An uncountable countable set
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From: Virgil on 16 Oct 2006 16:06 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?
From: Virgil on 16 Oct 2006 16:11 In article <4533d0ff(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45319846(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> Functions can exist at points at which their limits do not. > >>> There are even functions with domain R which are discontinuous at every > >>> rational argument but continuous at every irrational one. > >> That sounds vaguely interesting. Can you give an example? > > > > It is standard fare for anyone who knows any analysis. > > > > Let f: R --> R, be such that for each irrational x f(x) = 0, > > and for each rational x whose expression in lowest terms is p/q, > > let f(x) = 1/q. Then that function is provably continuous at each > > irrational and provably discontinuous at each rational except 0. It is > > provably the case that for each real a, lim_{x --> a} f(x) = 0, which > > establishes the claim. > > Can you give an example of two irrational numbers, between which there > are no rational numbers? Irrelevant to the example above. For the function,f, defined above one has at every irrationals value of x, lim_Py -> x} f(y) = 0 = f(x) but at every rational value, x = p/q in lowest terms, lim_Py -> x} f(y) = 0 but f(p/q) = 1/q != 0
From: Virgil on 16 Oct 2006 16:22 In article <4533d18b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45319b8c(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Given any finitely numbered ball, we can calculate its entry and exit > >> times. However, we can also say that when it exits, there are more balls > >> in the vase than when it entered. If you had any upper bound to your set > >> of naturals, you'd see your logic makes no sense, but there is none. > > > > When expressed as functions of time, rather than the number of > > operations, there is no problem with having an empty vase at noon. > > When expressed in terms of iterations, the conclusion is quite the > opposite, so you cannot claim to be working with pure unquestionable > logic. You must choose which logical construction of the two is valid, > if either. As the problem is stated in terms of the times at which events occur, expressing things in terms of time is natural and indicated. When expressed in terms only of events, there is no longer any requirement that there even be an event of completing all the insertions-removals, or if there is, that it be close in time to any other event. You cannot dismiss time and still have it. > >> Before noon, there are balls. At noon, there are not. What happened? > > > > They were one by one removed. > > "One by one" and all removed, but no last one. Vigilogic at its worst. It is apparently TOlogic, that if one moves from point A to point B one must first cover half the distance then half the remaining distance, and so on ad infinitum, so that one never reaches point B. It is Virgilogic that one can go from one such midpoint to the next "one by one" but still reach point B in finite time.
From: Han.deBruijn on 16 Oct 2006 16:27 imaginatorium(a)despammed.com schreef: > 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. Well, why don't you just say it !? Google will record it, patiently. And I will show to my children and grandchildren what the main "arguments" of mainstream mathematics have been: name calling. Han de Bruijn
From: Virgil on 16 Oct 2006 16:28
In article <4533d1fc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45319d93(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45310688(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <452fbf0e(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> So let us leave them coupled but merely change the coupling so that the > >>>>> nth ball is inserted, say , 1/2^n minutes before it is removed. Both > >>>>> the > >>>>> insertions and the removals are still all completed before noon, and it > >>>>> is obvious that the vase is empty at noon. > >>>>> > >>>>> > >>>> Then you are inserting balls one at a time, and removing them as you > >>>> insert the next. What does that have to do with the original problem? > >>> The only necessary constraint on insertions of balls into the vase and > >>> removals of balls from the vase is that each ball that is to be removed > >>> must be inserted before it can be removed, and, subject only to that > >>> constraint, the set of balls remaining in the vase at the end of all > >>> removals is independent of both the times of insertion and of the times > >>> of removal. > >> WRONG!! :) > >> > >> There is the additional constraint that ten other balls (or nine, for > >> the first) must be inserted before it can be removed. > > > > That "constraint" is irrelevant, as it by the clock that the insertions > > and removals are determined, not merely by the insertion or removal of > > other balls. And how can earlier insertions make for fewer balls at any > > time as TO claims it does. > > Already explained, but not explained by you why the salient feature I > mention is irrelevant. TO's "salient feature: of adding balls later is negated by adding those balls earlier, which obviously cannot decrease the number of balls left left at noon. > > > > >> To argue that the adding of ten balls can be coupled with the removal of > >> one and get an eventual result of zero is just plain silly. > > > > To argue that adding balls earlier but removing them as before leaves > > fewer of them is even sillier. > > No, I already explained that. Apparently you didn't pay attention. I saw no explanation that made sense. How can adding balls earlier without changing how they are removed produce fewer balls at noon? > > >>>>> When infinitely many are inserted and all of them removed, what is > >>>>> obvious to TO is false to logic. > >>>> Your take on logic is very, shall we say, provincial. > >>> You may say what you like, however it remains correct. > >> Define "correct". > > > > Correct in this context means an analysis in accord with the > > constraints of the original problem. > > > > Which TO's analysis is not. > > When you claim that stated features of the process are "irrelevant" by > edict, you're not one to speak. They are irrelevant by logic. it is TO's claims which are made by edict but are not in accord with the original problem. |