An uncountable countable set
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From: Tony Orlow on 21 Aug 2006 23:08 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Dik T. Winter wrote: > > <snip> > >>> Division by 0 can not be handled in a ring. >> According to MathWorld, indeed, there is an exception for 0 in the >> optional Multiplicative Inverse condition. Only required due to the >> aversion to infinitesimals. Alas! Why cannot we tie this all together? >> >> http://mathworld.wolfram.com/Ring.html > > Thanks for that link - it includes a motivation for the name 'ring'. My pleasure. > > As for "tying together", well if you take the axioms of a ring (even I > think the absolute minimal set of axioms: group under addition, > semigroup under multiplication, distributive law), you have (for any x, > y, in the ring): > > x . 0 = x . ( y - y ) = x . y - x . y = 0 > > Note that we only rely on the *additive* properties of inverses, plus > the distributive law to get this. Therefore there cannot be a nonzero p > such that x . 0 = p, which is what we would need to have a > multiplicative inverse of zero. That is all very true of absolute 0 and oo. Where you instead substitute a measurable infinity for oo, and its inverse for 0, then that infinitesimal value is greater than 0, and the equation doesn't hold. e=(y+e)-y > > Of course, you can append an object to a ring and call it Bigun (or > anything else) and investigate the resulting structure (see javascript > and my lens calculators for a practical example), but this structure > will not be a ring. Big'un already exists in 2's complement as 1000... It's its own additive inverse, and not 0. > > You may huff and puff at this if you like. I am not sure that will topple the walls of the Garden, so maybe I'll march around them with drums or something. ;) > > Brian Chandler > http://imaginatorium.org > Smiles, Tony
From: Tony Orlow on 21 Aug 2006 23:27 Dik T. Winter wrote: > In article <ecb84s$b1q$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > > With the addition of infinitesimals and specific infinite quantities, > > > > even division by 0 can be handled, probably even for complex numbers, > > > > > > Division by 0 can not be handled in a ring. > > > > According to MathWorld, indeed, there is an exception for 0 in the > > optional Multiplicative Inverse condition. Only required due to the > > aversion to infinitesimals. Alas! Why cannot we tie this all together? > > No. Required because if 0 has also to have an inverse in a ring, trivially > there is only one ring. The ring consisting of the single element 0 with > the standard operations. Because, suppose we have a ring. It can be > proven that in a ring 0*a = 0 for every a in that ring. But if 0 has > an inverse, say b, we also have 0*b = 1, and so 0 = 1. > > On the other hand, do not confuse 0 with the infinitesimals. And there > is no aversion to the infinitesimals. Look at the surreals, look at > non-standard analysis. Infinitesimals abound. Still, in order to be > logically consistent, 0 has a special role. Yes, true. Absolute 0, the origin, is a single point, and infinitesimals, neighboring points, are something ever so slightly different. But then again, oo as a concept and a limit is something absolute and different from, say, the number of points per unit of space, or some other infinite unit. To have infinitesimal numbers, you need to have non-absolute infinities also. The ring is broken by the reluctance to allow absolute oo to balance 0 in this respect. So, I agree, I think. Tell me if I'm wrong. :) Tony
From: Tony Orlow on 21 Aug 2006 23:37 Dik T. Winter wrote: > In article <ecb927$ce6$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > > > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > ... > > > > > > > > > Indeed, I agree with WM's logic concerning the identity relationship > > > > between element count and value in the naturals. He's quite correct in > > > > that regard. > > > > > > Well, you and he are not. The logic is flawed. > > > > How so, precisely? > > Because you by the same logic the ordinal number of the set of naturals > is *larger* than the largest natural, or *smaller*. No, I say the ordinal is larger than the greatest finite, since that's how the ordinals are defined, but that they, and he cardinals, really don't properly represent the naturals. As far as I' concerned, the size of the set of naturals is equal to the largest of them, which doesn't exist properly. Claiming the set is larger than any finite I consider bogus. > > > > > I'm a devoted post-Cantorian delver into > > > > infinity. While he takes the argument put forth as proving that the set > > > > of naturals is finite, he does so with the assumption that infinite > > > > naturals cannot exist. > > > > > > And his proof is not a proof. > > > > When any positive infinite value is considered greater than any finite > > value, and you prove inductively the p(x) is true for x > finite y, > > well, yes, it's a proof. > > You can not prove it inductively based on properties for finite x. > The induction axiom only is about finite x. That is what I am suggesting changing. When starting with the assumption that oo>n for n e N, any proof of f(x)=g(x) for x>n should also apply to oo. > > > > > For my part, I agree that the set of finite > > > > naturals is finite, though unbounded, > > > > > > In that case you are not using standard mathematical terminology. I > > > have