An uncountable countable set
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From: Tony Orlow on 23 Aug 2006 17:27 Virgil wrote: > In article <ecdsfu$om3$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> imaginatorium(a)despammed.com wrote: > >>> Therefore [ in a ring] there cannot be a nonzero p >>> such that x . 0 = p, which is what we would need to have a >>> multiplicative inverse of zero. >> That is all very true of absolute 0 and oo. Where you instead substitute >> a measurable infinity for oo, and its inverse for 0, then that >> infinitesimal value is greater than 0, and the equation doesn't hold. > > That constraint holds in absolutely every ring. > > Until TO defines the entire arithmetical structure of his system and > proves there is a model satisfying that definition, the zero object has > no multiplicative inverse in any arithmetic containing non-zero elements. > >>> Of course, you can append an object to a ring and call it Bigun (or >>> anything else) and investigate the resulting structure (see javascript >>> and my lens calculators for a practical example), but this structure >>> will not be a ring. >> Big'un already exists in 2's complement as 1000... It's its own additive >> inverse, and not 0. > > In what axiom system, TO? > > TO has not shown his alleged 'bigun' is even possible in ZF or NBG and > he has no system of axioms of his own. > > So in what axiom system, TO? You ask for the chicken, I sit on the egg.
From: Tony Orlow on 23 Aug 2006 17:28 Virgil wrote: > In article <ecdtie$q90$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Dik T. Winter wrote: >>> In article <ecb84s$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. :) > > You are wrong! Are you Dik, that you should judge whether he and I agree? Pullllease!!! > > What is your model? What are the axioms, undefined terms and basic > definitions? Absent these, TO is peddling nothing but hot air. My air is warming up.
From: Tony Orlow on 23 Aug 2006 17:31 Albrecht wrote: > Virgil schrieb: > >> In article <1156163893.101419.232240(a)p79g2000cwp.googlegroups.com>, >> "Albrecht" <albstorz(a)gmx.de> wrote: >> >>> Infinite sets are self contradicting. >> Not in ZF or NBG. What are the axioms of Storz's system? > > There is no relevance in which system the axiom is found. > E.g. the Axiom A: "Axiom A is wrong", is self contradicting, regardless > of which other axioms are used, I think. The same holds for the axiom > of infinity. > > Best regards > Albrecht S. Storz > Hi Abrecht - From that statement I would guess you ascribe to the notion of universal consistency within mathematics, that not only must each theory be internally consistent, but the axioms of each theory must not contradict those of any other, since math describes one universe. How far off am I from understanding your position? :) Tony
From: Tony Orlow on 23 Aug 2006 17:35 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: >>>> Dik T. Winter wrote: >>> <snip> > >>> 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. > > Huh? I didn't mention "oo", which is an undefined symbol in a ring in > general. I also didn't mention "absolute 0", and have _no_ idea what > such a thing would mean, unless it is simply zero. Simply zero and nothing else, like an infinitesimal or something. > >> .... 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. > > I don't understand what you mean by "substitute". Look, I find it very > difficult to keep track of what you're trying to show at any particular > stage, since you seem to leap around tinkering with this and that, > hoping to make your intuitions come true. What are we doing now? You > have been told that a ring cannot have an inverse of zero unless it is > the trivial single-element ring. Several proofs of this have been > given, none more than about two lines long. Are you trying to argue > that this is "wrong" in some sense, or are you trying to claim that you > have something "better", in which zero does have an inverse (an > "O-ring" I suppose). > > We could consider something much simpler than an infinite set here. > Suppose Z5 is the field of order 5. It contains five elements (I think > of it as the "five number circle"), we'll call 0, 1, 2, 3, 4, and here > are the addition and multiplication tables: > > + 0 1 2 3 4 > 0 0 1 2 3 4 > 1 1 2 3 4 0 > 2 2 3 4 0 1 > 3 3 4 0 1 2 > 4 4 0 1 2 3 > > x 0 1 2 3 4 > 0 0 0 0 0 0 > 1 0 1 2 3 4 > 2 0 2 4 1 3 > 3 0 3 1 4 2 > 4 0 4 3 2 1 > > Of course 0 has no inverse (that is, no nonzero element appears in the > multiplication row or column for 0). Suppose you see this as a problem; > please explain (very slowly, and you should be able to be explicit I > hope) how you would modify Z5 to make an O-ring in which 0 does have an > inverse. > > In particular, if this starts by "substituting" something, you could > show the elements in the O-ring after the substitution. You're tiring. Consider 2's complement and 1000... > > >> e=(y+e)-y > > What's this equation? What is 'e'? > > >>> 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. > > Please explain how something "existing in 2's complement" affects the > elements of (for example) Z5. I've given up hoping you'll try to > explain how "Bigun" has a 1 at the left end of a string that extends > leftward without end. > Left of the 1 are all 0's. Like I say, you're tiring. You're smart enough to get it, but spend too much time making fun of it to make sense of it. So, struggling against that's a waste of my time. > > Brian Chandler > http://imaginatorium.org >
From: Tony Orlow on 23 Aug 2006 17:40
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> 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. > > So should we understand that in Tonyspeak "absolute 0" just means what > mathematicians call zero. What is the point of using different words? > The point is that infinitesimal values are considered 0 in the standard math, and where we're discussing infinitesimal, it's prudent to distinguish between them and innfitesimal values. > >> 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. Yes, non-absolute infinites are "specific" infinites, by which you've asked me what I meant in the past. I mean non-absolute, not-all-the-way-out-there infinities, but specific points infinitely far from the origin. > > What do you mean by a "ring" being "broken"? What would it mean for > "absolute oo" (whatever that is, exactly) to "balance" 0? Who is > reluctant? Suppose I agree not to be reluctant: what happens then? > Then you put aside your taught assumptions for a moment, and consider a different approach. I'll hold my breath in the meantime. > > Brian Chandler Tony |