An uncountable countable set
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From: Tony Orlow on 17 Sep 2006 11:38 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>> Mike Kelly wrote: >>>> >>>>> [ ... snip ... ] It's not clear to me that providing finite examples then >>>>> saying "obviously this holds for infinite cases too" without any >>>>> justification whatsoever should be at all convincing to anyone. >>>> It may be not clear to any mathematician, but it is clear to any >>>> scientist. The reason is that infinities do not really exist. >>>> They only exist as an attempt to make the "very large" rigorous >>>> in some sense. The moment you forget this, you get into trouble. >> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find >> it objectionable to say that this also applies to any infinite value, if >> such a thing existed, given that any infinite value would be greater >> than any finite value, and therefore greater than 2? >> >>> But we are discussing whether there exists a uniform distribution over >>> the naturals. If you don't think this claim means anything at all then >>> why do you dispute it? If you reject the existence of the set of >>> natural numbers then you reject the set theory probability is based on. >>> So why bother to argue against individual theorems? You don't accept >>> *any* of probability theory. >> Just because someone disagrees with the transfinite portions of set >> theory doesn't mean they reject all of set theory. Clearly those of us >> who object do so on the basis of the conclusions drawn in infinite case, >> which derive from the axiom of infinity and/or the axiom of choice. > > So, which do you reject? The axiom of infinity or the axiom of choice? > >> As >> far as probability goes, it certainly depends on the concept of sets, >> since probability more or less measures a subset of events with respect >> to the entire set of possible events. However, the same question remains >> as with the rest of transfinitology - is the cardinality generalization, >> based solely on raw bijection, really the most appropriate >> generalization from the finite to the infinite for sets? > > Irrelevant. To what? > >> Do we need to >> know the last element and exact range to derive a probability for >> something as simple as "n is a multiple of 3"? > > No, but we need to know that it is possible to define a uniform > distribution on the set. Which requires an average, which requires a range. > >>> It seem your argument is based on the idea that infinites do not exist >>> in physical reality. But mathematics is abstract, so this seems an >>> absurd objection. >> I think if Wolfgang and Han were offered a more sensible treatment of >> the infinite case, they might find it more palatable. > > Uh, like yours you mean? Snicker. Han rejects completely the existence > of a "completed infinity". This is pretty close to the opposite of what > you're trying to do. That is true, for the reason I just mentioned. The standard streatment is nonsensical. > >>> If you refuse the idea of infinite sets, what does it mean to you to >>> say a function has domain and range R? >>> >> What does it mean to you, if not that one can use that range as a >> constant for infinite sets? Why can't we say that, over the entire range >> of R, the naturals have twice the density of the evens, and so are twice >> as large a set? > > You're free to do so. The evens do have have a density of 1/2 in the > naturals (not in R, I'm not sure you meant to say R. I think they both > have 0 density in R...). If you want to think of this as meaning the > set is twice the "size" then you're free to do so. I said "the range of R", meaning over the real line. > >> Why should set theory contradict so basic an understanding? > > It doesn't. You really don't get it, do you? I've told you half a dozen > times : cardinality doesn't claim to be the only or the "correct" > generalisation of size to infinite sets. > If it claims that the vase is empty, when sequences tell us the vase is not, then a set is not a generality of a sequence. If set theory claims to subsumes all of math, then it cannot contradict any other part of math. TOny
From: Tony Orlow on 17 Sep 2006 11:43 Mike Kelly wrote: > Tony Orlow wrote: >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> Mike Kelly wrote: >>> >>>> Given that any second-year student of probability theory knows that >>>> there are no uniform distributions over countable sample spaces, [ ... ] >>> This "given" is most disturbing. Mainstream mathematics is so certain >>> about its own right that no sensible debate is possible. >> There IS no LUB on the finites, omega notwithstanding. Omega's a >> phantom. That's why you can't get any average value or any uniform >> probability distribution. > > Vaguely correct, minus reflexive whining about Omega. > >> In general, it doesn't make sense to talk >> about probability without a uniform probability distribution over a >> finite set. > > That's absurd. I don't think you meant to say what you said here. Of > course there are non-uniform probability distributions and probability > distributions on infinite sets. Okay. I misspoke. But what about uniform probability distributions on infinite sets in general? > >> However, since probability is really a percentage, > > That is, a real number between 0 and 1. Yes. > >> any >> subset which is a finite fraction of the whole can certainly have a >> probability associated with it: that fraction. > > Only if a uniform distribution can be defined on the whole. Why? A probability IS a fraction. A random number n has x chance of being in any subset of N which is x portion of N. > >> This discussion could not have occurred, say, regarding the primes, >> because over the infinite range of R, n has 0% chance of being prime, >> rather than a 1/3 chance. Still, as every natural has an equal chance, >> in theory, of being selected from the vase o' balls, every natural has a >> chance which is not strictly 0. And the same goes for a random natural >> being prime. >> >> The agreement that I think Han and I came to in "Calculus XOR >> Probability" was that such probabilities are infinitesimal. > > Probabalities are never infinitesimal. They are real numbers between 0 > and 1. > Everything between 0 and 1 is a real number. An infinitesimal is a real less than any finite real, the recioprocal of any infinite real, which is greater than any finite real. The reciproacl of anything greater than 1 lies in [0,1].
