost. Don't understand the relation between the comma-separated clauses of the first sentence. > A > > ball is inserted in the vase or removed from the vase only at a time > > that is -1/n for some pofnat n. There is no pofnat m such that -1/m = > > 0. Therefore no ball is either inserted or removed at noon. (This > > really is elementary, you know.) > > Well then, nothing can change at noon that was true at every time before > noon, when there is a growing positive number of balls in the vase. What > changed at noon? Every time *before* noon was a time at which a ball was still to be removed. Give me a (real, genuine, numerical) value of a time before noon, and I will give you the number of a ball that has yet to be removed from the vase. *At* the time noon, there is no ball that has yet to be removed. (In normal logic, this means the vase is empty.) > Nothing, since nothing happened "at" noon. No ball movement, no. > So, the > number of balls in the vase at noon is growing is growing, No, the number of balls at any time, however small, *before* noon is growing. And the smaller the time before noon, the crazier it is growing. > >> ... since the distinction between > >> iterations disappears. How do you know there are countably many > >> iterations, and not some uncountably number? You don't. > > > > Of course I do. > > Ah, Zeno told you.... Try being less obnoxious. In the end you might make yourself look less silly. > The problem explicitly says balls with natural numbers > > (pofnats) on them. > > And this set ends where? Nowhere. Well, actually, at noon. Isn't that a > tad artificial, and somewhat contradictory? Does the infinite series (gosh, it's amazing, but I believe there are tiny fragments of mathematics you have actually managed to grasp) 1 + 1/2 + 1/4 + 1/8 + 1/16 + .... "end" anywhere. No it doesn't. But if you represent this as an unending sequence of blocks laid in a row, the blocks do not extend indefinitely, ever though there is no last one of them. > The sequence either of insertions or of removals is > > immediately mapped onto the pofnats. > > The sequence of both intertwined amounts to a linearly increasing sum, > when kept in their stated order. > There can - by definition - only > > be a countable sequence of pofnats. (Actually, in non-mumbo-jumbo, > > "uncountable sequence" is a contradiction in terms.) > That definition is bunk, when discussing sequences where there are > clearly infinitely-indexed iterations, such as occur at noon. Oh, dear, oh dear. "Occur at noon"? Means what? There are no "infinitely-indexed iterations", because no ball is ever put in the vase unless it has a pofnat written on it. DO YOU UNDERSTAND THAT? > Then absolutely nothing happens at noon, since that would require > infinite n, to change the fact that, at every moment from 11:59 until > then, there were a nonzero number o balls in the vase. Sorry, I'm lost. Nothing happens at one time to change the fact that at another time something was true? No of course it doesn't. > When only the > pofnats are involved, so only are finite times before noon, during which > this fact holds true. Do you have a refutation of that argument, > specifically? No, of course not, because it's true. For any time *before* noon, there are balls yet to be removed. Yes, of course the number of ball
From: Lester Zick on 17 Oct 2006 14:36
On 17 Oct 2006 10:37:08 -0700, imaginatorium(a)despammed.com wrote: >Tony Orlow wrote: [. . .] >> Oh. What was Aleph_0 again? > >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". So now aleph is an empirical concept, Brian? I mean unless you consider children too lazy or stupid to intuit the subtle virtue of set theoretic assumptions underlying their ability to count one by one. ~v~~ |