An uncountable countable set
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From: MoeBlee on 16 Oct 2006 18:25 P.S. I lost the context, but somewhere you (Orlow) posted: "ZF and NBG don't handle sequences or their sums, but only unordered sets" Z set theory defines and proves theorems about ordered tuples, finite and infinite sequences, and infinite summations and infinite products and many other things like that. MoeBlee
From: Tony Orlow on 16 Oct 2006 19:45 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: > > <snipple-snapple> Don't eat at the computer.... > >>>>> Consider the function step0: R -> R mapping x to 0 if x<0 and mapping x >>>>> to 1 if x>=0. >>>> A discontinuous function at x=0. > > <snip> > .....or cut your hair on the keyboard. >>>> A function with such a declared discontinuity has two limits at that >>>> point, depending on the direction of approach. So, what else is new? > >>> Ah. Is a "declared discontinuity" somehow significantly different from >>> a simple discontinuity? I mean, is there such a thing as an "undeclared >>> discontinuity" to which different rules apply? (I've no idea: this is >>> not normal mathematical terminology you see.) >> You have defined your step function with an explicit discontinuity. >> There is no explicit discontinuity in the gedanken under discussion. The >> discontinuity is introduced with the application of omega. > > Of course step0() is a step function, and it has a discontinuity. But > all I have done is *define* it, which I can do without any step in the > definition called "declaring a discontinuity". When you say "for x<0" as opposed to "for x>=0", well, you have spliced two different continuous functions together at a point where they are unequal. > > In the ball/vase thought experiment, of course there is a discontinuity > at noon - even just from the words of the description, it is clear > there is an abrupt change from ever-more-frenetic vase filling/emptying > to a state of calm where nothing changes. Yes, supposedly the infinite process completes. That's an end to the domain of the function, not a discontinuity within the domain. Duh. > > 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'...) You can say that, but that's rather Freudian of you. > > 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.... > > I suppose we agree that sleight(), which increases without limit as > negative x->0, has a discontinuity at zero. Is this a "declared > discontinuity"? Uh, yeah, you specifically named 0 as the point of discontinuity, where you splice together two continuous functions at a point of inequality. You're smarter than this.... > > <snip> > Uh, don't trim your nose hairs over the keyboard either... >>> According to your "view" then, there is no discontinuity at noon - is >>> that right? The number of balls identified by natural numbers increases >>> without limit, and despite the fact that there is no ball not removed >>> before noon, at ten past an unlimited number of them are somehow still >>> lurking in the vase? >> The process at noon is not well defined, > > 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. 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? Nothing, since nothing happened "at" noon. So, the number of balls in the vase at noon is growing is growing, not only at an infinite rate, but at an infinitely increasing, and accelerating, and jerking, etc, rate, covering all infinite n that could possibly exist. And all that in a single moment. Suddenly, iterations are no longer distinguishable or even vaguely measurable. Ah, the unscrutable design of Zeno. What an engineer. Undubitably. > >> ... 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.... 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? 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. No finite iterations occur at noon, when there is an unbounded sum of balls accumulating, and yet, at noon you claim something changes all that. That's where you get onto the infinite speedway, but take a sharp turn into the bushes. >> You base your >> argument on all iterations being finite, > > Yes, because the problem explicitly says that every ball inserted or > removed is marked with a pofnat. > 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. 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? >> ... but there is no least upper >> bound to the finites, > > Congratulations. You got something right. Uh, yeah. Your turn. > >> because there is no least infinite. > > Uh, no. Uh, yeah, there's not, where any kind of useful arithmetic can be done. There is no upper bound to the pofnats (which you so cutely > call "the finites"), That includes the finite reals, to which there is also no upper bound. The salient feature of the number is that it's finite, not that it's an integral number of increments from 0. because they go on for ever. To every fini
From: Tony Orlow on 16 Oct 2006 19:48 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Tony Orlow schreef: > >> Han de Bruijn wrote: >>> So the axiom of infinity says that you can get everything from nothing. >>> This is contradictory to all laws of physics, where it is said that you >>> pay a price for everything. E.g. mass and energy are conserved. >> Han, you can't really be looking for conservation of energy or momentum >> or mass in abstract mathematics, can you? This axiom basically defines >> the infinite linear inductive set. Given this method of generation, >> there should be things we can say about the set, no? > > So to speak, Tony. In physics and economics, you can't get something > for nothing. Nothing just gives nothing. You must have _something_ to > start with. I find the idea absurd that natural numbers can be built > by putting curly braces around the empty set. > > Han de Bruijn > 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.
From: David Marcus on 16 Oct 2006 19:51 MoeBlee wrote: > David Marcus wrote: > > Ross A. Finlayson wrote: > > > There is no universe in ZF, ZF is inconsistent. > > > > What exactly is the inconsistency, please? > > Of course he'll never show you an inconsistency. "Set theory is > inconsistent" is just one among the phrases that he's fond of > muttering. Apparently true. -- David Marcus
From: David Marcus on 16 Oct 2006 19:53
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? -- David Marcus |