An uncountable countable set
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From: Virgil on 27 Jun 2006 23:14 In article <1151441901.524144.290280(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > While that numbers may not have a known or even knowable decimal > > expansion, the fact that you have named that one validates its existence. > > So the number pipifax which I just named does exist too? > > What is its approximate size? > > Regards, WM It's about that big!
From: Virgil on 27 Jun 2006 23:22 In article <1151442095.686342.310050(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Which digits in 0.111... = sum_{n in N} 10 ^(-n) are not indexed by > > members of N? > > Best you can see it here: > > 1 0.1 > 2 0.11 > 3 0.111 > ... ... > n 0.111...n > ... ... > omega 0.111...omega Your "n 0,111...n" does not fit the previous pattern, as it contains digits other than 1's, and omega is not a member of the sequence 0,1,2,... So your example is cooked, and is meaningless. > > It is true that the set of natural numbers is not an actually infinite > set but a potentially infinite i.e. always finite but not limited by a > finite threshold. In ZFC and NBG, which are the only systems under discussion here, there is no such potentiality not actualized. > > > > 0.0 > 0.1 > 0.11 > 0.111 > ... > The diagonal number 0.111... can *only then* be different from any > number of the list, if it has more digits 1 than any list number. Actually, it is a deliberately ambiguous representation of an arbitrary member of the list, sort of like a variable.
From: Virgil on 27 Jun 2006 23:28 In article <1151442257.587808.324200(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > We cannot write or read an unending string. The assumption, such > > > strings existed, is wrong. > > > > The assumption that for every rational number we can describe precisely > > the unending string representing it is right. > > So some endless strings exist, even if only in the imagination. > > But that is where all mathematics exists! > > Yes. And because it exists only there, nothing does exist, which does > not precisely exist there. Such a nothing is an unending string. You > may believe or even be convinced that it does exist, but you are wrong. > Numbers like 5 or 1/3 do, but numbers like pi do not. No numbers exist outside of the imagination. All numbers that we can imagine exist within it. Though whether there is a viable arithmetic for them depends on out imagining a lot more. > > > > > > > > > > All numbers you will ever produce can be listed. That is enough. > > > > In what list? Give me any listing which you claim lists every real, and > > I will give you a rule producing one nonmember of that list for every > > subset of N. > > And that number can also be inserted in the list "g". "That number" is not just one number, it is a massive collection of numbers with one for each subset of the power set of the set of naturals. And that set can be specified by a fairly simple rule of construction. > > Regards, WM
From: Virgil on 27 Jun 2006 23:29 In article <1151442361.464301.177450(a)u72g2000cwu.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Daryl McCullough schrieb: > > > >f: 1 --> {1} and a --> { } with K_f = { }, > > >g: 1 --> { } and a --> {1} with K_g = {1}. > > >Here we have certainly no problem with lacking elements. > > > > The claim made by Cantor is that for any set A, there is > > no surjection from A to P(A). Of course, if you construct > > a new set B by adding extra elements to A, then there might > > be a surjection from B to P(A). But why is that relevant? > > To show you that the condition K is impossible, independent of any > surjectivity. > Map |R (including |N) on P(|N) with the only condition that a natural > number has to be mapped on that set K e P(|N) which contains all > natural numbers which are not mapped on sets containing them. You see: > It is impossible a condition. There are all sorts of impossible conditions, most of which are irrelevant to anything. What is the alleged relevance of this one?
From: mueckenh on 28 Jun 2006 10:31
Daryl McCullough schrieb: > mueckenh(a)rz.fh-augsburg.de says... > > >Daryl McCullough schrieb: > > >> So the answer is no? There is no function f enumerating > >> all the real numbers? So the reals are uncountable. > > > >There is a mathematical form containing all real numbers of [0, 1]. In > >binary representation it is given here. Why not call it a function in > >an extended sense? > > The question is this: does it map the naturals onto the real numbers > in [0,1] such that every real number is in its image? > > In particular, with your representation, which natural number maps > to the real 1/3? It is not necessary to know this natural number (which remains unknown or does not exist) in oder to prove that the infinite branches of the tree are countable. Regards, WM |