From: Mike Kelly on 17 Sep 2006 12:23 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>> Mike Kelly wrote: > >>>> > >>>>> [ ... snip ... ] It's not clear to me that providing finite examples then > >>>>> saying "obviously this holds for infinite cases too" without any > >>>>> justification whatsoever should be at all convincing to anyone. > >>>> It may be not clear to any mathematician, but it is clear to any > >>>> scientist. The reason is that infinities do not really exist. > >>>> They only exist as an attempt to make the "very large" rigorous > >>>> in some sense. The moment you forget this, you get into trouble. > >> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find > >> it objectionable to say that this also applies to any infinite value, if > >> such a thing existed, given that any infinite value would be greater > >> than any finite value, and therefore greater than 2? > >> > >>> But we are discussing whether there exists a uniform distribution over > >>> the naturals. If you don't think this claim means anything at all then > >>> why do you dispute it? If you reject the existence of the set of > >>> natural numbers then you reject the set theory probability is based on. > >>> So why bother to argue against individual theorems? You don't accept > >>> *any* of probability theory. > >> Just because someone disagrees with the transfinite portions of set > >> theory doesn't mean they reject all of set theory. Clearly those of us > >> who object do so on the basis of the conclusions drawn in infinite case, > >> which derive from the axiom of infinity and/or the axiom of choice. > > > > So, which do you reject? The axiom of infinity or the axiom of choice? > > > >> As > >> far as probability goes, it certainly depends on the concept of sets, > >> since probability more or less measures a subset of events with respect > >> to the entire set of possible events. However, the same question remains > >> as with the rest of transfinitology - is the cardinality generalization, > >> based solely on raw bijection, really the most appropriate > >> generalization from the finite to the infinite for sets? > > > > Irrelevant. > > To what? > > > > >> Do we need to > >> know the last element and exact range to derive a probability for > >> something as simple as "n is a multiple of 3"? > > > > No, but we need to know that it is possible to define a uniform > > distribution on the set. > > Which requires an average, which requires a range. What does "requires an average" mean? Loosely speaking, to define a uniform distribution to select an element from a set one assigns a contsant probability to each element such that they all sum to 1. No such contstant exists for countable sets. > >>> It seem your argument is based on the idea that infinites do not exist > >>> in physical reality. But mathematics is abstract, so this seems an > >>> absurd objection. > >> I think if Wolfgang and Han were offered a more sensible treatment of > >> the infinite case, they might find it more palatable. > > > > Uh, like yours you mean? Snicker. Han rejects completely the existence > > of a "completed infinity". This is pretty close to the opposite of what > > you're trying to do. > > That is true, for the reason I just mentioned. The standard streatment > is nonsensical. You really do have a hugely inflated opinion of yourself. The standard treatment makes a huge amount of sense. That you are unable to grasp it points to a problem with you, not with the treatment. If you additionally think that anyone but you is ever going to find sense in your treatment, I'd call you delusional. > >>> If you refuse the idea of infinite sets, what does it mean to you to > >>> say a function has domain and range R? > >>> > >> What does it mean to you, if not that one can use that range as a > >> constant for infinite sets? Why can't we say that, over the entire range > >> of R, the naturals have twice the density of the evens, and so are twice > >> as large a set? > > > > You're free to do so. The evens do have have a density of 1/2 in the > > naturals (not in R, I'm not sure you meant to say R. I think they both > > have 0 density in R...). If you want to think of this as meaning the > > set is twice the "size" then you're free to do so. > > I said "the range of R", meaning over the real line. Well, this is your own terminology that really doesn't mean anything much to me. "The number line" is a visual aid used in kindergarten, not a central part of formal mathematics. You've certainly never defined it. The evens have a density of 1/2 in the naturals. Nobody disputes this. If you want to think of this as meaning the naturals are twice the "size" of the evens, you're free to do so. What exactly is your problem? > >> Why should set theory contradict so basic an understanding? > > > > It doesn't. You really don't get it, do you? I've told you half a dozen > > times : cardinality doesn't claim to be the only or the "correct" > > generalisation of size to infinite sets. > > > > If it claims that the vase is empty, when sequences tell us the vase is > not, then a set is not a generality of a sequence. Nonsensical sentence as far as I can tell. What would it mean for a set to be "a generality of a sequence"? > If set theory claims > to subsumes all of math, then it cannot contradict any other part of math. Set theory doesn't claim to subsume all of math. People use it in (almost) every area of math because it works extremely well. Meanwhile, you continue to argue against a strawman. Cardinality doesn't claim to be the best or only generalisation of size to infinite sets - only the most general. It doesn't deny the possiblity of other ways of measuring "size". You really can't bring yourself to acknowledge that all your arguments against cardinality are totally lacking in substance. Fine. Continue wasting your time railing against something you don't even understand. -- mike.
