Re: Cantor and the binary tree



Gottfried Helms wrote:

> I always understood "transfinite number" not being a "count-of-elements"
> (which would be impossible with uncountable sets) but being an index of
> the *type* of infinity (already in mind, that that types have
> somehow an order).

It was Cantor's pride to have extended the domain of numbers having
found arithmetic laws connecting the new numbers, the second (zweite
Zahlenklasse) and even higher classes of numbers, as he expressed it.

> in the -possible- case, that your tree represents a rule to describe
> a continuum(*1): since then it is referring to a "number of all nodes",
> which cannot be given, if it possibly is "un-countable".

We have a law yielding all possible infinite strings of bits.
We have a law connecting each spring-off, i.e., the source of a newly
separated path with a node B:
/
B
/\
This is the basic element of the tree and does nowhere change. Hence it
holds all over the tree. It yields a one-to-one relation between nodes
and paths. Nobody can reasonably even rise the question whether there
could be more paths than nodes. And if it is meaningful to talk about
infinity and to count infinity, then it is absolutely clear that nodes
and paths have same number.

> The concept "number of" may be assigned to describe an (important)
> property of a finite set, but if we deal with mathematical objects,
> which are constructed to be infinite, it is better to switch to the
> terminus "cardinality" (or maybe a better one), to not apriori throw
> away the ability to include uncountable(*1) infinities in our
> considerations.

The idea of a bijection of a set with N is a convincing one, in many
respects. But the idea of considering the basic element
/
B
/\
of a tree is at least as convincing. To do the last step, we could even
neglect the terminus "node". Then I assert: There are not more
separated paths in the tree than paths which separate themselves
somewhere in the tree. Set theorists say: There are more separated
paths in the tree than are separated in the tree.

Regards, WM

.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... Where in set theory is it defined as such? ... Then the axiom of infinity supplies an undefined set. ... > is taking the union. ... The set of all nodes of the tree and the set of all nodes of the union ...
    (sci.math)
  • Re: Paths
    ... I think we know both that infinity is nonsense, ... is not in the tree. ... leads to a contradiction (i.e. conflicting with previously specified ... Even if so an invalid does not "disprove" its existence. ...
    (sci.math)
  • Re: Paths
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    (sci.math)
  • Re: Answer to OJ
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    (sci.logic)
  • Re: Review of Mueckenheims book.
    ... The axiom of parallels grants the existence of only one to a given ... Then the axiom of infinity supplies an undefined set. ... The set of all nodes of the tree and the set of all nodes of the union ...
    (sci.math)