An uncountable countable set
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From: Ross A. Finlayson on 22 Oct 2006 06:17 Ross A. Finlayson wrote: > Lester Zick wrote: > > On 19 Oct 2006 17:37:49 -0700, "Ross A. Finlayson" > > <raf(a)tiki-lounge.com> wrote: > > > > >Lester Zick wrote: > > >... > > >> > > >> So what is it exactly that "set" theory allows us to do in mathematics > > >> that we couldn't already do without it? Define infinity? Define > > >> regularity? Define choice? Define ordinals? Define cardinals? You seem > > >> to be of a psychological frame of reference prevalent among modern > > >> mathematikers that arithmetic in the form of set theory represents > > >> some kind of TOE. > > >> > > >> ~v~~ > > > > > >Lester, > > > > > >Only the null axiom theory could be the TOE. > > > > Sorry, Ross, but I don't rely on axioms and I have no idea what the > > "null axiom theory" is or is supposed to explain. It seems that every > > time I turn around someone is propounding some new axiom which is > > supposed to explain something people can't explain without assuming > > some new and completely trivial assumption. > > > > ~v~~ > > Hi Lester, > > I thought you already knew that the null axiom theory, A theory, refers > to an axiomless system of natural deduction. > > I promote _less_ axioms. > > Also I thought you knew that the null axiom theory escapes Goedelian > incompleteness and is the theory of everything. > > Snark is a boojum. Define definition. > It was really great when Poker Joker defined definition. EF is just a range over the rational numbers. The lists are uncountable so, one of them is the list. Thus, the naturals are a countable, uncountable set. Let's see, the lists are uncountable: what does it mean to say the lists are uncountable? Starting with different elements of the reals where "countably" many antidiagonals can be found, there are uncountably many lists, and, they're a countable collection of countable lists. Ross
From: Ross A. Finlayson on 22 Oct 2006 07:51 Naturals: countable AND uncountable. It's like, where the sets are Cantor-Bernstein, but not Cantor-Schroeder-Bernstein. Infinite sets are equivalent. Where you can put any number on the list, the transfer principle can be applied. Where you can't, the reals aren't a set. I wonder how the anti-anti-Cantorians will deny this one. Maybe they won't and become post-Cantorians. Did you know the halting problem is not a problem? If there's an infinite program there are infinite programs. I guess that's negateable. Let's see, foundations of mathematics, check. Ross
From: Virgil on 22 Oct 2006 14:51 In article <453b326d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4539000e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > Claimed but not justified. TO's usual technique! > > You didn't justify yours. It's clearly nonsensical. It pretends there's > a time between noon and all times before noon. I only claim there is a time between any time before noon and noon. > >>>>>> You really > >>>>>> don't understand the implications of the Zeno machine, do you? > >>> As I am not using one, that is irrelevant. > >> So, now you're doing it in linear time? Let me know when you're done.... > > > > It is the problem that uses linear time, 60 seconds to 1 minute, and so > > on. > > The iterations do not occur in linear time. They occur in linear time but not at equally spaced intervals in that time. Time being linear merely means that all times can be lined up in order. > >>> I can grasp logic well enough, but from TO I have not seen any. > >> You have to step out of the cave to see it in the light, Virgil. They're > >> only birds...... > > > > TO has obviously been standing under those birds at the wrong time too > > often. > > heh heh. good one, Virge. Try showering, TO. if you use enough soap. it will even get the bird do off.
From: Virgil on 22 Oct 2006 14:54 In article <453b32a5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Does anything occur in the vase at noon? If not, then it should have the > same state as before noon. At what time before noon? Since there are different states at different times before noon, TO must pick a time at which he claims the state is the same as at noon if he wishes to claim that there is such a time before noon.
From: Virgil on 22 Oct 2006 15:10
In article <453b34c5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > It happens over the interval during which balls are being inserted AND > > REMOVED. > > That would be before noon, then. However, at all times before noon, > there are balls in the vase. So, it couldn't occur during that time > interval. So which balls are left AT noon, TO? > > As WE do not have any infinitely-numbered balls and the problem does not > > allow for any either, they must be inserted by TO, so exist only in his > > private version. > > > > If we cannot insert infinitely numbered balls, then the experiment > cannot continue until time 0=1/2^n, otherwise known as noon. It can everywhere outside of TOmania. > > > > > > >> All > >> naturally numbered balls will be gone at that point, but the vase will > >> be far from empty. > > > > But it will be empty of all the balls which we are allowed to count. > > If we are only allowed to count the finite iterations, then we are not > ALLOWED to get to noon. WE are allowed, and required, to get to noon, even if TO is hung up in some chronological limbo. > > >>> Hmm. So your "those balls" must have been introduced into the vase in > >>> this mysterious zone "between before noon and noon". But, see, in > >>> mathematics, it's quite clear there is no such zone - here's a proof. > >> If there is no such zone, and nothing changes AT noon, then the state > >> must be the same as it was immediately BEFORE noon. > > > > That assumes an unwarranted "continuity". Since the number of balls as a > > function of time is already endowed with infinitely many discontinuities > > having noon as a cluster point, there is no reason to assume any better > > behaviour at noon. > > Uh, are you saying time itself is not continuous? We are saying that the number of b balls in the vase as a function of time is discontinuous, which even an idiot like TO should be able to see. > >> If at every time > >> before noon there are balls in the vase, then there are still balls in > >> the vase, because nothing happened, at noon. > > > > Assumes "continuity" at a point where continuity cannot occur. > > > > > > In time? It is numbers of balls in the vase as a function of time which is not a continuous function, as even fools can plainly see. >You have a countably infinite number of balls disappearing in a > moment, or even between moments. Can that "occur"? TO may have that, but everyone else has them "disappearing" from the vase one at a time. > > > > > > >> If something DID happen at > >> noon, then it involved infinitely-numbered balls, and the vase has an > >> uncountable number of balls. > > > > Only in TOland. Nowhere else do any such "infinitely-numbered balls" > > exist. > > So, one can have a finitely numbered ball n such that 1/2^n=0? That's a > fancy trick. TO may want to have something idiotic like that but no one else sees any need for such idiocies. > > In the world as described by the problem, there is no need for any of > > TO's dreams. > > No, just for time discontinuities and denial of the fact that it's a > simple infinite series while obfuscating the situation with a Zeno > machine, and for what? To sound magical and smart? Oy. As any discontinuity in time is the direct result of a discontinuity in TO's thinking process, and putative existence any Zeno machine is equally a result of that same hiccup in TO's mentation, TO is the only one who pays any attention to them, or needs to. |