An uncountable countable set
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From: stephen on 8 Sep 2006 11:27 imaginatorium(a)despammed.com wrote: > stephen(a)nomail.com wrote: >> imaginatorium(a)despammed.com wrote: >> > stephen(a)nomail.com wrote: >> >> >> >> Tony's argument seems to be a upside-down inside-out version >> >> of one of Zeno's paradoxes. >> >> >> >> ...111111 (in binary) = 1 + 2 + 4 + 8 + ... >> >> >> >> All the numbers on the right are finite, so the sum must >> >> be finite. Apparently a sum can only be infinite if one of the >> >> terms in the sum is infinite. An infinite sum of finite terms >> >> cannot possibly be infinite in Tony's mind. >> >> > Not entirely sure about this. The problem above is that we are >> > restricting the string >> > ...11111 >> > so that every 1 is in a finite position. The position numbers never >> > achieve true infinity. >> >> Yes, which means that each term in the sum >> 1 + 2 + 4 + 8 + ... >> never achieves true infinity. Apparently because no single >> term in the sum is infinite, the sum itself cannot be infinite. >> If you had a 1 in an infinite position p, then the sum would >> contain the term 2^p, which would be infinite, because p >> is infinite, and the sum would be infinite. > Hmm, I don't think you're quite getting into the spirit of this. I am trying to figure out what Tony's misconceptions are. Occassionaly I just make fun of them, but at other times I try to see if I can figure out his intuition. > In > Tony's scheme (insofar as I understand it, and of course it appears to > me to be as contradictory as it does to you), I believe you can count > numbers 1, 2, 3, ... and so on, and go on for ever. Yes, I have the same understanding of Tony's scheme. > But at some foggy > point (perhaps after about ever has elapsed) you find that you have > arrived in the infinite zone, and are counting numbers which, in an > infinite binary notation, have nonzero digits at least close to the > (nonexistent) left end. One of these numbers is called Big'un, and if > you write the sum > 1 + 2 + 4 + 8 + ... + 2^Big'un + 2^(Big'un+1) + ... > then it _is_ infinite (in Tony's scheme), because at least one of the > numbers in the sum has achieved genuine infinity. This seems to be the same understanding I have. > So in Tonyspeak, you have to quantify 1+2+4+... by saying what sort of > zone the dots extend to. In this particular case we are talking only about the finite values. So 1+2+4+... for all the finite values. According to Tony this must be finite, because the sum of finite values, even an endless sum of finite values, must be finite. > I think. The point of my post is that Tony's intuition is sort of the opposite of the intuition that makes Zeno's paradoxes so compelling. The idea that an infinite number of finite numbers can "sum" to a finite value is a bit counterintuitive. The idea that you can keep adding to something, and that it keeps getting bigger, but that it remains finite, has puzzled (some) people for millenia. Tony has twisted this around, and insists that an endless (infinite) sum of finite numbers must be finite. Tony's intuition is based on the idea that numbers are processes. ...11111111 = 1 + 2 + 4 + 8 + 16 + ... (all finite powers of 2) If I add the numbers on the right one at a time, I will always have a finite number: 1, 3, 7, 15, 31, ... Tony therefore concludes that the right hand side must be finite. There is no step at which it "becomes" infinite. This is true. Neither is there a step at which I have added all the numbers on the right. ...11111111 equals the sum of all the numbers on the right. So according to Tony's logic (if he was consistent), there is no step at which the sum on the right becomes the value on the left. So either the two are not equal, in which case the fact that the right hand side is "finite" tells us nothing about the value on the left. Or the right hand side "becomes" infinite on the same step that it "becomes equal" to the left hand side. Stephen
From: mueckenh on 8 Sep 2006 15:12 David R Tribble schrieb: > mueckenh wrote: > >> All we can attach to it is the number of elements known > >> or existing. Disregarding physical constraints ... > > > > David R Tribble schrieb: > >> I was not aware that abstract mathematical concepts (e.g., sets) > >> had any physical constraints. > > > > mueckenh wrote: > > Then you should learn it. It you are unable to physically (i.e. in > > written form or in your mind) distinguish all the elements of a set, > > then the set does not exist. > > Nope, that still does not explain why abstract mental concepts > are limited by physical constraints. I don't believe it. All your mental concepts are nothing but electric loads and currents in your brain. > > Exchange the last igit of this number P by 6 and find out whether the > > new number is larger than P or not. > > If I'm allowed to use base-16 notation, I can do that rather easily. Do it. > (You're probably not aware that there is an formula for finding > any hexadecimal digit of pi.) I am aware of this. And I have seen that the examples given are in the range of less than 10^20. What do you think why this is so? And even if you could determine the digit number 10^100, then many of the previous digits necessarily are lacking. > David R Tribble schrieb: > >> What exactly is it about your set theory that physically constrains > >> your sets to being less than some arbitrarily chosen maximum size? > > > > mueckenh wrote: > > Not arbitrarily cosen. There is no means in the universe to surpass > > this amount of information. > > How do you know? You have proof? By observation of the observable. What is outside of our observable realm cannot be used by us for storing our bits. Regards, WM