From: Mike Kelly on 17 Sep 2006 12:35 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> Mike Kelly wrote: > >>> > >>>> Given that any second-year student of probability theory knows that > >>>> there are no uniform distributions over countable sample spaces, [ ... ] > >>> This "given" is most disturbing. Mainstream mathematics is so certain > >>> about its own right that no sensible debate is possible. > >> There IS no LUB on the finites, omega notwithstanding. Omega's a > >> phantom. That's why you can't get any average value or any uniform > >> probability distribution. > > > > Vaguely correct, minus reflexive whining about Omega. > > > >> In general, it doesn't make sense to talk > >> about probability without a uniform probability distribution over a > >> finite set. > > > > That's absurd. I don't think you meant to say what you said here. Of > > course there are non-uniform probability distributions and probability > > distributions on infinite sets. > > Okay. I misspoke. But what about uniform probability distributions on > infinite sets in general? They don't exist on countably infinite sets. One can have continuous uniform distributions on real intervals [a,b]. > >> However, since probability is really a percentage, > > > > That is, a real number between 0 and 1. > > Yes. > > > > >> any > >> subset which is a finite fraction of the whole can certainly have a > >> probability associated with it: that fraction. > > > > Only if a uniform distribution can be defined on the whole. > > Why? A probability IS a fraction. A random number n has x chance of > being in any subset of N which is x portion of N. Only if there is a uniform distribution on N. There isn't. > >> This discussion could not have occurred, say, regarding the primes, > >> because over the infinite range of R, n has 0% chance of being prime, > >> rather than a 1/3 chance. Still, as every natural has an equal chance, > >> in theory, of being selected from the vase o' balls, every natural has a > >> chance which is not strictly 0. And the same goes for a random natural > >> being prime. > >> > >> The agreement that I think Han and I came to in "Calculus XOR > >> Probability" was that such probabilities are infinitesimal. > > > > Probabalities are never infinitesimal. They are real numbers between 0 > > and 1. > > > > Everything between 0 and 1 is a real number. An infinitesimal is a real > less than any finite real, the recioprocal of any infinite real, which > is greater than any finite real. The reciproacl of anything greater than > 1 lies in [0,1]. *sigh*. Probabilities are *standard* real numbers between 0 and 1. -- mike.
From: David R Tribble on 17 Sep 2006 13:08
Tony Orlow wrote: >> If you remove an element, the proper subset should ALWAYS be smaller by >> 1. That is the case for me. For a theory to claim a proper subset is the >> same "size" as the proper superset is an immediate deal-breaker for me. > Tony Orlow wrote: >> N maps to S using f(n)=n+1. The inverse of that function is >> g(x)=x-1. > David R Tribble wrote: >> Which proves that every n in N has an x in S. > Tony Orlow wrote: >> So, over the range of 0 to N, |S|=|N|-1. > David R Tribble wrote: >> Funny how you don't define what |X| is. You're using standard >> symbols but obviously with a different meaning, since "|X|" means >> "cardinality of X" when X is a set. Your IFR bijection proves that >> |S| = |N|. > Tony Orlow wrote: >> |X| means size of, like the absolute value of a real. > What is its relation to your IFR (bijection) function? Your entire argument rests on the assumption that the "size" of a set is equivalent to some mapping function, that |S| ~ f. What is that equivalence relation? In set theory, the cardinality |A| of set A is defined as equal to the cardinality |B| of another set B if there exists a bijection between the two sets. Your mapping function f above proves that such a mapping exists between S and N, so by definition the two sets have the same cardinality, i.e., |S| = |N|. You obviously have something else in mind for "size of a set" which depends on your IFR "mapping with measure" function in a different way than cardinality does. What is it? |