An uncountable countable set
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From: Tony Orlow on 15 Sep 2006 15:14 David R Tribble wrote: > Mike Kelly wrote: >>> There is no bijection between the naturals and the set of all >>> binary strings. > > David R Tribble wrote: >>> Finite binary strings (only). Tony thinks that the set of all possible >>> finite bitstrings is the powerset of the naturals (the finite bit >>> positions or indices), so there are more (finite) bitstrings than >>> bit positions. >>> >>> Yet he admits that there is a bijection between the two. He >>> concludes that this demonstrates a flaw or inconsistency in set >>> theory. >>> >>> In a similar vein, he thinks he has a bijection between the naturals >>> (as binary strings) and all the subsets of N. Likewise, he thinks that >>> this proves that set theory is flawed. >>> >>> Obviously he's confused between powerset cardinalities (|P(S)| = 2^|S|) >>> and binary bitstrings (bit n = 2^n), probably because of the similarity >>> in notation. > > Tony Orlow wrote: >> You don't have it quite right. While there are countably infinite bit >> strings in the power set of the naturals which don't correspond to any >> standard finite natural, I contend that these strings cannot represent >> infinite values, given that all bit positions are finite, and that they >> qualify as finite naturals, ... > > They can't be finite naturals if they have an infinite number of > nonzero digits. That would mean that s=1+2+4+8+... is a finite > quantity and does not diverge, which is obviously false. It diverges as n->oo, but remains finite as long as n is finite. If it never reaches that infinite limit for n, then 2^n-1 is never infinite either. > > Also, if this were true, then ...111 would be the largest binary > natural, but, as you have said yourself, there is no such thing. Certainly that value cannot exist as any kind of point on the real line. Finding the largest finite that's not infinite is like trying to find the smallest which is not infinitesimal. There's no such animal,and yet, this string more or less expresses the concept. > >> ... since each has a successor and precedessor, >> generated the same way as for standard finite naturals. > > If they have an infinite number of digits they cannot be "generated" > the same way that finite naturals are. Specifically, incrementing such > a value cannot be the same operation as incrementing a natural. Indeed it is. Find the least significant (rightmost) 0 bit, and invert all bits (0->1, 1->0) from that position rightward. Tada! Increment. > >> So, when these strings are included as finite naturals, indeed, >> that set of strings bijects with the power set of the naturals. > > That's half right. The set of infinite bitstrings bijects with the set > of infinite subsets of N. But infinite bitstrings cannot represent > finite naturals. They can represent binary reals in [0,1), though. > They also cannot represent infinite values, since none of their finite bit positions allows it, and that's all they have. So, what do they represent, then? You say countably infinite. I say unboundedly large but finite, that is, potentially but not actually infinite. Tony
From: Virgil on 15 Sep 2006 15:14 In article <1158313360.862711.19070(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > That does not exist. At least it is not described by n. III is a > > > representation of 3. > > > > Now you shift back to representation, pray remain consistent. > > Representation is number. There is no difference. Numerals have no > "soul". As numerals are merely names, by that logic "Mueckenh" has no soul either. > > > > Nope. I will only to need to know successors. Anyhow, you read much more > > in the successor of the Peano axioms than is present. The successor is > > defined without even any knowledge of addition at all. So > > succ(George V) = Edward VIII, succ(Edward VIII) = George VI and > > succ(George VI) = Elizabeth II within the set of British kings and queens. > > I do not think what way of addition you would propose for that. Of > > course, this successor function does not satisfy all of Peano's axioms, > > but I hope you get the idea. > > I was talking of *counting* (remember: Zahl and zaehlen) which requires > natural numbers which require the ability to add 1 after you have run > out of the numbers known by heart. We are talking about what precedes counting and is a necessary basis for it, without which counting is impossible, i.e., a starting point and a way always to get to the next point. It is called tallying, and is still used, in which one simply makes a mark, or moves a pebble, for each object tallied.
From: Virgil on 15 Sep 2006 15:18 In article <1158313932.806912.250350(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > 2) or is completely determined by a series of digits. > > > > Question. A terminating or a non-terminating series? > > There are only terminating series. There is no infinity in reality and > useful mathematics. There is a non-terminating series of digits for every rational number whose denominator is not a factor of some power of the base used to represent integers. In decimal, notation, for example, 1/3 = 0.333....
From: Virgil on 15 Sep 2006 15:23 In article <1158324386.661753.251440(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Index = natural number. There is no infinite index, because indexing is > > > identifying. > > > > That does *not* explain why the list contains all numbers that can be > > indexed. > > Let m be a unary number with m 1's, and n a list number with n 1's. > If E n >= m, then m is in the list and can be indexed completely by > list numbers. > If not E n >= m, then m is not in the list and cannot be indexed > completely by list numbers. "Mueckenh" demands the impossible. If m = n, "Mueckenh" demands both X and not X simultaneoulsy. This sort of mental carelessness demonstrates why "Mueckenh" is so often wrong.
From: Virgil on 15 Sep 2006 15:30
In article <1158325325.937083.324210(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > Han de Bruijn wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > David R Tribble schrieb: > > > > > > > >>Tony Orlow wrote: > > > >> > > > >>>Wolfgang's and my position is that N is unbounded but finite, > > > >> > > > >>"Unbounded but finite" is a contradiction, meaning "not finite but > > > >>finite". I'm sure you and Wolfgang think this double-think makes > > > >>sense, but the rest of us don't. > > > > > > > > Your position only reflects the miseducation in mathematics during the > > > > last decades. > > > > > > > > Actual or finished infinity is a contradicton. Surpassed infinity is a > > > > contradiction. > > > > > > > > Unbounded but finite is mathematical reality. Think of the set of all > > > > natural numbers which have been realized by writing down these numbers. > > > > Think of the set of known prime numbers. Think of the set of written > > > > novels. Think of the set of postings. > > > > > > > > These sets are unbounded because they can be extended without end. > > > > Nevertheless they are always finite. > > > > > > Sorry for jumping in so late. But VM is quite right, of course. We have > > > encoutered utterly absurd consequences of thinking otherwise, like the > > > mainstream "theorem" that the probability of a natural being a multiple > > > of 3 doesn't exist. While the obvious truth is that it is equal to 1/3 . > > > > > > This topic has been discussed at length in a thread called "Calculus XOR > > > Probability". Let Google be your friend, eventually. > > > > > > Han de Bruijn > > > > So you still don't know what "probability" means. How predictable. > > Sorry, would you agree with me that the probability to take a natural > number divisible by 3 from a box contaning the numbers from 1 to 9 is > just 1/3? > Would this probability change if the box contained the numbers from 1 > to 900 or to 90000? > Would it significantly change if the box contained the numbers from 1 > to 100000? > > If you agree with me up to that point, then it is clear that the chance > remains the same if we have all natural numbers in that box (provided, > "all natural numbers" is a meaningful expression). What holds for bounded/finite sets need not hold for unbounded/infinite ones. In particular, one cannot even define a uniform probability for drawing members from a countably infinite set. The mathematics of > the infinite can only be derived from the mathematics of the finite > (because nobody has an idea what "the infinite" is). We have a better idea of what "infinite" sets are like than "Mueckenh" thinks. "Mueckenh" could join us if he allowed himself to think unfettered by his unprovable assumptions. Otherwise the > limit of the sequence 1/n might be 100. Nobody could prove that false. > > Regards, WM |