Re: Cantor and the binary tree

Hi,

It's interesting that you note that there is one more path than node,
where the number of nodes is considered to be countable and paths
uncountable.

In the model of ubiquitous ordinals, the powerset is order type is
successor and f(x)=x+1 is the mapping between set and powerset for
which the missing element is the empty set.

The vacuity of the empty set is misleading. The empty set is an
element of any powerset by the usual definition, and regardless of
whether that particular set thus vacuously or trivially satisfies a
given expression of a regularity axiom, that it otherwise does not is
not obvious.

It is basically about that one element, the successor, that induction
guarantees to exist, towards the inductive impasse, or from it.

About set theory, and its consistency, there's a difference between
saying quantification over sets implies a universal set and thus the
regularity axiom of ZF is inconsistent and saying any set theory is
inconsistent.

As far as the tree construction goes, you might notice that the binary
representation of the tree that would represent an integer is as well
infinite, except that some of the 1 nodes would be leaf nodes, or
rather, each node has four child nodes, two each of one and zero where
one of each of those is a leaf and the other not.

About having a pair of nodes for each zero or one decision, with a leaf
node indicating the implicit infinitely trailing zeros (or ones) of a
rational that has a denominator that is a power of two or one of the
integers, it is basically the same thing as an internal node with
implicitly following afterwards the right or left branch of each
descendent in the balanced binary tree.

The tree to represent natural integers via integral moduli is still no
different than the treeto represent the reals via integral moduli .
While that may be so, it might be more convenient to have a distinction
between the leaf and non-leaf nodes to indicate where the path ends.

I think that's an interesting way to consider the construction of the
real numbers, as binary or decimal or equivalently (in the sense of
definitions) Cauchy/Dedekind or continued fractions, but as a point-set
there are also some considerations of the continuity of the real number
line that transcend (in the sense of transcendental numbers) those
definitions. That deserves more explanation.

I am glad to see you move from your ultrafinitist viewpoint to instead
one that acknowledges the infinite. However, if you proceed that way,
then your viewpoint will make obsolete your papers, which I admit to
you I don't think are very strong. Please be careful and take your
time on the argument to not expose weaknesses that are readily
exploited. Furthermore I wish you good luck and recommend my words on
these matters.


These infinite things are not finite: they're unending, there's always
one more.

Ross F.

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