An uncountable countable set
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From: imaginatorium on 23 Oct 2006 14:47 MoeBlee wrote: > Lester Zick wrote: > > You and he just have different perspectives on the problem and nothing > > in what you have to say has any relevance to what Tony believes any > > more than what Tony believes has any relevance to what you believe. > > Differing perspectives are welcome. But that is different from simply > saying incorrect things about the technical points and also from giving > the kind of woozy arguments he gives for his non-axiomatic mathematics. Uh, do you have a technical definition for "woozy" at this point? Sounds rather fascinating... Brian Chandler http://imaginatorium.org
From: Tony Orlow on 23 Oct 2006 14:49 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. > > Why are these "infinitely-numbered balls" inserted, then? Because the experiment of inserting ten and removing one, repeatedly, has extended to a time that requires the numbers on the balls to be infinite. The rules > quite explicitly say that no ball is inserted unless it has a (finite) > natural number written on it. Then every ball is inserted and removed at a finite time before noon, but noon itself is never reached. 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? The rules are not being ignored. The experiment cannot continue until noon because that requires infinite naturals, but if it stops before noon it is not completed. During all the time that the experiment can continue without allowing infinite values of n, there is a steadily growing number of balls in the vase, nine per iteration, each more quickly than the last. But, noon cannot be reached with the pofnats. > > 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)? At noon, all finitely numbered balls have been inserted and removed, but in order for noon to arrive, an uncountably infinite number of balls with infinite values have to have been inserted. 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? No, that's idiotic. At noon there is a condensation point in time due to the Zeno machine where a finite process is being executed infinitely quickly. That's different from Zeno's Paradox, where the distance covered is equal to the time elapsed, both of which shrink exponential, so there is a finite limit to a process with an infinite number of steps. Here, each step has the same finite difference, and the Zeno Time Machine causes an infinite number of successions in a moment. So, you get uncountable infinity at t=0. Finite naturals can't get you there, and where they do get you, it's obvious the vase is far from empty. > > Brian Chandler > http://imaginatorium.org > Tony
From: Virgil on 23 Oct 2006 14:49 In article <453caf87(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> You have agreed with everything so far. At every point before noon > >>>>>> balls > >>>>>> remain. > >>>>> To be precise, the assertions above all imply that at every time t = > >>>>> -1/n, where n is a natural number, there are balls in the vase. > >>>>> > >>>>> But that *alone* does not even include every time t before noon; let > >>>>> alone every time t. For example, notice that nowhere above do you or I > >>>>> /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a > >>>>> positive finite number". > >>>>> > >>>>> We assert something specific about t = -2/4, and something specific > >>>>> about t = -1/3, but nowhere do we directly state somthing about t = > >>>>> -2/3. > >>>>> > >>>>> On the other hand, given the problem statement, I think we would both > >>>>> /agree/ that there "should be" an obvious (perhaps even unique) > >>>>> well-defined answer to the question : "what is the number of balls in > >>>>> the vase at time t = -1/pi?" > >>>>> > >>>>> Assuming in the remaining statements that one agrees with the previous > >>>>> statement, this leads us to the question: what are the unstated > >>>>> assumptons that allow to agree that this must be the case? > >>>>> > >>>>> I attempted to describe those assumptions in my previoius post. Did you > >>>>> read those assumptions? If so, do you agree with those assumptions? > >>>> At this point I don't recall your previous post. I've been off a bit. > >>> Well, allow me to repeat them here (with two minor changes): > >>> > >>> In order to interpret the problem > >>> > >>> "At each time t = -1/n where n is a (strictly positive) natural number, > >>> we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > >>> vase, and remove the ball labelled n from the vase. What is the number > >>> of balls in the vase at time t=0?" > >>> > >>> I make the following simple (and I would claim, fairly uncontroversial > >>> and natural) assumptions: > >>> > >>> --- (object permanence) > >>> > >>> (1) When we speak of a time t, we mean some real number t. > >>> > >>> (2) If a ball is in the vase at any time t0, there is a time t <= t0 > >>> for which we can say "that ball was placed in the vase at time t". > >>> > >>> (3) If a ball is placed