An uncountable countable set
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From: Virgil on 20 Aug 2006 12:44 In article <1156082511.793900.314620(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > The sequence of edges leading up to any edge is finite. So a finite > > sequence of 0's and 1's, 0 for left child, 1 for right child, uniquely > > identifies each edge, but it takes an endless sequence of 0's and 1's to > > identify any endless path. > > > > Thus > > ? do you know what you are talking about? Better than "Mueckenh", by all the evidence. > > > in infinite binary trees, the set of edges is countable but the set > > of endless paths is not.
From: Virgil on 20 Aug 2006 12:51 In article <ec9qh6$762$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Dik T. Winter wrote: > > > Transfinite cardinalities > > > don't really measure anything, though, do they? > > > > They measure something, but not what you want them to measure. > > As far as I can see, they "measure" with a fluid yardstick, based on > non-logical arbitrary axioms. That is because TO can't see very far, and is immune to the persuasions of honest logic but easily swayed by the siren songs of his intuition. > If cardinalities measured set size at all > accurately, then they would always reflect the addition or removal of > elements with a change in value. It's actually really not hard to > achieve that. Except that neither TO nor anyone else has yet managed it without producing fatal inconsistencies. > > That is an area I don't know too much about That area about which TO knows so little encompasses most of logic and mathematics.
From: Tony Orlow on 20 Aug 2006 13:12 David R Tribble wrote: > Tony Orlow wrote: >> ... it's inductively provable likewise that the size of a successor >> ordinal is one greater than its max element. That would make aleph_0 1 >> greater than the max finite natural. > > Except that Aleph_0 is not a successor ordinal. It's not even an > ordinal. And the fact that there is no maximum finite natural. > > So your "inductive proof" has a couple of holes in it. > Apologies. I mean omega. The concept with the limit ordinals is that they are the first after the last of something that doesn't end, the next biggest thing. But, when all successor ordinals are exactly one larger than their max element, then limit ordinals are something else entirely. The declaration of omega as that thing right after all the finites itself leads to silliness. > >> The only way around this is the declaration of the limit ordinals, >> but that's just a philosophical monkey wrench. > > Ordinals and cardinals are necessities if we want to talk about > set "order" and "size" in any kind of logical, well-defined way. Not limit ordinals and transfinite cardinalities, and in the finite case, a count's a count. You don't need to call 1 an ordinal sometimes and a cardinal at others. It's just a natural, a count. > > What do you call the least ordinal that is greater than all finite > ordinals? You don't have a name for it, do you? obviously, that's omega, if you entertain such a useless idea. I don't. There is no least infinity, any more than there's a greatest finite, where addition or subtraction changes the value. It always should. > > What do you call the "size" of a countable set with no end? > You don't have a name for it, do you? No, I find focused concentration on the Twilight Zone, the "boundary" between finite and infinite, to be a rather fruitless exercise. There is no such distinct boundary. > > What do you call the "size" of an uncountable set? You don't > have a name for it, do you? Uh, yeah. Infinite. Infinite sets can be finely ordered formulaically. > > If you are going to keep taking this political approach to > mathematics, condemning what's already established and functional > but not providing any workable alternatives, you might want to > consider running for office instead. > It's not functional, and I have provided functional alternatives. But, maybe I should run for office anyway. Maybe we can outlaw this thing.... ;) haha Tony
From: Tony Orlow on 20 Aug 2006 13:22 David R Tribble wrote: > Dik T. Winter wrote: >>> Let L be the list of all natural numbers >>> Let K be the sequence of digits such that for each p in L, K[p] = 1. >>> What is *wrong* with that definition? > > Tony Orlow wrote: >>> Sorry to interject, but could you tell me how long a sequence K is? If >>> it's finite, then you only have a finite list L, and if it's countably >>> infinite, then you have an uncountable list of naturals? > > Virgil wrote: >>> How does having a countable list of naturals require an uncountable list >>> of naturals? > > Tony Orlow wrote: >> Is K a list of natural numbers? Pay attention. K is the sequence of >> digits required to represent the naturals in binary. The set of binary >> strings of length n has size 2^n. If n is aleph_0, then your countably >> infinite string of digits produces uncountably many possible strings. >> We've been through this. You cannot cull bits or strings in any way to >> reduce this uncountably infinite set to produce the countably infinite >> set of naturals you claim exists. In other words, there is no TYPE of >> number which could represent the size of K. > > Yes, we've been though all this before and you still won't listen to > reason. > > It takes ceil(log2(k)+1) binary digits to encode each natural k, which > we'll call D(k). So: > D(1) = 1, D(2) = 2, D(3) = 2, D(4) = 3, etc. > > Now add up all those D(k) for all natural k (all k in N), and call it > t, the total number of binary digits for all naturals: > t = sum D(k), for all k in N. Why would you sum them all, when each sequence of bit positions include those before it? > > Obviously, since each k is a finite natural, then each D(k) is a finite > natural as well, i.e., each natural k requires a finite number of > binary digits to represent it. Adding each finite D(k) to the finite > sum D(1)+D(2)+...+D(k-1) up to that point yields a countable finite > total number of digits for each k. Right, in your theory, which yields an uncountable number of binary strings, ala 2^aleph_0. > > There are a countably infinite number of naturals. Since the sum > of a countable number of countable values is itself countable, t is > countable. > So, what is ceil(log2(aleph_0)+1) again? Is that countably infinite? You have missed the point entirely, as usual. Oh well. TO
From: Virgil on 20 Aug 2006 14:12
In article <ec9r9u$7tb$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Dik T. Winter wrote: > > In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > Dik T. Winter wrote: > > ... > > > > Eh? It holds for every finite number but does not hold for an > > > > infinite > > > > number of finite numbers? What do you mean with that? If there are > > > > infinitely many finite numbers, I would say that as it holds for > > > > every finite number, it also holds for infinitely many finite numbers. > > > > > > If the formula applies to an infinite number of finites, then does the > > > sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? > > > > I think there is a misunderstanding. The formula > > sum{i = 1 .. n} i = n * (n + 1) / 2 > > holds for every finite number n, so it holds for infinitely many finite > > numers n (as there are infinitely many finite numbers n). But we can > > not switch to an infinite sum (that is something different). > > Well, we can. TO claims a lot but does not deliver. > If we turn to what we know about infinite series, we can > apply notions such as, if every term in series A is greater than its > corresponding term in series B, then the sum is obviously greater. Only if such a sum "exists". None of the standard ways of defining such sums of infinitely many terms require arbitrary "sums" to have a value. > > True. WM and I and others want numbers to behave like numbers TO wants numbers which are different from each other to behave as if they were not different from each other(TO wants, in effect, even numbers to behave like odd numbers and vice versa). Each type of number has its own properties, and requiring one type to behave as if it were a different type, as TO keeps doing, is stupid. > and cardinalities and ordinals simply do not. The finite ones behave like, and actually are, natural numbers. > We find them useless and a > digression from the study of quantity and representation which we see as > being the foundations of math. TO does not "find" mathematics at all, and TO needs several years of concentrated mathematical study before he will be properly prepared to take of the "foundations of math". > It irks people like us when set theory > claims to be the foundation of math It irks those of us who know a bit about mathematics that those who know so little about mathematics as TO set themselves up as authorities on what they know so little about. |