no idea what a finite but unbounded set is. > > > > Yes, that's contradiction in terms given the Dedekind definition of an > > infinite set. WM's point is very subtle. > > A *subtle* contradiction in terms? Yes. Given the Dedekind set-theoretic definition, the set is infinite because it is bounded only by finiteness, which offers no clear boundary. For sets truly infinite in membership, it applies, but it misses the subtlety of the Twilight Zone, the lack of boundary between finite and infinite. > > > I know that because I have > > raised it a bunch of times over the last year+, as I think has he. But, > > where there is a constant finite distance on a line between points, and > > no point is infinitely distant from any other, there is a finite range, > > and only a finite number of such disjoint intervals can occupy that > > space. > > Yes, you are arguing that, and WM is arguing that, but offering no proof. Not according to set-theoretic principles, no, but according to numerical and quantitative principles, yes. > > > > > but that there exists an infinite > > > > set of naturals, which includes infinite values. > > > > > > That is alright with me, only, do not call them naturals, because that is > > > extremely confusing. > > > > Yes, I think "hypernaturals" as a superset of the naturals is best, > > though that name has been used, and may carry some unwanted ssumptions > > with it. Still, for now, "hypernaturals"? > > That is probably less confusion, although it also has been used. > > > Dik - have you given much thought to infinite values? > > I have studied the surreals quite thoroughly. And does that lead you to any thoughts as to how one might improve the precision of comparisons between the sizes of infinite sets? Thanks, Tony
From: Tony Orlow on 22 Aug 2006 00:07 Albrecht wrote: > Tony Orlow schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> Franziska Neugebauer schrieb: >>> >>>> You have agreed to that this limit does not exist ("There is no L in >>>> N"). So the sum is at least _not_ _finite_ (not in omega). >>>> >>> There is no L, nowhere. It is wrong to say that L is in |N and it is >>> wrong to say that L is larger than any n e |N. Actual infinity does not >>> exist! I told you that already several times. >>> But refuse to understand what "to exist" means. In no case infinitely >>> many differeces of 1 can exist unless infinitely many diferences of 1 >>> do exist. But that means an infinite size. >>> >>> Regards, WM. >>> >> Well, here you answer my question just posed to you in my last post. >> There is no infinite number, in your opinion. But, how do you reconcile >> this with the infinity of reals in any unit interval? If, between any >> two distinguishable reals there is a third distinguishable from the >> first two, then does this not imply that we can subdivide the unit >> interval infinitely, yielding an infinite set of interval endpoints, aka >> reals? Isn't the set of reals in (0,1] infinite, and how do you >> characterize this number? >> > > Hi Tony, how do you do? Hi Albrecht - Very well, thanks, and you? How have you been? Nice to see you in the arena. :) > > Maybe the following thoughts may help to find an answer to your > question: > > A difference is, if we say: > - the line (0,1] consists of infinite many points (reals) > or > - we can found infinite many points (reals) on the line. > > The first statement is wrong. A point has no extension. Okay, I see where you're coming from, but if the infinity of points cannot contribute to the length of the line segment due to a point's lack of extension, and the line must contain something else to give it this extension, then that doesn't change the fact that there are an infinity of distinct points within that segment. Does it? Just because points aren't all a line segment consists of doesn't mean there aren't an infinite number of them in there. > So, you can put > as many points together as you want, you can't build up an extension as > e.g. a line of lenght 1. I think that's debatable. No finite number of contiguous points will be finitely measurable. But, an uncountable number of points can occupy finite space with little else to discern. I agree there are problems with point-set modes of thought, as I discovered while refuting a counterexample to infinite-case induction. Ah, another staircase example.... > We must conclude that we can only say: we can find infinite many points > on a line. This is a potential infinity. If we can find them there, then there they exist, already, actually. > > Now the analogue phrases are: > - the set of natural numbers consists of infinite many numbers > or > - we can found infinite many natural numbers. I don't agree with either one, unless your including infinite hypernaturals, personally. > > Here, the first statement is as wrong as above. So, the second > statement must be the right one. The infinity of the amount of natural > numbers is a potential one. An infinite set doesn't exist. Well, that second set is finite but unbounded, in my world, whereas the uncountable infinities are actual. There is an actual natural infinity, but it's not standard. > Infinite sets are self contradicting. That depends how they're treated. I'm not at all fond of ordinals, cardinals, or any other part of transfinite set theory. There are good alternatives. > > Greetings > Albrecht S. Storz > Felicitations! Tony
From: Virgil on 22 Aug 2006 01:43
