On Jul 15, 4:08 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 15 Jul., 21:17, MoeBlee <modem...@gmail.com> wrote: > > > > You cannot be too unmathematical to miss the fact that every B_k > > > contains a countable set of infinite paths, > > > "contains". Element or subset? > > In this case that is irrelevant, because the countably set is empty.
WM makes no sense. He talks about B_k "containing" some other thing. Then I ask is "contain" meant to mean 'has as elements' or 'has as subsets'. Then he says it doesn't matter, because the thing mentioned that B_k "contains" is a countable empty set.
WM needs to just put his claim in plain mathematical terms. One of these (or if neither, then exactly what?):
For all natural numbers k, there exists a countable set S such that every member of S is an infinite path and S is a member of B_k.
or
For all natural numbers k, there exists a countable set S such that every member of S is an infinite path and S is a subset of B_k.
Of course, though, vacously if S is the empty set then S is a subset of B_k.
And the first is false (to be pedantic (hello, poster Transfer Principle): false in any model of Z set theory, or, less formally, contradicts the ordinary mathematical treatment of this subject mattter).
Then, so what?
> B_k contains zero infinite subsets of nodes
No member of B_k is an infinite set of nodes. and No subset of B_k is an infinite set of nodes.
> and it contains zero > infinite paths if B_k is considered as a set of paths.
Earlier WM was using B_k as a set of nodes. Now he's asking to "consider" it as a set of paths.
WM needs to agree with my vedry simple definition of 'the complete binary tree' or give his own NON-ostensive definition. Then he needs to give a precise definition of the function B on the set of natural numbers.
Anyway, in the sense WM was using 'B' previously (as each B_k is a certain set of nodes), then, of course, no infinite path is a member of B_k and no infinite path is a subset of B_k.
> (Under > "countable set I understand all sets with a cardinal numbers less than > 2^aleph_0.)
That's not equivalent with the definition of 'countable' in ordinary mathematics (except by assuming the continuum hypothesis), so WM is using his own special definition of 'countable'. I don't see the point in using such an unusual definition.
Instead, an ordinary defintion is:
x is countable <-> (x is finite or x is equinumerous with w)
> > Anyway no set that has as a member an infinite path is an element of, > > nor a subset of, any B_k. > > My invention consists mainly in the way to look at the Binary Tree: It > can be considered as an ordered set of nodes but it can also be > considered as an ordered set of finite initial segments of infinite > paths.
It can be considered lots of ways. WM needs to state a precise definition (not an ostensive definition) and then prove what ever other equivalences or isomorphisms he wishes to use.
(1) Give a precise mathematical definition (not an ostensive definition) of 'the complete binary tree'. Or accept the very simple definition given by MoeBlee
(2) Give precise mathematical definitions of 'node' and 'path'.
(3) Give a precise mathematical definition of the function B on the set of natural numbers.
But more fundamentally:
(1a) State precisely what mathematical principles and rules of inference are to be allowed in this context of formulations and arguments regarding the complete binary tree.
(2a) State precisely what is meant by claims that set theory or ZFC or the axiom of infinity is "self-contradictory" if what is meant is not the usual definition that a set of formulas is inconsistent iff there is a formula P such that both P and ~P are theorems of said set of formulas.
Those are quite reasonable requests for a mathematical discussion. That WM refuses to fulfill such a reasonable set of requests is consistent with his general dogmatism and unreasonability.
