An uncountable countable set
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From: MoeBlee on 20 Sep 2006 16:44 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >>> It isn't. What does "successor to N" even mean? > >> Ask von Neumann. It is the set of all naturals, > > > > What are you talking about? The successor of w is not w. The successor > > of w is wu{w}. > > The size of the set x is the value, and it contains every natural less > than x. Is this not the generalizable scheme? Whatever that T-means, it doesn't change that w is not the successor of w. > >> and any ordinal being > >> the set of all preceding naturals, > > > > What are you talking about? An ordinal is the set of all preceding > > ordinals, not necessarily the set of all preceding naturals. > > Aren't finite ordinals and cardinals and naturals the same thing? Perty > much. the set of finite ordinals = the set of finite cardinals = the set of natural numbers. Yes. But that doesn't entail taht an ordinal is the set of all preceding naturals. > I am well aware of the predecessor discontinuities which allow for > "well" ordering. A set has a well ordered or it does not, and doesn't need to be "allowed" to be well ordered. MoeBlee
From: Virgil on 20 Sep 2006 16:46 In article <4511474c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Given N is the standard naturals: > >> > >> Actually infinite(s) = E seS A neN index(s)>n > >> Potentially infinite = A seS index(s)eN ^ ~E neN index(s)<>n > > > > First, that doesn't even make sense as a definition, since you have a > > free variable 'S' (upper case 'S') that is in the definiens but not in > > the definiendum. > > Sorry, hastily written. The 's' in the definiendum should be an 'S', > obviously, and the second rule isn't quite right, on second thought. > > >> Actually infinite(S) <-> E seS A neN index(s)>n > >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > > > > > Next, what is the logistic system? > > First order logic > > What are the non-logical primitives? > e > > > index > > > What are the non-logical axioms? > > These are definitions of potential vs. actual infinity. Consider them > axiomatically stated if you wish. > > What is the definition of 'index'? > > The count of elements up to and including a given element. Then there will be elements of "actually infinite sets" for which no index can be defined. > The number of > steps between and element and the first element in a linear order, plus 1. Then there will be elements of "actually infinite sets" for which no index can be defined. > > > What is the definition of '>' (if it's different from the standard > > 'greater than' relation on natural numbers)? What is the definition of > > '<>'? > > Forget '<>' for now - I removed it. '>' is the standard order relation > on real quantities such that a>b ^ b>c -> a>c. That presumes the reals, with all their structure. What axiom system does TO presume that allows him access to all the properties of the reals without an axiom system to support them?
From: MoeBlee on 20 Sep 2006 16:50 Tony Orlow wrote: > I have nothing against set theory in general. It's the transfinite > portions which are schlock. They lead to nothing useful, or even sensible. Then you won't have the axiom of infinity. Or you won't have both the axiom of infinity and the axiom schema of replacement. Please tell us already what axioms you do have. MoeBlee
From: Virgil on 20 Sep 2006 16:52 In article <451149ef(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Consider the equally spaced staircase from (0,0) to (1,1), as the number > of steps increases from 1 without bound. Is it the same as the diagonal > line? Inductively we can prove that the length of the staircase is 2 at > every step. Does it really suddenly become sqrt(2) in the infinite case? There is no "infintie case", there is only a limit case. If the cases for a finite number of steps are sets of points, so is the limit case. If the finite cases are sets of segments with specific directions determined by their endpoints, the limit case will only contain pairs of identical points which do not determine any direction at all, and so is ill defined.
