Posts:
791
Registered:
9/1/10
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Re: ZFC and God
Posted:
Jan 26, 2013 11:45 PM
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On Jan 26, 3:30 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <06a85bef-99c1-4104-862c-27351c153...@f6g2000yqm.googlegroups.com>,
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> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 26 Jan., 16:06, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > > WM <mueck...@rz.fh-augsburg.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.
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> > Sorry, it makes a difference. 0.1000... is an infinite path in the
> > Binary Tree.
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> 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 parent-child 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 parent-child linked
> nodes.
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> > > Let's state the definition explicitly then:
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> > > 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.
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> > > And we define a sequence d_j so that
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> > > d_j = 7 if j > k or t_j(j) != 7
> > > d_j = 6 if j <= k and t_j(j) = 7.
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> > > As before, we can notice the following facts:
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> > > d_j is defined for every j in N.
> > > d_j = 7 or d_j = 6 for every j in N.
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> > > 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
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> > > d = sum_i=1^oo d_i & 10^-i
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> > > has no finite decimal representation.
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> > > Now, please tell me what is unclear about these obvious facts?
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> > 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.
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> > 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?
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> Then it is not a Complete Infinite Binary Tree.
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> > 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.
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> Outside of Wolkenmuekenheim , the definition of a path is any such
> binary tree is that is it a maximal parent-child 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.
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> 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
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