From: mueckenh on 8 Sep 2006 15:15 Dik T. Winter schrieb: > > You intermingle numbers (paths) with digits. I can state the n-th digit > > of any path you want. > > I do not. The digits are countable because I can immediate state the > n-th digit when n is given. The paths are not asserted to be countable, because you could biject them to N, but because we know that they are only half of the number of the edges. > The paths are not countable because I can > not state the n-th path when n is given. The edges within a given path > are countable because you can state which is the n-th edge when n is > given. All the edges are countable, because you can state what is the m-th edge on the n-th level. That is the definition of countability. > > > On the other hand, you *claim* that your set of paths is countable, > so you should be able to state what the tenth path is. But you refuse > to do so. There are other proofs of 2 + 2 < 10 than executing the addition. If I know that 2 + 2 < 5, then I can conclude that 2 + 2 < 10. Why should some logic conclusion like that be forbidden in set theory? > > > > Give my a tree of infinite paths consisting of 0's and 1's, and I show > > > > that there are not less edges than paths. > > > > > > Indeed. If all edges terminate, also all paths terminate, and both are > > > countable, and 1/3 is not in the tree. If edges do *not* terminale, > > > > What are you talking about? Every edge terminates because it is the > > connectio between subsquent nodes of a path. > > And so all paths terminate, and 1/3 is not in your tree. But I told you already: The paths do not terminate. The paths are taken from the diagonals in Cantor's list. > No, I did not know the strange exponentiation used in ordinals. Apparently > it can be defined as the set of functions from a set with ordinal number > omega can be mapped to a set with ordinal number 2, with the proviso that > only finitely elements are mapped to the second element. Also my paths consist of only finite elements, i.e., edges at finite places. Nevertheless the paths are infinite. Regards, WM
From: mueckenh on 8 Sep 2006 15:18 Dik T. Winter schrieb: > > > > > > > > A law is derived from the natural properties of arithmetics. > > > > > > Oh. What law derived from the natural properties of arithmetics is he > > > talking about when he gets the completely determined set of all integral > > > finite numbers? > > > > That is but his conviction. > > I ask you what law, but you are not willing to answer? Again, what law is > he using (note that in English law generally refers to Theorem). Cantor held the opinion that there are some natural "laws" or "rules" , not theorems, but Grundwahrheiten, so say truths, which are valid for the natural numbers and which cannot be changed without leading to rubbish. Something like axioms but not arbitrarily stated but derived from "nature" or "reality". One of them is 2 + 2 = 4 (in decimal notation). Another one, according to his opinion, is the existence of infinitely many finite numbers. Regards, WM
From: mueckenh on 8 Sep 2006 15:20
David R Tribble schrieb: > mueckenh wrote: > >> All we can attach to it is the number of elements known > >> or existing. Disregarding physical constraints ... > > > > David R Tribble schrieb: > >> I was not aware that abstract mathematical concepts (e.g., sets) > >> had any physical constraints. > > > > mueckenh wrote: > > Then you should learn it. It you are unable to physically (i.e. in > > written form or in your mind) distinguish all the elements of a set, > > then the set does not exist. > > Consider the set S formed from the elements > 0 > and > y, where y = x+1 for each x in S > > For every member x in S, I know there is also a member x+1 in S. > So I can distinguish every element of the set. I can also choose > x to be as large as I wish - for x=10^1000, I know that 10^1000+1 > is also in S. So set S must exist. > > I really don't see where the physical limitation is for visualizing the > elements or the set. More to the point, I don't see how the phyics > of the real world have any limiting effect on abstract concepts. What you are arguing is only the beginning of infinity, because you are unable to see more than this little realm. Try to determine the natural number which consists of the 10^100 digits following the first 10^1000 digits of pi. Your unability is not due to lack of time. This is only one of the infinitely many numbers which you cannot deal with (because it is not a number). Regards, WM |