Re: Would it matter if ZF was inconsistent?

On Jul 17, 12:16 pm, LudovicoVan <ju...@xxxxxxxxxxxxx> wrote:

Below I have tried some transfinite-inductive definitions, leveraging
the infinite cartesian product to define the infinite case. For
illustration, there is also an implementation in Prolog that works for
the finite cases, with some sample output.

The predicate nodes(n) defines the set (actually, the list) of nodes
at level (depth) n in the binary tree, i.e. those nodes with n-1
parents up to (but excluding) the root. In particular, each node
encodes the whole path from the root to it, i.e. paths and nodes are
equivalent. For instance, an element of nodes(3) is [1, [0, [1, []]]],
corresponding to the path R-1-0-1.

The predicate tree(n) defines the set (actually, the list) of nodes in
a tree of level (depth) n. In particular, we will call *leaf nodes*
that subset of tree(n) that is the set nodes(n).

If I understand your specifications, your leaf nodes are not leaf
nodes of the complete (infinite) binary tree but rather of certain
subsets of the complete (infinite) binary tree and resultingly leaf
nodes, under your definition, turn out just to be nodes of the
complete (infinite) binary tree. Of course, whether a node is a leaf
node depends on what tree we're talking about. In this context, the
tree under discussion has been the complete (infinite) binary tree,
which has no leaf nodes, though, of course, certain subtrees have leaf
nodes (as, again, 'leaf node' is defined PER a given tree).

And the set paths(n)
in tree(n) (i.e. the set of paths of length n in the tree of depth n)
is equivalent to the set nodes(n), i.e. the set of leaf nodes in tree
(n).

If I understand your specifications, each tree(n) is a finite tree.
The matter of previous discussion had not been as to finite trees. But
perhaps that is what you are getting at with the following:

Extending the definitions to the transfinite, we have in particular
that the set of infinite paths in tree(w) corresponds to the set nodes
(w): i.e., again, the set of leaf nodes.

Perhaps your conviction is based on whatever your method is of
"extending the definitions to the transfinite", which may not be
equivalent with the actual set theoretical definition of the infinite
binary tree.

If you ever wish to see a rigorous treatment and definition in set
theory, just let me know, and I'll recommend some books for you
(anyone of which is adequate for the task).

MoeBlee

.

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