in the vase at time t1 and it is not removed > >>> from the vase at some time t where t1 <= t <= t2, then that ball is in > >>> the vase at time t2. > >>> > >>> (4) If a ball is removed from the vase at time t1, and there is no time > >>> t such that t1 < t <= t2 when that ball is placed in the vase, then > >>> that ball is not in the vase at time t2. > >>> > >>> ---- (obedience to the problem constraints) > >>> > >>> (5) If a ball is placed in the vase at some time t, it must be in > >>> accordance with the description given in the problem: it must be a ball > >>> with a natural number n on it, and the time t at which it is placed in > >>> the vase must be -1/floor(n/10). > >>> > >>> (6) If a ball is removed from the vase at some time t, it must be in > >>> accordance with the description given in the problem: it must be a ball > >>> with a natural number n on it, and the time t at which it is removed > >>> from the vase must be -1/n. > >>> > >>> (7) If n is a natural number with n > 0, then the ball labelled n is > >>> placed in the vase at some time t1; and it is removed from the vase at > >>> some time t2. > >>> > >>> --- (very general definition of "the vase is empty at noon") > >>> > >>> (8) the number of balls in the vase at time t=0 is 0 if, and only if, > >>> the statement "there is a ball in the vase at time t=0" is false. > >>> > >>> --- > >>> > >>> Perhaps you would add other assumptions (9), (10), etc.; but my > >>> question is: > >>> > >>> Given the problem statement, do you agree that /each/ of these > >>> assumptions, /on its own/, is reasonable and not just some arbitrary > >>> statement plucked out of thin air? > >>> > >>> If not, which assumption(s) is(are) not reasonable or is(are) > >>> unneccessarily arbitrary? > >>> > >>> <snip> > >> Those all look reasonable to me as I read them. I don't see any > >> statement regarding the fact that ten balls are added for every one > >> removed, though that can be surmised from the insertion and removal > >> schedule. That's the salient fact here. You never remove as many as you > >> add, so you can't end up empty. > > > > What about #5? It says that every ball in the vase has a natural number > > on it. Do you agree with that? > > That is in the problem statement. Therefore, nothing transpires at noon, > since -1/n<0 for all n e N. That statement offers no problems to those who do not require anything beyond the conditions of the original problem. > > > > >> Either something happens an noon, or it doesn't. Where do you stand on > >> the matter? > > > > What does "something happens" mean, please? I really don't know what you > > mean. > > > > ??? Do you live in the universe, or in a static picture? When "something > happens" o an object, some property or condition of it "changes". That > occurs within some time period, which includes at least one moment. In any physical world, something happening, or changing, requires an interval of time of strictly positive duration to occur. Nothing can "happen" instantaneously in that world. So what does TO mean by "something happening" instantaneously in a mathematical world, for which we have no physical world analog? > There is no moment in this problem where the vase is emptying, > therefore, that never "occurs". The process of "emptying" may not occur, in the sense of the number of balls decreasing from one moment to another at any time before noon, but the result does, in the sense of there being no ball which has not been removed, at noon. > If you are going to insist that time is > a crucial element of this problem, then you should at least be familiar > with the fact that it's a continuum, and that events occurs within > intervals of that continuum. Then the whole problem is cooked, since no ball can be inserted or removed instant
From: Virgil on 23 Oct 2006 14:55 In article <453cb03a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. And where do these allegedly infinitely numbered balls come from? I do not recall any of them being mentioned in the original gedanken, so that TO is creating his own separate gedanken. Note that whatever TO may require in his version of the gedanken, his requirements do not alter that no such thing occurs in the original. TO takes the childish position that if he cannot have things his own way playing by the rules, he will change the rules to get his own way.
From: Tony Orlow on 23 Oct 2006 15:00
Lester Zick wrote: > 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. > What is log2(0)? >>> 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. Technically, the number of reals in the unit interval (0,1] is Big'un. That's also the infinite length of the real number line, in unit intervals. The unit infinitesimal is Lil'un, or 1/Big'un. Now they're all specific and related to spatial measure and quantity. :) > > 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~~ The formulas that lend themselves to L'Hospital's Rule usually cannot be simplified any further to resolve that problem. Subtracti |