An uncountable countable set
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From: Tony Orlow on 23 Oct 2006 08:06 David Marcus wrote: > Tony Orlow wrote: > >> At every time before noon there are a growing number of balls in the >> vase. The only way to actually remove all naturally numbered balls from >> the vase is to actually reach noon, in which case you have extended the >> experiment and added infinitely-numbered balls to the vase. All >> naturally numbered balls will be gone at that point, but the vase will >> be far from empty. > > By "infinitely-numbered", do you mean the ball will have something other > than a natural number written on it? E.g., it will have "infinity" > written on it? > Yes, that is precisely what I mean. If the experiment is continued until noon, so that all naturally numbered balls are actually removed (for at no finite time before noon is this the case), then any ball inserted at noon must have a number n such that 1/n=0, which is only the case for infinite n. If the experiment does not go until noon, not all naturaly numbered balls are removed. If it does, infinitely-numbered balls are inserted.
From: imaginatorium on 23 Oct 2006 10:57 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > >> At every time before noon there are a growing number of balls in the > >> vase. The only way to actually remove all naturally numbered balls from > >> the vase is to actually reach noon, in which case you have extended the > >> experiment and added infinitely-numbered balls to the vase. All > >> naturally numbered balls will be gone at that point, but the vase will > >> be far from empty. > > > > By "infinitely-numbered", do you mean the ball will have something other > > than a natural number written on it? E.g., it will have "infinity" > > written on it? > > > > Yes, that is precisely what I mean. If the experiment is continued until > noon, so that all naturally numbered balls are actually removed (for at > no finite time before noon is this the case), then any ball inserted at > noon must have a number n such that 1/n=0, which is only the case for > infinite n. If the experiment does not go until noon, not all naturaly > numbered balls are removed. If it does, infinitely-numbered balls are > inserted. Why are these "infinitely-numbered balls" inserted, then? The rules quite explicitly say that no ball is inserted unless it has a (finite) natural number written on it. We've had variations on the rules that say that the demon, after going to put a ball in the vase, double-checks, and if the ball doesn't have a pofnat written on it, throws it away. Why in your version of the experiment are the rules just ignored when it seems to give you the answer you want? Also, suppose for the sake of argument, that there _are_ these "infinitely numbered" balls. Are you saying that there is a point at which all of the "finitely numbered" balls have been removed (leaving the vase empty, which isn't what you are hoping for)? Or are you saying there comes a point at which a ball with a number "near the end" of the pofnats is being removed, and at the same time the balls being put in are actually "infinitely numbered"? That appears to imply that there exists a pofnat, call it B, such that B is finite, but 10*B is infinite. Is that right? How does this square with even your confused understanding of the Peano axioms? Brian Chandler http://imaginatorium.org
From: Lester Zick on 23 Oct 2006 10:59 On Mon, 23 Oct 2006 07:57:22 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Sun, 22 Oct 2006 05:38:11 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Fri, 20 Oct 2006 14:12:27 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> MoeBlee wrote: >>>>>> Tony Orlow wrote: >>>>>>> Also, upon which axioms is the definition of cardinality based? >>>>>> The usual definition is: >>>>>> >>>>>> card(x) = the least ordinal equinumerous with x >>>>>> >>>>>> The definition ultimately reverts to the 1-place predicate symbol 'e' >>>>>> (and the 1-place predicate symbol '=', if equality is taken as >>>>>> primitive). For the definition to "work out" ('work out' is informal >>>>>> here) in Z set theory, we usually suppose the axioims of Z set theory >>>>>> plus the axiom of schema of replacement (thus we're in ZF) and the >>>>>> axiom of choice (thus we're in ZFC). However, there is a way to avoid >>>>>> the axiom of choice by using the axiom of regularity instead with a >>>>>> somewhat different definition from just 'least ordinal equinumerous >>>>>> with'. Also, we could adopt a "midpoint" between the axiom schema of >>>>>> replacement and the axiom of choice by adopting the numeration theorem >>>>>> (AxEy y is an ordinal equinumerous with x) instead, which would be a >>>>>> method stronger than adopting the axiom of choice, but weaker than >>>>>> adopting both the axiom of choice and the axiom schema of replacement. >>>>>> As to the more basic axioms of Z, for the definition to "work out", I'm >>>>>> pretty sure we need extensionality, schema of separation (or schema of >>>>>> replacement if we go that way), union, and pairing (pairing is not >>>>>> needed if we have the schema of replacement). I'm not 100% sure, but my >>>>>> strong guess is that we don't need the power set axiom for this >>>>>> purpose. And we don't need the axiom of infinity. >>>>>> >>>>>> Why don't you just a set theory textbook? >>>>>> >>>>>> MoeBlee >>>>>> >>>>> I'm reading Non-Standard Analysis instead. Robinson agrees there's no >>>>> smallest infinity, >>>> Just out of curiosity, Tony, what's his rationale? >>> I quoted it for MoeBlee, and you can see, but basically, he extends N to >>> *N by applying the basic truths about finite numbers to the infinite. >>> Since it is true for all n in N that, except for 0, every element has a >>> predecessor n-1, then this must also be true for any infinite n. We >>> assume some smallest infinite n. Since n-1<n, and since n-1 is infinite >>> if n is infinite, we have n-1 being a smaller infinity than n, which is >>> a contradiction. For any smallest n we can assume, there is a smaller >>> n-1, so there is no smallest infinite, just like there is no greatest >>> finite. >> >> Thanks for the explanation, Tony. I can see what the argument amounts >> to and basically I agree. But