An uncountable countable set
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From: Virgil on 8 Sep 2006 16:20 In article <1157742956.066595.98530(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > 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. It's not what M knows that hurts him,. its what he "knows" that ain't so. > > > 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. Not in my definition. There are no 'levels' in my definition. > > > > > > 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. None which do not ultimately depend on being able to "do the addition". > If I know that 2 + 2 < 5, then I can conclude that 2 + 2 < 10. Proving "2 + 2 < 5 " requires that one be able to"do the addition". > Why should some logic conclusion like that be forbidden in set theory? Why should the logical conclusion that there cannot be any surjection from the set of edges to the set of paths be forbidden in set theory?
From: Virgil on 8 Sep 2006 16:22 In article <1157743126.475610.189460(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. If you are citing Cantor as authoritative, then you must also accept Cantor's orders of infiniteness as well.
From: Dik T. Winter on 8 Sep 2006 22:48 In article <1157742956.066595.98530(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. You keep confusing the situation where edges do terminate (and so all paths do terminate) with the situation where edges do not terminate (and so all paths do not terminate). > > > 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. Say, which of my statements do you contradict here? > > 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? What is the relevance with respect to my question? > > > 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. You second sentence makes no sense. > > 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. It is well known that the set of *finite* sequences of digits is countable. You are telling nothing new here. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David R Tribble on 8 Sep 2006 22:57 Tony Orlow wrote: >> ....11111 binary (all bit positions finite) > Mike Kelly wrote: >> That isn't a natural number, Tony. > Tony Orlow wrote: > Are you sure? Pay close attention. > > For any finite bit position n, it and all predecessors can only sum to a > finite bit string value of 2^(n+1)-1. Since there are only finite bit > positions in the string, it can never achieve any infinite value at any > position in the unending string of bits. Therefore the value must be finite. Breathtaking. You have the annoying habit of ignoring the completely obvious when it suits you. If ...111 is a binary number with an infinite number of digits, why is it a finite value? Shouldn't an infinite value have an infinite number of digits? > Furthermore, since any such number does have a predecessor and > successor, in this case ....1110 and ...0000, respectively, it fits in > the successorship model. The only concept this breaks is that 0 is now a > successor as well, creating an infinite ring of successorship. Other > than that, it works as a natural, ... If ...111 is a finite value, then there must be another finite value that is larger than it. You claim that this is 0, but that gives you a contradiction: 0 = ...111+1 > ...111 > 0, or 0 > 0. (We can now use this to prove that any finite number is larger than itself: k = 0+k = (...111+1)+k > ...111+k > 0+k = k, so k > k.) This should be a rather big clue that something's wrong with your concept, and that it does not, in fact, "work as a natural". > and in fact, this is the way signed > integers work in your very computer. Computers only deal with fixed-length finite numbers, modulo 2^B for some word width B. That does not provide a very useful model for the entire set of naturals. > So, while ...111 may not be considered a standard natural, I see no > reason why it should not be considered, say, an extended natural. Which presumably requires some extended Peano axioms in order to exist?
From: Dik T. Winter on 8 Sep 2006 23:00
In article <1157743126.475610.189460(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. Yes, in English such things are called axioms. Although I disagree that changing them leads to rubbish. Changing them leads to at most a different kind of mathematics. > Something like axioms but not arbitrarily stated but derived > from "nature" or "reality". Ah, like the parallel postulate of Euclides. > One of them is 2 + 2 = 4 (in decimal > notation). Another one, according to his opinion, is the existence of > infinitely many finite numbers. Well, if you want to use only Cantor's view you should not attack current set theory. In current set theory that is an axiom, and so I think my translation with the word "axiom" is quite correct. In current set theory it is an axiom that there exist infinitely many finite numbers. And the reason it is an axiom is because it can not be proven using the other axioms. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |