Re: Cantor and the binary tree

In article <1118165176.503868.261870@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > > > Ok, so you are stating that some 100-digit combinations are not a
> > > > natural number. Could you specify such a 100-digit combination that
> > > > does not correspond to a natural number?
> > >
> > > If I specifed it, it was a natural number. But it is impossible to
> > > specify all of them.
> >
> > Not if one has an infinite amount of time in which to do it. One need
> > only keep adding 1.
>
> And one must not forget the previous results.
> >
>
> > The set of natural
> > > numbers is not static. To any number claimed to be the largest, one can
> > > create a larger one.
> >
> > Either it is static or it is not a set!
>
> In principle you are right. There is no infinite Cantorian set at all.

Wat WM says does not occur in WM's universe does not constrain what
transpires in the ideal world of manthematics.
> > > >
> > > > In math, counterintuitivity is admissable, falsity not.
> > >
> > > Would be nice to apply that insight to the binary tree.
> > > The basic element of the binary tree is the branching where a path is
> > > separated.
> > > /
> > > B
> > > /\
> > > Separated means separately visible, distinct from the others.
> > > You need not define which of the two paths going out it is. All we need
> > > to know is that the number of separated paths has increased by 1. The
> > > number of separated paths is equal to the number of branchings.

In a maximal binary tree, each maximal path passes through, and thus
determines, an infinite set of nodes. There are uncountably many such
infinite sets of nodes.
> > >
> > > Set theory requires: The number of separated paths is larger than the
> > > number of branchings. That is not counterintuitive. That is wrong.
> >
> > Only in WM's world, in which mathematics does not exist.
>
> Double the number of nodes.

"Doubling" an infiniteset has no effect on its cardinality.
.

Relevant Pages

  • Re: Answer to OJ
    ... depiction of a single non-root, non-leaf node in a binary tree, but I ... One edge goes in (to all but the root node) and two edges come out. ... infinite tree without needing every path. ... with infinitely many right branchings need occur in order to cover all ...
    (sci.logic)
  • Re: Cantor and the binary tree
    ... >> foggy and ambiguous for mathematics to exist in. ... >> Not if one has an infinite amount of time in which to do it. ... >>> Would be nice to apply that insight to the binary tree. ... >>> number of separated paths is equal to the number of branchings. ...
    (sci.math)
  • Re: Cantor Confusion
    ... What is "half" of an infinite number? ... infinite binary strings in each complete infinite binary tree. ... One wonders which branchings WM denies in his incomplete trees. ...
    (sci.math)
  • Re: infinity
    ... >>> FINITE number greater than any finite you specify. ... > infinite, because there is always a larger finite value than any finite value ... they are equivalent in everyday English. ... then it would be equivalent to saying it's infinite. ...
    (sci.math)
  • Re: Approaching the infinite binary tree
    ... But there are infinite paths in the complete binary tree that do ... It is your dogma, not ours, that you are being bit by. ... and all of them can be bijected with N. The set of even naturals, ...
    (sci.math)
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