I've become extremely skeptical of the >> combination of finites and infinites in arithmetic operations in >> general. I'm beginning to suspect there is no such thing as trans >> finite arithmetic. I think arithmetic works with finites and calculus >> with infinites. And the rest is just so much mathematical pretense. > >Unfortunately, transfinitology exists, despite the fact that it makes no >sense underneath the hood. When it comes to arithmetic on them, it's one >big kludge. But, there are forms of infinite numbers upon which one can >define arithmetic. They just have nothing whatsoever to do with omega or >the alephs. Not sure what you're talking about here, Tony. Lots of things exist in the sense of having been defined. That doesn't make them true and doesn't mean they form any basis for the truth of other things defined on them. There's no shortage of things other than infinity on which to define arithmetic. >> Yet I've also been considering what it looks like you're trying to do >> with trans finite arithmetic.In particular it occurs to me that if one >> takes +00 to be larger than any positive finite -00 correspondingly >> must be smaller than any negative finite such that your concept of >> circularity among arithmetic numbers might be combined in the >> following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only >> difference would be that whereas +00 represents the number of >> infinitesimals, -00 would represent the size of infinitesimals. Thus >> we'd have a positive axis with the number of infinitesimals and a >> negative axis with the size of infinitesimals. At least that's the >> best I can make of the situation. > >Well, I rather think of 1/oo as the size of infinitesimals, or more >precisely, for any specific infinite n, 1/n is a specific infinitesimal >value. When it comes to the number circle, in some ways oo and -oo can >be considered the same so the number line forms an infinite circle, but >in others, such as lim(n->oo) as opposed to lim(n->-oo), there is a very >clear difference between the two. I think it's a bit like the >wave-particle dualism for physical objects, and may actually be directly >connected. Well now you're just back to the idea of arithmetic as some kind of a TOE, Tony. It's very simple. The only mechanical definition for 00 is any finite/0. And if that product can't be defined than neither can 00. There is no specific size to infinitesimals because they're an process not a static thing. Any series of infinitesimals varies in size continuously. There's a reciprocity between number and size for any infinite series but no "circle" between them. On the other hand if you want to do transfinite arithmetic you might ask yourself what the results of 00-00 or 00/00 are. The latter can be addressed through application of L'Hospital's rule but I don't know any way to address the former. ~v~~
From: Randy Poe on 23 Oct 2006 11:03 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > >> At every time before noon there are a growing number of balls in the > >> vase. The only way to actually remove all naturally numbered balls from > >> the vase is to actually reach noon, in which case you have extended the > >> experiment and added infinitely-numbered balls to the vase. All > >> naturally numbered balls will be gone at that point, but the vase will > >> be far from empty. > > > > By "infinitely-numbered", do you mean the ball will have something other > > than a natural number written on it? E.g., it will have "infinity" > > written on it? > > > > Yes, that is precisely what I mean. If the experiment is continued until > noon, The clock ticks till noon and beyond. However, the explicitly- stated insertion times are all before noon. > so that all naturally numbered balls are actually removed (for at > no finite time before noon is this the case), There is no finite time before noon when all balls have been removed. However, any particular ball is removed at a finite time before noon. > then any ball inserted at > noon must have a number n such that 1/n=0, However, there is no ball inserted at noon. > which is only the case for > infinite n. If the experiment does not go until noon, not all naturaly > numbered balls are removed. The experiment goes past noon. No ball is inserted at noon, or past noon. - Randy
From: stephen on 23 Oct 2006 11:36
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> David Marcus wrote: >> > Tony Orlow wrote: >> > >> >> At every time before noon there are a growing number of balls in the >> >> vase. The only way to actually remove all naturally numbered balls from >> >> the vase is to actually reach noon, in which case you have extended the >> >> experiment and added infinitely-numbered balls to the vase. All >> >> naturally numbered balls will be gone at that point, but the vase will >> >> be far from empty. >> > >> > By "infinitely-numbered", do you mean the ball will have something other >> > than a natural number written on it? E.g., it will have "infinity" >> > written on it? >> > >> >> Yes, that is precisely what I mean. If the experiment is continued until >> noon, so that all naturally numbered balls are actually removed (for at >> no finite time before noon is this the case), then any ball inserted at >> noon must have a number n such that 1/n=0, which is only the case for >> infinite n. If the experiment does not go until noon, not all naturaly >> numbered balls are removed. If it does, infinitely-numbered balls are >> inserted. <snip> > Also, suppose for the sake of argument, that there _are_ these > "infinitely numbered" balls. Are you saying that there is a point at > which all of the "finitely numbered" balls have been removed (leaving > the vase empty, which isn't what you are hoping for)? Or are you saying > there comes a point at which a ball with a number "near the end" of the > pofnats is being removed, and at the same time the balls being put in > are actually "infinitely numbered"? That appears to imply that there > exists a pofnat, call it B, such that B is finite, but 10*B is > infinite. Is that right? How does this square with even your confused > understanding of the Peano axioms? > Brian Chandler > http://imaginatorium.org 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. Stephen |