Posts:
822
Registered:
9/1/10


Re: ZFC and God
Posted:
Jan 26, 2013 11:45 PM


On Jan 26, 3:30 pm, Virgil <vir...@ligriv.com> wrote: > In article > <06a85bef99c14104862c27351c153...@f6g2000yqm.googlegroups.com>, > > > > > > > > > > WM <mueck...@rz.fhaugsburg.de> wrote: > > On 26 Jan., 16:06, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > > > WM <mueck...@rz.fhaugsburg.de> writes: > > > > On 26 Jan., 02:50, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > > > > >> I asked how you define terminating decimal representation. How is > > > >> that meaningless? > > > > > Sorry, where did you ask? > > > > You've snipped the question three times, in the thread directly > > > preceding this post. > > > Please excuse me, but there are very many text that I have to read. > > Sometimes I overlook something. Nevertheless, I answered it: > > > > > The latter is not quite correct, because a terminating decimal > > > > representation has nothing behind its last digit d_n, neither zeros > > > > nor any other digits. (But of course, we can expand every terminating > > > > decimal by a finite set of further decimals d_j = 0 for every j with n > > > > <j <m, m in N.) > > > > I don't know why you want to avoid using the usual convention that > > > 0.1 = 0.1000...., but okay. It makes no difference. > > > Sorry, it makes a difference. 0.1000... is an infinite path in the > > Binary Tree. > > Since 0.1 = 0,1000... is presented as a decimal, it is not at all the > same as the binary 0.1000..., and while the binary 0,1000... can be > identified with a path in a Complete Infinite Binary Tree, no such > idenification is relevant here. > > > But in the original question I distingusihed the Binary > > Tree constructed by all finite initial segments of infinite paths and > > the Binary Tree constructed by (all) infinite paths. Of course it > > makes not a difference with respect to nodes. But, according to the > > belief of matheologians, it makes a big difference with respect to > > paths. > > If, as is usual, one defines a path in any binary tree as a maximal > sequence of parentchild linked nodes in the node set of that tree, then > a Complete Infinite Binary Tree cannot have any finite paths, as no > finite set of nodes can be a maximal sequence of parentchild linked > nodes. > > > > > > > > > > > > > > Let's state the definition explicitly then: > > > > Let x be a real number in [0,1]. We say that x has a terminating > > > decimal representation iff there is a natural number k and a > > > function f:{1,...,k} > {0,...,9} such that > > > > x = sum_i=1^k f(i) * 10^i. > > > > Right? > > > Right. > > > > Now, let {t_i} be a list of all the finite decimal representations of > > > reals, that is, each t_i is a finite decimal representation, and every > > > finite decimal representation is in the list. For each t_i, let k_i > > > be the "length" of t_i. > > > > And we define a sequence d_j so that > > > > d_j = 7 if j > k or t_j(j) != 7 > > > d_j = 6 if j <= k and t_j(j) = 7. > > > > As before, we can notice the following facts: > > > > d_j is defined for every j in N. > > > d_j = 7 or d_j = 6 for every j in N. > > > > Clearly, d_j is *NOT* a finite sequence. Moreover, since the sequence > > > d_j does not end in trailing 0s or 9s, the real number d defined by > > > > d = sum_i=1^oo d_i & 10^i > > > > has no finite decimal representation. > > > > Now, please tell me what is unclear about these obvious facts? > > > It is unclear why you apparently are unable to understand, that we are > > working in the set of terminating decimals. Therefore the diagonal > > cannot be actually infinite, although there is no last digit. > > The WM must be working in Wolkenmuekenheim again, as a sequence with no > last term is not finite, and outside of Wolkenmuekenheim "not finite" > and "infinite" mean the same thing. > > > > > Can't you understand that the Binary Tree constructed by all > > terminating paths has no last level and is nevertheless not actually > > infinite because it does not contain any actually infinite path like > > that of 1/9, 1/7, 1/3, and many, many more? > > Then it is not a Complete Infinite Binary Tree. > > > > > You have to distinguish these both cases, because the most important > > argument of matheologians is the following: The tree constructed by > > means of all finite paths does not contain the actually infinite > > paths. > > Outside of Wolkenmuekenheim , the definition of a path is any such > binary tree is that is it a maximal parentchild connected set of nodes. > This means that every path must start at the root node and every finite > path must end in a terminal node having no children. > > But in a Complete Infinite Binary Tree every node has exactly two child > nodes so such terminal nodes are not possible is such a tree. > > > Therefore I restrict the discussion to all finite paths. > > NONE of which can exist n any COMPLETE Infinite Binary Tree. > > > And > > obviously the diagonal cannot be longer, because its digits consist > > only of digits of the finite paths. > > Only in Wolkenmuekenheim. > > Outside of Wolkenmuekenheim, all paths in all Complete Infinite Binary > Trees are necessarily infinite sets of nodes. > > It the diagonal nevertheless> appears infinite to you, then this fact only shows that the most > > important argument is wrong. > > > Regards, WM > >  I know this is all very interesting to all of you but what does Colonel Sanders have to do with God other than his restaurant? Thanks, Martin

