An uncountable countable set
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From: David Marcus on 23 Oct 2006 18:29 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > Also, supposing for the sake of argument that there are "infinitely > > number balls", if a ball is added at time -1/(2^floor(n/10)), and removed > > at time -1/(2^n)), then the balls added at time t=0, are those > > where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > > then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic > > on these infinite numbers works), so those balls are also removed at noon and > > never spend any time in the vase. > > Yes, the insertion/removal schedule instantly becomes infinitely fast in > a truly uncountable way. The only way to get a handle on it is to > explicitly state the level of infinity the iterations are allowed to > achieve at noon. When the iterations are restricted to finite values, > noon is never reached, but approached as a limit. Suppose we only do an insertion or removal at t = 1/n for n a natural number. What do you mean by "noon is never reached"? -- David Marcus
From: David Marcus on 23 Oct 2006 18:47 MoeBlee wrote: > Tony Orlow wrote: > > In Chapter III, section 3.1.1, he states: > > > > "There is no smallest infinite number. For if a is infinite then a<>0, > > hence a=b+1 (the corresponding fact being true in N). But b cannot be > > finite, for then a would be finite. Hence, there exists an infinite > > numbers [sic] which is smaller than a." > > I'll try to take a look at the book soon, but I very strongly suspect > that he's using 'infinite' not to refer to cardinality, but rather to > position in certain orderings. I'm sure that's right. He's extending the real numbers. I haven't read Robinson, but Enderton has a section on nonstandard analysis. He considers the statements {"r < x" | r in R}. By compactness, there is a model. The trick is to keep track of which statements that are true in the nonstandard model are also true in the standard reals. > That is fine, as long as we understand > the terminology in the context. As far as I know, it doesn't contradict > that there is a least infinite cardinality. Correct. He's doing analysis, not set theory. -- David Marcus
From: David Marcus on 23 Oct 2006 18:49 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> 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 really mean to say that there is no vase at noon or do you mean > > to say that the vase is not empty at noon? > > If noon exists at all, the vase is not empty. All finite naturals will > have been removed, but an infinite number of infinitely-numbered balls > will remain. "If noon exists at all"? How do we decide? -- David Marcus
From: David Marcus on 23 Oct 2006 18:51 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <4533d315(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> Then let us put all the balls in at once before the first is removed and > >>>>>>> then remove them according to the original time schedule. > >>>>>> Great! You changed the problem and got a different conclusion. How > >>>>>> very....like you. > >>>>> Does TO claim that putting balls in earlier but taking them out as in > >>>>> the original will result in fewer balls at the end? > >>>> If the two are separate events, sure. > >>> Not sure what you mean by "separate events". Suppose we put all the > >>> balls in at one minute before noon and take them out according to the > >>> original schedule. How many balls are in the vase at noon? > >>> > >> empty. > > > > Suppose we put ball n in at 1/n before noon and remove it at 1/(n+1) > > before noon. How many balls in the vase at noon? > > At all times >=-1 there will be 1 ball in the vase. And, the ball that is in the vase at noon, what is the number on the bal? "Infinity"? -- David Marcus
From: David Marcus on 23 Oct 2006 18:54
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <4533d315(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> Then let us put all the balls in at once before the first is removed and > >>>>>>> then remove them according to the original time schedule. > >>>>>> Great! You changed the problem and got a different conclusion. How > >>>>>> very....like you. > >>>>> Does TO claim that putting balls in earlier but taking them out as in > >>>>> the original will result in fewer balls at the end? > >>>> If the two are separate events, sure. > >>> Not sure what you mean by "separate events". Suppose we put all the > >>> balls in at one minute before noon and take them out according to the > >>> original schedule. How many balls are in the vase at noon? > >> empty. > > > > Why? > > Because of the infinite rate of removal without insertions at noon. OK. Just to recall, this vase has all the balls put in at one minute before noon, then taken out on the usual schedule. How many balls are in this vase at times before noon? -- David Marcus |