An uncountable countable set
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From: stephen on 16 Oct 2006 20:18 David Marcus <DavidMarcus(a)alumdotmit.edu> 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? Tony has said that he does not like the axiom of infinity and the axiom of choice. Unlike many people who object to the axiom of infinity, Tony believes in infinite sets. He has more infinities than most. As for the axiom of choice, does the analysis of the balls in the vase problem use the axiom of choice? Stephen
From: MoeBlee on 16 Oct 2006 20:45 stephen(a)nomail.com wrote: > Tony has said that he does not like the axiom of infinity and > the axiom of choice. Unlike many people who object to the > axiom of infinity, Tony believes in infinite sets. Okay, so that leads to the second part of my message. If he rejects the axiom of infinity (does he actually?), then how does he get the existence of infinite sets? Of course, this is meaningless for me to ask, since he has no theory - just jumbles of undefined and circularly defined verbiage. MoeBlee
From: Virgil on 16 Oct 2006 20:46 In article <45341984(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> 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. > > > ....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. There is a discontinuity in the number of ball s in the vase every time a ball is moved in the gedanken under discussion. > > > > 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. Making, in all, one discontinuous function. Just as one pieces together the differing numbers of balls in the vase at different times to make one discontinuous function. > > > > > 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. Does TO claim that there is no such thing as a time after noon at which the number of balls in the vase can be considered? I see nothing in the statement of gedanken to prohibit the function being defined at and after noon. > > 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.... Such "splicing" is quite standard and legitimate in mathematics. For example consider the "floor" function which for each real number equals the largest integer not greater than that real number, and has a discontinuity at every integer value in the reals. > >>> 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. WE can say quite precisely what happens AT noon. Nothing at all happens because everything has already happened at times before noon, including the removal of every ball inserted at a time before noon at a later time still before noon. > > 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. That seems like a great deal of nothing. But whatever happens cannot be happening AT noon since every "happening" is strictly before noon. > 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. We admit that in the immediate forenoon whomever is moving those balls around will be very busy, but by noon he is done with it at can take his lunch at ease knowing that it is over and that every ball he has inserted into the urn he has later removed, leaving the status quo ante. > > > > >> ... 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,
From: Virgil on 16 Oct 2006 20:48 In article <45341a3a$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Well, I think that, while the empty set may easily be taken to represent > 0, 1 is not the set containing 0. That doesn't seem, even at first > glance, like a very accurate model of what 1 is. If TO is not happy with the set representing 1 containing a single item does TO want the set representing 1 to contain more or less that single item?
From: Virgil on 16 Oct 2006 20:51
In article <eh17g4$pn$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> 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? > > Tony has said that he does not like the axiom of infinity and > the axiom of choice. Unlike many people who object to the > axiom of infinity, Tony believes in infinite sets. He has more > infinities than most. As for the axiom of choice, does > the analysis of the balls in the vase problem use the axiom > of choice? Not if one allows that the set of finite ordinals is already well ordered so that any non-empty subset of it has a determinable first member. |