From: Mike Kelly on 20 Sep 2006 16:54
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Han de Bruijn wrote: > >>>> Mike Kelly wrote: > >>>> > >>>>> Han de Bruijn wrote: > >>>>> > >>>>>> Mike Kelly wrote: > >>>>>> > >>>>>> > >>>>>>> Han de Bruijn wrote: > >>>>>>> > >>>>>>> > >>>>>>>> Mike Kelly wrote: > >>>>>>>> > >>>>>>>> > >>>>>>>>> Han de Bruijn wrote: > >>>>>>>>> > >>>>>>>>> > >>>>>>>>>> Mike Kelly wrote: > >>>>>>>>>> > >>>>>>>>>> > >>>>>>>>>>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>>>>>>>>> > >>>>>>>>>>> > >>>>>>>>>>>> Mike Kelly wrote: > >>>>>>>>>>>> > >>>>>>>>>>>> > >>>>>>>>>>>>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>>>>>>>>>>>> to schoolkids. > >>>>>>>>>>>> Look who is talking. Good luck explaining alpha_0 to schoolkids. > >>>>>>>>>>> Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > >>>>>>>>>>> this is a bad analogy, because school kids don't need to know about > >>>>>>>>>>> cardinals but they do need to know how to work with natural numbers. My > >>>>>>>>>>> point, if you really missed it, was that Tony's ideas of "infinite > >>>>>>>>>>> natural numbers" don't match up to our "naive" or "intuitive" idea of > >>>>>>>>>>> what numbers should be - how we were taught to do arithmetic in school. > >>>>>>>>>>> I for one don't understand what the hell an "infinite natural number" > >>>>>>>>>>> is. And yet supposedly the advantage of his ideas are that they're more > >>>>>>>>>>> intuitive than a standard formal treatment. > >>>>>>>>>> My point is that the pot is telling the kettle that it's black (: de pot > >>>>>>>>>> verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than > >>>>>>>>>> Tony's "infinite natural number". > >>>>>>>>> Your analogy is terrible, as usual. > >>>>>>>>> > >>>>>>>>> My point was that Tony's "infinite natural numbers" are not compliant > >>>>>>>>> with everyday arithmetic. Aleph_0 is part of a formalisation that leads > >>>>>>>>> to an arithmetic that works exactly as we expect it to. > >>>>>>>> "... that works exactly as we expect it to". Ha, ha. Don't be silly! > >>>>>>> So, what part of the arithmetic on natural numbers defined rigorously > >>>>>>> as sets doesn't match up to the "naive" arithmetic we were taught at > >>>>>>> school? > >>>>>> I thought you meant the arithmetic with transfinite numbers. No? > >>>>> In what way is the arithmetic of transfinite numbers part of everyday > >>>>> arithmetic??? > >>>> Precisely! > >>> What the hell are you talking about? Arguing with someone who can't > >>> speak English is getting aggravating. > >> This isn't a language issue. Han is saying that transfinitology has > >> NOTHING to do with everyday arithmetic. > > > > Even though set theory leads to an arithmetic on natural numbers that > > is identical to everyday arithmetic? > > I have nothing against set theory in general. It's the transfinite > portions which are schlock. They lead to nothing useful, or even sensible. But you've never studied mathematics, so how on Earth would you know how mathematically useful transfinite set theory is? "It contradicts my intuition and is therefore not useful". Heh. Bonus question : is your set theory going to lead to more useful results? > >> That's the point. It doesn't fir > >> into mathematics. The conclusions are absurd. To quote George Boole, > >> inventor of the system which allows you to confirm the deductive > >> consistency of your axiom systems, in his "An Investigation Into The > >> Laws Of Thought": > >> > >> "Let it be considered whether in any science, viewed either as a system > >> of truths or as the foundation of a practical art, there can properly be > >> any other test of the completeness and fundamental character of its > >> laws, than the completeness of its system of derived truths, and the > >> generality of the methods which it serves to establish." > >> > >> Where the conclusions are incorrect, > > > > What does it mean for a conclusion to be incorrect? That it is not a > > logical consequence of one's assumptions? > > It means that it contradicts the greater part of our knowledge in the > area, so there's something wrong, either with the assumptions/data, or > the rules of inference. The first is a discovery problem, and the second > a matter of mathematics and logic. I would suggest that, while there > area couple of unresolved questions in logic, that the fault with > transfinitology lies in its starting assumptions. Given that the only axiom of ZFC you claim to reject is Choice, I don't see this can be the case. Choice isn't necessary to derive a lot of reuslts you seem to find objectionable. > >> where what is considered the > >> "foundation" of mathematics contradicts many particular areas of > >> mathematics, > > > > For example? > > > >> it can only be properly rejected as reflecting the > >> fundamental truths upon which math is founded. > > > > (I think you meant to say "not reflecting") > > > > What fundamental truths would those be? > > > >>> I claim that Aleph_0 is part of a formalisation that leads to an > >>> arithmetic on natural numbers that works just how naive arithmetic > >>> works. Do you disagree? > >>> > >> Yes, wholeheartedly. In finite arithmetic, when you add a nonzero > >> quantity, you increase the value - not so in transfinitology. You can > >> remove elements, divide the set in half, double it, add elements, all > >> without changing what is supposed to be the measure of the set. That's > >> not how it works in the finite realm. > > > > That's not how it works in "the finite realm" in set theory, either. > > Apparently you're completely missing my point, too. > > What? No, you are missing MY point. It's not CONSISTENT with the rest of > mathematics. How is it not consistent? Apparently your argument is "infinite sets behave differently from finite sets". Stupid, stupid, stupid. >It's a FICTION. What does it mean for a piece of mathematics to be a fiction? > > I pointed out that your "infinite integers" contradict with everyday > > arithmetic. > > Not all operations possible with finite numbers are possible with the > T-riffic numbers, but I don't see as they contradict finite numbers in > any way. Double standards huh? It's ok for your infinite whatevers to behave differently from finite whatevers. Why? > > Han responded wit |