Re: Review of Mueckenheims book.

In article <1175413415.007852.74290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

On 29 Mrz., 20:53, "William Hughes" <wpihug...@xxxxxxxxxxx> wrote:

24. Every path in T_U contains an infinite set of nodes.

No.

Then WM is claiming that infinite trees are not infinite trees.

But it is easy enough in either ZF or NBG to model such a tree, for
example:

Let N = {0,1,2,3,...} be the von Neumann naturals, so that for each n e
N, n = {m e N: m e n}.

Then every ordered partition (L,R) of n denotes a node of level n, that
node reached by branching left or right at level m according to whether
M is a member of L or R.

The number of nodes at level n equals the number of ordered partitions
of N which equals Card(P(n)) = 2^n.

For the root node one has L = R= {}
For the root's left child one has L= {0}, R= {}
and the root's right child one has L= {}, R= {0}


At the next level one has the 4 grandchildren
({0,1},{}) and {({0},{1}) and ({1},{0}) and ({},{0,1}).

and so on.

Thus at level n there are 2^n nodes, the cardinal of the power set of
N_n = {0,1,2,...,n-1}

Then A path in such a tree is merely a similar partition of N into an
ordered partition of N itself.

Now can WM find any reason why this model does not satisfy any
requirement of treeness?

Except, of course, that it proves him wrong.
.

Loading