In article <ecdpc8$jl1$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ecb1o9$h$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Virgil wrote: > > > >>> Is TO off on his "largest natural" kick again? > >>> > >> Because it's a set of consecutive naturals starting at 1. > > > > It doesn't matter where it starts, the issue is whether it ends with a > > "largest natural". In standard mathematics it does not. > > TO seems to switch postions erratically on the issue. > > > Let's put it this way. The max and the size of the set are equal. If one > exists, the other exists as well, since it's the same. If one does not > exist, then neither does the other. That may be the case in TO's as yet non-existent mythical system, but it is explicitly false in ZF and NBG. > > I have never said I think there is a largest natural. I have said that > some of your assumptions lead to that conclusion. Which false claim TO cannot justify. The assumptions of ZF, ZFC and NBG require the existence of sets which simultaneously are endless with no maximum member but having both an ordinal and a cardinal size. If TO's intuition misleads him to suppose otherwise, he has been advised of his stupidity. > >>>>> There goes TO's delusion again. What ever makes TO think that a process > >>>>> which cannot not end must end? > >>>>> > >>>> If it doesn't end what makes Virgil think he can tack a count on it? > >>> I can "tack on " bijections of endless sets by defining those bijections. > >>> And in standard mathematics that is one way of counting. > >>> > >> But since there is no end to counting, there is no particular count. > > > > If one can prove that some function is a bijection between some ordinal > > and the set to be counted, the counting has been completely achieved by > > that proof, at least for those not off in TO's never never land of > > TOmania. > > For those that ascribe to the von Neumann model of the naturals, sure. I > don't. Since TO cannot find anything wrong with the vN model except that he does not like it, that hardly constitutes evidence. > > Which is sufficient justification for refusing to allow TO's extension > > of induction beyond standard induction. > > Or, alternatively, sufficient justification for rejecting > transfinitology and the notion of a specific boundary between finite and > infinite. TO is the one who demands a "specific boundary" between finite and infinite, what we rerquire and what we have is specific definitions of what constitutes a finite set and what constitutes an infinite set. > > >>> The only acceptable forms of induction in ZF or NBG are finite > >>> induction and transfinite induction, neither of which has TO shown he > >>> knows how to apply. > >> But I am explicitly suggesting the alternative, that if f(x)>g(x) for > >> all x>y, then it is true for all infinite x. This leads to a clearer set > >> of conclusions. > > > > TO's alternative also leads to inconsistencies with the status quo ante, > > which is sufficient reason for rejecting it. > > Huh? You mean, if I reject your axiom system, and concoct another, the > fact that yours came first means that contradictions between the two are > the fault of mine? What I mean is that you cannot impose your axioms/assumptions on top of our ZF, ZFC of NBG and then object to the contradictions which arise as being solely due to our axioms. Our axioms sets do not allow your form of induction. Assuming your version in addition to ours gives rise to contradictions, so we reject your assumptions as contradictory. If ever TO completes his own system of axioms, including if he wishes his version of induction, we shall see whether it has obvious inconsistencies. Ours don't. > Hmmmm..... Nah. If my ideas all fit together and > provide at least as robust a system as the status quo, then seniority > really doesn't count. Having a system does count, and while we have several, TO has none. > > > > As TO has shown no talent for recognizing truth, and even seems to have > > a talent for avoiding it, his judgements on where mathematical "truth" > > lies are, at best, untrustworthy. > > > > Define "truth". The tautologies of formal logic are truths. Those are the only ones mathematicians qua mathematicians can be sure of. Anything else is uncertain. > >> No, transfinitology doesn't satisfy my spiritual needs. > > > > For spiritual needs, one needs something a little less literal minded > > than mathematics. > > What makes you think mathematics is literal? It's certainly not > concrete, but very abstract. What the numbers refer to is anyone's guess > when they're doing the algebra. That's the beauty. It only becomes > literal upon application. Absolutely backwards! Mathematics is at its most literal minded when most abstract. To the extent that it becomes applied it loses its literal mindedness. > > > > >>> What does one do for sets which are not ordered, or ordered sets which > >>> have no max?, both of which are abundant. > > > > > >> Examples? Symbolic sets (languages) need not be ordered to be measured, > >> necessarily, and sets which have no distinct range have no distinct > >> size, unless the elements themselves have no relative measure. > > > > The multidimensional point sets of multidimnsonal spaces, such as real > > vector spaces, are not ordered and are not orderable in any way > > consistent with their geometric properties. > >. It's not linear Thus, being non-linear, it does not fit on any number line. By which TO admits his error. |